aa r X i v : . [ m a t h - ph ] S e p C HAS ING H AMILTON IA N S TRUCTURE INGYROKINE TI C THEORY
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ISSERTATION P RESENTED TO THE F ACULTYOF P RINCETON U NIVERSITYIN C ANDIDACY FOR THE D EGREEOF D OCTOR OF P HILOSOPHY R ECOMMENDED FOR A CCEPTANCEBY THE D EPARTMENT OF A STROPHYSICAL S CIENCES P ROGRAM IN P LASMA P HYSICS A DVISER : H. Q IN S EPTEMBER bstract
Hamiltonian structure is pursued and uncovered in collisional and collisionless gyrokinetictheory. A new Hamiltonian formulation of collisionless electromagnetic theory is presented thatis ideally suited to implementation on modern supercomputers. The method used to uncover thisstructure is described in detail and applied to a number of examples, where several well-knownplasma models are endowed with a Hamiltonian structure for the first time. The first energy- andmomentum-conserving formulation of full-F collisional gyrokinetics is presented. In an effort tounderstand the theoretical underpinnings of this result at a deeper level, a stochastic
Hamiltonianmodeling approach is presented and applied to pitch angle scattering. Interestingly, the collisionoperator produced by the Hamiltonian approach is equal to the Lorentz operator plus higher-orderterms, but does not exactly conserve energy. Conversely, the classical Lorentz collision operator isprovably not Hamiltonian in the stochastic sense.iii cknowledgements
It’s been much harder to learn how to disengage from my thesis research than it was to becomeengrossed in it. In part, this is due to the complete autonomy conferred on me by my advisor, Hong;it’s a symptom of working on very little, other than topics dear to my heart. In equal measure, myaddictive personality is to blame. For all of the days I spent happily fiddling with Poisson bracketsand variational principles, there were nights when I wished I could forget about such things, andjust sleep.For the pleasant times I’ve had as a paid free thinker at Princeton, I need to express my extremegratitude to the PPPL staff who continually encouraged and cultivated my interests in hard-to-market, mathematically-oriented basic theory problems. Hong Qin, John Krommes, Bill Tang,Roscoe White, Ilya Dodin, Nat Fisch, Cynthia Phillips, Doug Darrow, Amitava Bhattacharjee, andGerrit Kramer: thank you, my interactions with you have been incalculably helpful, and genuinelyenjoyable. Barbara, I can’t thank you enough for your detailed and patient descriptions of howto do normal things at the lab without suffering the wrath of Uncle Sam. Beth, you’ve made theprocess of attempting to graduate a real pleasure.For pulling me away from research when I really needed to be, I need to thank my close friends.Jono and Seth, I can’t imagine better housemates than you guys; you’ve made living in Princetona lot less stressful and lonely than it might have been in your absence. Alex, while I only hadthe pleasure of being your housemate for a year, it was a year well-spent; I wish you, Jack, Jono,Seth and I lived together sooner. Lei and Yao, all of the time you spent patiently introducing meto Chinese food in the US and in China hasn’t been wasted; our numerous visits to the Chineserestaurants around Princeton kept me sane and well-fed. Lee, you’ve been my brother at Princeton.I’m sad our career paths will no longer be coincident, but I’m excited to see where you end up.Mike, you’ve helped me appreciate and laugh about many of the absurd aspects of academia andivife in general. Lan, our runs have been a necessary distraction for me (and I hope your foot getsbetter.)Looking ahead, I’m glad that I’ll be spending the next few years fairly close to my parents,brothers, and dogs in Saratoga Springs. I relish the thought of being near enough to see my parentsretire, my brothers settle into jobs they enjoy, and my dogs grow out of puppyhood.vf it’s in a word, or it’s in a look, you can’t get rid of the Babadook.vi ontents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Bibliography 173 viii hapter 1Introduction
On microscopic scales, the physics of plasmas is Hamiltonian in nature. Neglecting quantum, rela-tivistic, and radiative effects for simplicity’s sake, the microscopic description of a plasma consistsof a separate instance of the Lorentz force law for each plasma particle along with Maxwell’sequations to couple everything together. This system of equations can be derived from a varia-tional principle, which in turn can be used to derive a microscopic Hamiltonian functional andPoisson bracket. These equations governing microscopic plasma physics have been called theKlimontovich-Maxwell system, and their Lagrangian formulation is described for instance inQin et al. (2014).With Hamiltonian structure ingrained so deeply in the foundation of the subject, plasma theoryought to be some grand exercise in the broader theory of Hamiltonian systems. And when viewedfrom a great distance, it is! However, for those in the trenches, studying plasma theory on a day-to-day basis, things seem different. The equations governing microscopic plasma theory are sohopelessly complicated that reduced plasma models are typically preferable to the Klimontovich-Maxwell model. These reduced models are obtained by carefully and cleverly applying Occam’srazor in order to tame the mathematical morass presented by the microscopic equations of motion.Sometimes, for instance in the case of ideal magnetohydrodynamics, the reduced model is prov-1
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NTRODUCTION . However, often times the connectionbetween the reduced model and Hamiltonian mechanics is hazy at best. The most striking examplesof this divorce from Hamiltonian mechanics arise when collisions must be accounted for withinthe confines of a continuum model. For instance the Landau collision operator probably cannotbe derived from a conventional variational principle. The same is true of the more-fundamentalBalescu-Lenard collision operator. More generally, there is a common feeling amongst plasmaphysicists that “dissipative dynamics are not Hamiltonian.” Even when collisions are neglected,there are reduced models that either fail to be Hamiltonian in nature, or so far have resisted at-tempts to be cast in Hamiltonian form.This thesis is devoted to revealing some new connections between Hamiltonian mechanicsand the particularly interesting reduced plasma model known as gyrokinetics. I believe that itillustrates two general points in connection with the “Hamiltonian dichotomy” between reducedplasma models and the microscopic plasma model just described. First, Hamiltonian mechanicscan be surprisingly useful in the study of reduced plasma models; the benefits of exploiting theHamiltonian formalism to formulate and study reduced models can be unpredictable. Second, theconnection between collisional plasma models and Hamiltonian mechanics is surprisingly deep;while collision operators do not fit within the traditional Hamiltonian framework, they may verywell fit within a stochastic Hamiltonian framework L´azaro-Cam´ı and Ortega (2008). The firstpoint is covered roughly by Chapters 2 through 4, while the second point is discussed in Chapter5. In Chapter 2, I present the results of an attempt to cast collisionless electromagnetic gyrokinet-ics in Hamiltonian form, a theory that already enjoys several
Lagrangian formulations. The earliestof the Lagrangian formulations are given in Sugama (2000); Brizard (2000b,a), while more recentadditions can be found in Pfirsch and Correa-Restrepo (2004); Squire et al. (2013). Given the typ- In the case of ideal MHD, as well as many other Hamiltonian reduced models, it is still unclear how the reducedmodel’s Hamiltonian structure is related to the microscopic Hamiltonian structure.
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NTRODUCTION should exist; the riddle was how to findit. In the process of searching for this Hamiltonian structure, and with guidance from ProfessorMorrison, I found it technically convenient to slightly reformulate electromagnetic gyrokinetics inorder to work with a manifestly gauge-invariant theory along the lines of Morrison (2013). Sur-prisingly, this reformulation turned out to have several features that make it especially well-suitedto simulation on modern supercomputers. This is the first example in the thesis of a surprisingconsequence of pursuing Hamiltonian structure in reduced plasma models.In chapter 3 I give an extended account of the theoretical machine used to derive the gyrokineticPoisson bracket from Chapter 2. This machine, which eats (possibly degenerate) Lagrangians andspits out Poisson brackets, is very closely related to the Peierls bracket formalism Peierls (1952),as well as the Dirac constraint formalism used in Squire et al. (2013). Nevertheless, several of theexamples worked out in this chapter (besides electromagnetic gyrokinetics) are new. The first ex-ample that contains a new result concerns the Vlasov-Darwin system. This system had previouslybeen cast in Hamiltonian form in Krause et al. (2007) using position-canonical momentum coor-dinates on the single-particle phase space. The novelty of the example in this Chapter is that thederivation of the bracket is done using position-velocity coordinates on the single-particle phasespace (which leads to a different expression for the bracket.) The second novel example is con-cerned with deriving a bracket for the (quasi) neutral Vlasov system introduced by C. Tronci andE. Camporeale in Tronci and Camporeale (2015). Tronci and Camporeale provide a Lagrangianformulation of this reduced model, but stop short of passing to the Hamiltonian side. Thus, this Previous work on Lagrangian electromagnetic gyrokinetics was usually done in the Coulomb gauge.
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NTRODUCTION stochastic Hamil-tonian mechanics
L´azaro-Cam´ı and Ortega (2008). Where ordinary Hamiltonian mechanics isconcerned with one-parameter subgroups of the phase space symplectomorphism group, stochas-tic Hamiltonian mechanics is concerned with Brownian motion on the symplectomorphism group.The remainder of the chapter is then devoted to applying these ideas to the pitch angle scatteringproblem, which can be formulated as an example of stochastic acceleration. A surprise here is thatthere is a tension between energy conservation and the stochastic Hamiltonian formalism. The col-lision operator produced using the Hamiltonian approach is equal to the Lorentz collision operatorplus higher-order terms, but does not exactly conserve kinetic energy. Conversely, the classicalLorentz operator is provably outside the realm of stochastic Hamiltonian mechanics. As I discussat the end of the chapter, it seems likely that a way to overcome this problem is to slighly relax the
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NTRODUCTION hapter 2Hamiltonian formulation of the gyrokineticVlasov-Maxwell equations
Electromagnetic gyrokinetic theory (EMGT) is a model used to describe the turbulent transport ofparticles and heat induced by fluctuating electric and magnetic fields in strongly magnetized plas-mas. EMGT is, in many ways, a more utilitarian tool than the more-fundamental Vlasov-Maxwellkinetic theory (VMKT). However, VMKT enjoys two important advantages over existing formu-lations of EMGT. (I) When simulated on a computer, the VMKT field solve is local; advancingthe electromagnetic field in time at a given grid point only requires communication with nearbygrid points Bowers et al. (2009). (II) There is an energy principle for assessing the stability ofVlasov-Maxwell equilibria Morrison and Pfirsch (1989) (also see Kruskal and Oberman (1958);Holm et al. (1985); Morrison (1998); Andreussi et al. (2012, 2013) for similar energy principles inother contexts). In contrast, modern EMGT simulations require global Poisson-like field solves ateach time step. This prevents EMGT simulations from scaling as favorably Madduri et al. (2011)as VMKT simulations when the number of processing cores is increased at fixed problem size.6
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AMILTONIAN FORMULATION OF THE GYROKINETIC V LASOV -M AXWELLEQUATIONS
The gyrokinetic Vlasov-Maxwell equations are given by ∂f s ∂t = − L V gy s f s (2.1a) c ∂ D ∂t = ∇ × H − πc J gy (2.1b) c ∂ B ∂t = −∇ × E (2.1c) ∇ · D = 4 πρ gy (2.1d) ∇ · B = 0 . (2.1e) f s is the gyrocenter volume form of species s , V gy s is the gyrocenter phase space velocity, L V gy s denotes the Lie derivative along the gyrocenter phase space velocity, J gy is the gyrocenter currentdensity, ρ gy is the gyrocenter charge density, E , B are the fluctuating electric and magnetic fields,and D , H are the auxiliary electric and magnetic fields. The volume form f s is defined by re-quiring that the number of particles of species s in a region of phase space U be given by R U f s .The gyrocenter phase space velocity is specified by the time-dependent tensor form of Hamilton’s HAPTER
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AMILTONIAN FORMULATION OF THE GYROKINETIC V LASOV -M AXWELLEQUATIONS V gy s ω gy s = d K s − e s E · d X , (2.2)where ω gy s is the gyrocenter symplectic form, K s is the gyrocenter kinetic energy, and d X denotesthe vector line element in the space of gyrocenter positions. The gyrocenter symplectic form is thesum of the guiding center symplectic form Cary and Brizard (2009); Burby et al. (2013a) and thefluctuating magnetic flux, ω gy s = ω gc s − e s c B · dS, (2.3)where dS is the surface element in the space of gyrocenter positions. The gyrocenter kineticenergy is a functional of the fluctuating electric and magnetic fields, and is related to the gyrocenterHamiltonian by H gy s = K s + e s φ (an explicit expression for K s will be given near the end of thisChapter). The auxiliary fields D , H are related to E , B by using relations that emerge from theHamiltonian theory developed in Morrison (2013), i.e., the constitutive relations are given by D = E − π δ K δ E (2.4) H = B + 4 π δ K δ B , (2.5)where K ( f, E , B ) = P s R f s K s ( E , B ) .Following Morrison (2013) the system above constitutes an infinite-dimensional Hamiltoniansystem with dynamical variables f , D , and B , and Hamiltonian functional given by H ( f, D , B ) = K ( f, ˆ E , B ) + Z ˆ P · ˆ E d X + 18 π Z (cid:18) ˆ E · ˆ E + B · B (cid:19) d X , (2.6) HAPTER
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AMILTONIAN FORMULATION OF THE GYROKINETIC V LASOV -M AXWELLEQUATIONS ˆ E = ˆ E ( f, D , B ) is the electric field operator defined implicitly by the equation D = ˆ E ( f, D , B ) − π δ K δ E ( f, ˆ E ( f, D , B ) , B ) , (2.7)and ˆ P = ˆ P ( f, D , B ) is the gyrocenter polarization operator given by ˆ P ( f, D , B ) = 14 π ( D − ˆ E ( f, D , B )) . (2.8)The noncanonical Poisson bracket is given by [ F , G ] = N s X s =1 Z B gy s (cid:18) d δ F δf s − πe s δ F δ D · d X , d δ G δf s − πe s δ G δ D · d X (cid:19) f s + 4 πc Z (cid:18) δ F δ D · ∇ × δ G δ B − δ G δ D · ∇ × δ F δ B (cid:19) d X . (2.9)Here B gy s is the gyrocenter Poisson tensor, which is defined as follows. If z a is a coordinate systemon the gyrocenter phase space and α, β are -forms on the same space, B gy s ( α, β ) = α a β b { z a , z b } gy s ,where {· , ·} gy s is the gyrocenter Poisson bracket. Note that a Poisson bracket for electrostatic gy-rokinetics was given in Squire et al. (2013). The complexity of that bracket should be contrastedwith the relative simplicity of the bracket given here for electromagnetic gyrokinetics. This bracket,which has a form akin to that of Morrison (2013), is to our knowledge the first demonstration ofHamiltonian structure for any electromagnetic gyrokinetic theory. We arrived at this electromagnetic gyrokinetic system by modifying the standard variational deriva-tion of electromagnetic gyrokinetics Sugama (2000); Brizard (2000b,a); Brizard and Hahm (2007);
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AMILTONIAN FORMULATION OF THE GYROKINETIC V LASOV -M AXWELLEQUATIONS
The usual argument for invoking the Darwin approximation in EMGT is that doing so eliminateslight waves. This may seem to be an especially compelling argument from a computational pointof view. After all, the presence of traveling waves with phase velocity c leads to a very restrictiveCFL condition for explicit integration schemes. Therefore, avoiding the Darwin approximation aswe have done may appear objectionable in a practical sense.On the other hand, this numerical argument supporting the Darwin approximation is not asstrong as it appears. As is evident from the form of the GVM equations given above, avoidingthe Darwin approximation does not lead to Maxwell’s equations, but Maxwell’s equations in apolarized and magnetized medium. Therefore, the light waves supported by these equations do nottravel at the speed of light in vacuum. HAPTER
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AMILTONIAN FORMULATION OF THE GYROKINETIC V LASOV -M AXWELLEQUATIONS c in a gyrokinetic plasma(this is consistent with the notion of a so-called “gyrokinetic vacuum”). Using the long-wavelengthlimit of the gyrokinetic dielectric function, ω pi /ω ci , as a rough approximation, we find that lightwaves in the GVM equations propagate at the Alfv´en speed. Thus, the CFL constraint imposed bylight waves in the GVM equations is not nearly as strict as the usual argument might suggest .An even stronger case can be made for the computational viability of this new formulation ofelectromagnetic gyrokinetics. We first make the following simple observation. A familiar calcula-tion shows that if ∇ · D = 4 πρ gy and ∇ · B = 0 at t = 0 , then these equations will also be satisfiedfor all subsequent times. This means that the evolution of the magnetic field and the auxiliary elec-tric field is completely determined by the Amp`ere equation and the Faraday equation. Interestingly,it can be shown that this property arises as a direct consequence of employing a gauge-invariantgyrocenter Lagrangian; the quantity ∇ · D − πρ gy is the conserved quantity associated with gaugesymmetry by Noether’s theorem.Now suppose the Amp`ere and Faraday equations were used to advance D and B in time on acomputer. Employing a simple explicit scheme, the following steps would have to be taken at eachtime step. (1) Using the constitutive relations, compute E and H from the known values of D and B . (2) Compute ∇ × H and ∇ × E . (3) Using a finite difference approximation for the partialtime derivative, solve for the new D and B .Steps (2) and (3) clearly require only local operations, and so represent nearly embarrassinglyparallel computations. Again invoking the long wavelength limit, step (1) can also be seen to belocal. In this limit, there is a simple algebraic relationship between D and E (see Brizard (2013),for example) that can be inverted analytically. Thus, the entire field solve step in an explicit time Strictly speaking, it is only light waves that travel perpendicular to the magnetic field that experience a reducedpropgation speed. Those that travel along the magnetic field lines may still travel near the speed of light in vacuum.However, the numerical grids appropriate for gyrokinetic simulations are significantly elongated along the field lines,which substantially reduces the parallel CFL condition.
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AMILTONIAN FORMULATION OF THE GYROKINETIC V LASOV -M AXWELLEQUATIONS
We will now turn from numerical benefits offered by the GVM equations in order to discuss theiranalytical benefits. First, we mention the system’s conservative properties. An immediate con-sequence of the GVM Poisson bracket structure is conservation of the Hamiltonian functional(this follows from antisymmetry of the bracket). It is also not difficult to show that there isa conserved momentum functional for each rotation or translation symmetry of the backgroundmagnetic field. Finally, there is a large family of conserved functionals given by the Poissonbracket’s Casimirs. These are functionals C that Poisson commute with every other functional, i.e. ∀F , [ C, F ] = 0 . Systems of gyrokinetic equations (electromagnetic or electrostatic) with exactenergy and momentum conservation laws can also be derived using the standard variational ap-proach Scott and Smirnov (2010); Sugama (2000); Brizard (2000b,a); Pfirsch and Correa-Restrepo(2004); Squire et al. (2013). Indeed, this was the main motivation for developing the standard vari-ational formulations of gyrokinetics. However, variational approaches do not readily produce theCasimir invariants (nor has it been shown that the usual variational formulations of EMGT possessPoisson brackets and Casimir invariants at all).Many of the GVM bracket’s Casimirs are given as follows. Let Ω s = − ω gy s ∧ ω gy s ∧ ω gy s (2.10) HAPTER
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AMILTONIAN FORMULATION OF THE GYROKINETIC V LASOV -M AXWELLEQUATIONS function , F s , where f s = F s Ω s , (2.11)then C h = N s X s =1 Z T Q h s ( F s ) Ω s (2.12)is a Casimir for each function of a single real variable h s . Moreover, any functional of ∇· D − πρ gy is a Casimir, which is one way of seeing that Eq. (2.1d) is satisfied in the Hamiltonian formulationof the GVM equations.Another advantage the Poisson bracket formulation of the GVM equations provides, which avariational formulation does not, is immediate access to the theory of dynamically accessible vari-ations Morrison and Pfirsch (1989) (see also Morrison (1998); Andreussi et al. (2013)). Supposewe perturb a GVM equilibrium by switching on a small time-dependent term in the Hamiltonain,i.e. H → H + δ H t , where δ H t is a time-dependent functional that is non-zero only in a briefinterval of time after t = 0 . Using the Poisson bracket, we can give an energy principle for as-sessing the stability of this perturbation in the limit where the kick caused by switching on δ H t isinfinitesimal.In this limit, and accounting for the fact that the perturbation is generated by altering the Hamil-tonian, we find that the perturbed distribution function, auxiliary electric field, and magnetic field HAPTER
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AMILTONIAN FORMULATION OF THE GYROKINETIC V LASOV -M AXWELLEQUATIONS δf s = − L ξ s f s (2.13) δ D = − π J ( ξ, f ) + 4 πc ∇ × β (2.14) δ B = − πc ∇ × α , (2.15)where α , β are arbitrary vector fields on configuration space, the phase space fluid displacementvector ξ s is determined by Hamilton’s equations,i ξ s ω gy s = d χ s + 4 πe s α · d X , (2.16)with χ s an arbitrary function on gyrocenter phase space, and J ( ξ, f ) is the gyrocenter current den-sity generated by fiducial gyrocenters with phase space velocity ξ s and distribution f s . Appealingto the general theory of dynamically accessible variations (see e.g. Morrison (1998)), our pertur-bation will be stable if the free energy functional δ F ( α , β , χ ) is positive whenever δf s , δ D , and δ B are not each zero. The free energy functional is defined by δ F ( α , β , χ ) = 12 [[ H , S ] , S ] , (2.17)where the functional S = P s R χ s f s + R α · D d X + R β · B d X . Physically, δ F is the second-order change in the energy functional H produced by our perturbation. In fact, δ F functions asthe (conserved) Hamiltonian of the linearized GVM equations. HAPTER
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AMILTONIAN FORMULATION OF THE GYROKINETIC V LASOV -M AXWELLEQUATIONS δ F can be written in the form δ F = X s Z (cid:18) ω gy s ( V gy s , ξ s ) δf s + δK s δf s + e s c δ B · ( V gy s ) X × ( ξ s ) X f s (cid:19) + 18 π Z (cid:18) δ D · δ E + δ B · δ H (cid:19) d X . (2.18)Here X in a subscript denotes the X -component of a velocity field on phase space. The variations δK s , δ E , and δ H are given by δK s = δK s δ E [ δ E ] + δK s δ B [ δ B ] (2.19) δ E = ε − [ δ D ] + η [ δ B ] (2.20) δ H = η † [ δ D ] + µ − [ δ B ] . (2.21)where the linear operators ε , µ , and η are given by (cf. Morrison (2013)) ε = 1 − π δ K δ E δ E (2.22) µ − = 1 + 4 π δ K δ B δ B + (4 π ) δ K δ E δ B ε − δ K δ B δ E (2.23) η = 4 πε − δ K δ B δ E . (2.24)In principle, an energy principle for electrostatic gyrokinetics analogous to this one could be de-rived using the Poisson bracket given in Squire et al. (2013). However, the authors of that Refer-ence deemed the electrostatic gyrokinetic Poisson bracket too complicated to be practically useful,and so did not attempt deriving an expression for δ F .We have used this expression for δ F to prove that, in the long wavelength limit, the thermalequilibrium state in a uniform background magnetic field is stable. In this case, the gyrocenter HAPTER
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AMILTONIAN FORMULATION OF THE GYROKINETIC V LASOV -M AXWELLEQUATIONS K = 12 mv k + ω c J − mc (cid:18) v k c B ⊥ B o + E × ˆ bB o (cid:19) , (2.25)where J is the gyroaction, ω c is the signed gyrofrequency, B ⊥ = B − ˆ b ˆ b · B , and B o is the mag-nitude of the background magnetic field. This expression agrees with that given by Krommes inKrommes (2013) in the absence of magnetic fluctuations. The linear response functions ε − , µ − , η are therefore given by the constant matrices ε = 1 + 4 πc v A (1 − ˆ b ˆ b ) (2.26) µ − = 1 − πβ (1 − ˆ b ˆ b ) (2.27) η = 0 , (2.28)where β = P s m s n s h v k i s B o is the plasma β and h·i s denotes the velocity space average. Using theseexpressions and the assumption of thermal equilibrium, a straightforward, but tedious calculationleads to the following form for δ F , δ F = X s Z T (cid:18) L ξ s H os − T δB k B o (cid:19) f s + 18 π Z δ D · ε − δ D d X + 18 π Z δ B ⊥ · µ − · δ B ⊥ d X + 18 π Z (1 − πnT /B o ) δB k d X , (2.29)where n = P s n s is the total gyrocenter number density. As long as πβ and πnT /B o are eachless than , a condition that is generally satisfied, δ F is manifestly non-negative, which implieslinear stability. HAPTER
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AMILTONIAN FORMULATION OF THE GYROKINETIC V LASOV -M AXWELLEQUATIONS The Hamiltonian formulation of the GVM system given in this Chapter is completely determinedby two key quantities, the gyrocenter kinetic energy K s , and the guiding center symplectic form ω gc s . Suppressing species labels, the gyrocenter kinetic energy is given explicitly to second orderin the amplitude of the fluctuating fields, ǫ δ , by K ( E, B ) = H gc − ǫ δ h ℓ i + ǫ δ B gy ( h δ Ξ i , d h ℓ i )+ 12 ǫ δ (cid:28) B gy s (cid:18) L R [ δ ˜Ξ − d I (˜ ℓ )] , [ δ ˜Ξ − d I (˜ ℓ )] (cid:19)(cid:29) , (2.30)where R is the infinitesimal generator of gyrophase rotations times the local gyrofrequency, I is theinverse of the Lie derivative L R , angle brackets denote gyroangle averaging, and ˜ Q = Q − h Q i . Instandard guiding center coordinates, L R = ω c ∂∂θ , where θ is the gyrophase, which means I amountsto an antiderivative in gyrophase. It can be shown that the second-order gyrocenter kinetic energyhas the same general form as Eq. (129) in Brizard and Hahm (2007). The relevant correspondencesbetween our symbols and those of Brizard and Hahm (2007) are ℓ ↔ − K , B gy ab ↔ J abo , δ Ξ ↔ ∆Γ , and L R δ Ξ ↔ L R (¯Γ + Γ ) .From this expression, it is clear that the gyrocenter kinetic energy is determined by the threequantities H gc , ℓ , and δ Ξ . H gc denotes the guiding center Hamiltonian truncated at some desiredorder in ρ/L . The function ℓ and the -form δ Ξ are defined in terms of any choice of the guidingcenter Lie generators as follows. Decompose the guiding center transformation τ gc : T Q → T Q as τ gc = τ ◦ τ , where τ = exp( G ) (2.31) τ = · · · ◦ exp( G ) ◦ exp( G ) ≡ exp( ¯ G ) , (2.32) HAPTER
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AMILTONIAN FORMULATION OF THE GYROKINETIC V LASOV -M AXWELLEQUATIONS G k are the guiding center Lie generators. The leading-order guiding center transformation, τ , must be handled carefully in gyrokinetics because the fluctuating fields are allowed to haveshort perpendicular wave lengths. The -form δ Ξ = − ec (exp( − L ˜ G ) i G U ( L G ) + i ¯ G U ( L ¯ G )) B · dS, (2.33)where the function U ( x ) = e − x/ sinh( x/ / ( x/ , represents the perturbation to the guidingcenter Lagrange -form produced by the fluctuating electromagnetic fields. The function δ H = e (exp( − L ˜ G ) i G U ( L G ) + i ¯ G U ( L ¯ G )) E · dX (2.34)represents the perturbation to the guiding center kinetic energy caused by the same fields. Thefunction ℓ = δ Ξ( V gy o ) − δH, (2.35)where V gy o is the unperturbred gyrocenter phase space velocity.The Hamiltonian structure of the GVM equations reproduces that of the Vlasov-Maxwell sys-tem Morrison (1980, 1982); Marsden and Weinstein (1982) under the substitutions K → m v (2.36) ω gc → m d x i ∧ d v i . (2.37)It is also interesting to compare [ · , · ] to the bracket given in Morrison (2013). The only significantdifference comes from the manner in which the inductive electric field is built into the kineticequation. HAPTER
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AMILTONIAN FORMULATION OF THE GYROKINETIC V LASOV -M AXWELLEQUATIONS K and ω gc .If this were true, then the benefits that our bracket brings to electromagnetic gyrokinetics could beextended to certain kinds of laser-plasma interactions. The results presented in this Chapter were obtained in conjunction with Professor Philip Morri-son and Professor Alain Brizard. They are also posted on the arXiv at arXiv:1411.1790 and inBurby et al. (2015). hapter 3Boundary terms and Poisson brackets
The purpose of this Chapter is twofold. The first is to describe a systematic procedure to pass froman Euler-Poincar´e formulation Holm et al. (1998) of a physical system with advected parametersand dynamical fields to a Poisson bracket formulation for that same system. The second is to applythis procedure to the Euler-Poincar´e formulation of several reduced plasma models, including thegyrokinetic Vlasov-Maxwell system presented in the previous Chapter. The value of a generalEluer-Poincar´e → Poisson procedure stems from the fact that implementing approximations withinthe Lagrangian formalism is a well-developed art, whereas finding approximations that respect theJacobi identity is much more subtle.To pass from an Euler-Poincar´e formulation with a regular
Lagrangian to a Poisson bracketformulation, all that is necessary is the procedure outlined in Holm’s paper on Euler-Poincar´etheory Holm et al. (1998), which consists of two steps. First one passes from the parameterizedLagrangian description to a parameterized Hamiltonian description using the Legendre transform.Then one applies the theory developed by Marsden in his paper Marsden et al. (1984) on the Hamil-20
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OUNDARY TERMS AND P OISSON BRACKETS not employ the Dirac theory of constraints. Instead, we will apply a technique rooted in a carefulanalysis of the boundary terms that appear when varying an action functional without keepingendpoints fixed. In detail, our method consists of the following steps. step 1—
Identify an Euler-Poincar´e formulation for the system under consideration. In par-ticular, identify a parameter-dependent Lagrangian, L a : T Q × T G → R , where Q is the spaceof dynamical fields, G is a Lie group (usually a diffeomorphism group), and the parameter a is anelement of a vector space V ∗ upon which G acts. step 2— Eliminate the parameters by introducing a Lagrange multiplier. This method is de-scribed in Cendra’s
Lagrangian reduction by stages
Cendra et al. (2001). The result of this simple
HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS L : T Q × T G × T ( V × V ∗ ) → R that embeds the original dynamics ina slightly larger space. step 3— Identify the submanifold P o ⊂ T Q × T G × T ( V × V ∗ ) that serves as the aug-mented system’s phase space. Note that this step involves analyzing the initial value problemassociated with L ’s Euler-Lagrange equations. Possible gauge symmetries and degeneracies ofthe Lagrangian make this step non-trivial in general. step 4— Identify a Poisson bracket [ · , · ] P o and Hamiltonian H P o on the augmented system’sphase space using the boundary symplectic form methodology described in Marsden et al. (1998).This bracket is essentially a Pierles bracket. step 5— Observe that the Pierles bracket and Hamiltonian on the augmented phase space areinvariant under the action of the semidirect product S = G ⋊ V . Perform Poisson reduction usingthis symmetry, thereby identifying the Poisson bracket on the reduced phase space P = P o /S . Inthis step, the Lagrange multiplier will be eliminated by the Poisson reduction, meaning P can bethought of as the physical phase space for the system under consideration.In what follows, we will illustrate this technique by applying it to a number of examples.These include (i) the (generalized) Vlasov-Poisson system, (ii) the Vlasov-Darwin system, (iii)the gyrokinetic Vlasov-Maxwell system, (iv) the neutral Vlasov model Tronci and Camporeale(2015), and (v) force-free electrodynamics Gralla and Jacobsen (2014). HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS Let P be a N -dimensional symplectic manifold with symplectic form ω = − d ϑ ; note that P is not necessarily a cotangent bundle and ω is not necessarily a canonical symplectic form. Let V ∗ =Ω N ( P ) and G = Diff ( P ) denote the space of N -forms on P and the group of diffeomorphismsof P , respectively. A typical element of G will be denoted g ∈ G while a typical element of V ∗ will be denoted f ∈ V ∗ .The generalized Vlasov-Poisson system is defined by the parameter-dependent Lagrangian, L f o : T G → R , given by L f o ( g, ˙ g ) = Z P ( g ∗ f o ) ϑ ( ˙ g ◦ g − ) − H ( g ∗ f o ) , (3.1)where H : V ∗ → R is the generalized Hamiltonian functional. The generalized Vlasov-Poissondynamics follow from this Lagrangian by applying Hamilton’s principle to the following actionfunctional. Let P ( G ) and g ∈ P ( G ) denote the space of paths in G and a typical path in G ,respectively. The action functional S f o : P ( G ) → R is given by S f o ( g ) = Z t t L f o ( g ( t ) , ˙ g ( t )) dt. (3.2)The Euler-Lagrange equations associated with this action functional can be derived as follows.Let g = X ( P ) and ξ ∈ g denote the space of vector fields on P and a typical vector field, respec-tively. Notice that L f o ( g, ˙ g ) = ℓ ( ˙ g ◦ g − , g ∗ f o ) , (3.3) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ℓ : g × V ∗ → R is given by ℓ ( ξ, f ) = Z P f ϑ ( ξ ) − H ( f ) . (3.4)It follows that the first (fixed-endpoint) variation of the action functional is given by δS f o ( g )[ δ g ] = Z t t (cid:18)Z P δℓδξ ( ˙ η ( t ) + [ ξ ( t ) , η ( t )]) − δℓδf L η ( t ) f ( t ) (cid:19) dt = − Z t t Z P (cid:18) dd t δℓδξ + L ξ ( t ) δℓδξ − d δℓδf ⊗ f ( t ) (cid:19) · η ( t ) dt = − Z t t Z P (cid:18) ϑ ⊗ ˙ f ( t ) + L ξ ( t ) ( ϑ ⊗ f ( t )) − d (cid:18) ϑ ( ξ ( t )) − δ H δf (cid:19) ⊗ f ( t ) (cid:19) · η ( t ) dt = Z t t Z P f ( t ) (cid:18) i ξ ( t ) ω − d δ H δf (cid:19) · η ( t ) dt, (3.5)where ξ ( t ) = ˙ g ( t ) ◦ g ( t ) − (3.6) η ( t ) = δ g ( t ) ◦ g ( t ) − (3.7) f ( t ) = g ( t ) ∗ f o , (3.8)and the functional derivatives are evaluated at ( ξ ( t ) , f ( t )) . The Euler-Poincar´e equations are there-fore i ξ ( t ) ω = d δ H δf , (3.9)which should be augmented with the equation f ( t ) = g ( t ) ∗ f o . (3.10) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ˙ f ( t ) = − L ξ ( t ) f ( t ) . Note that the equation ˙ f ( t ) = − L ξ ( t ) f ( t ) from the previous section does not follow from the Euler-Lagrange equations associated with the parameter-dependent Lagrangian L f o because f o is treatedas merely a parameter in Hamilton’s principle. We can formally remedy this issue as follows.Let V = V ∗∗ and χ ∈ V be the dual to V ∗ and a typical element of V , respectively. Thespace V is properly the space of continuous linear functionals on V ∗ , which is naturally the setof distributional functions on P . Define the parameter-independent Lagrangian L : T G × T ( V × V ∗ ) → R by L ( g, ˙ g, χ, f o , ˙ χ, ˙ f o ) = L f o ( g, ˙ g ) + Z P χ ˙ f o . (3.11)When Hamilton’s principle is applied to the augmented action functional , S : P ( G × V × V ∗ ) → R ,given by S ( g , χ , f o ) = Z t t L ( g ( t ) , ˙ g ( t ) , χ ( t ) , f o ( t ) , ˙ χ ( t ) , ˙ f o ( t )) dt, (3.12)the resulting Euler-Lagrange equations are given by ˙ f o ( t ) = 0 (3.13) ˙ χ ( t ) = g ( t ) ∗ (cid:18) ϑ ( ξ ( t )) − δ H δf (cid:19) (3.14)i ξ ( t ) ω = d δ H δf , (3.15)where the functional derivatives are evaluated at g ( t ) ∗ f o ( t ) . We have thus succeeded in embedding the Euler-Poincar´e equations along with the advection equation into a larger system. We will refer HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS the augmented generalized Vlasov-Poisson equations (AGVP equations, for short).
We will now study the initial value problem associated with the augmented generalized Vlasov-Poisson equations. In particular, we would like to identify a submanifold, P o ⊂ T G × T ( V × V ∗ ) ,such that the AGVP equations define a first-order (infinite-dimensional) ODE on P o . In order toaccomplish this task, we will merely rearrange the AGVP equations given in the previous sectioninto the form of a first-order equation, and then deduce the allowed set of initial data.As they were written in the previous section, the AGPV equations are nearly expressed as afirst-order system. In order to achieve the desired form, we re-write Eq. (3.15) in terms of g ( t ) andsubstitute Eq. (3.15) into Eq. (3.15), giving ˙ f o ( t ) = 0 (3.16) ˙ χ ( t ) = g ( t ) ∗ (cid:18) ϑ ( X δ H /δf ) − δ H δf (cid:19) (3.17) ˙ g ( t ) = X δ H δf ◦ g ( t ) , (3.18)where the functional derivatives are evaluated at g ( t ) ∗ f o ( t ) , which is clearly a first-order systemof equations in the variables ( f o , χ , g ) . That is, there is a vector field Y on G × V × V ∗ such that ( g ( t ) , ˙ g ( t ) , χ ( t ) , f o ( t ) , ˙ χ ( t ) , ˙ f o ( t )) = Y ( g ( t ) , χ ( t ) , f o ( t )) . (3.19)The triple ( f o , χ, g ) belongs to the set G × V × V ∗ , which can naturally be identified withthe graph of the vector field Y on G × V × V ∗ that is defined by Eqs. (3.16), (3.17), and (3.18). HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS P o = { ( g, ˙ g, χ, f o , ˙ χ, ˙ f o ) ∈ T G × T ( V × V ∗ ) | ( g, ˙ g, χ, f o , ˙ χ, ˙ f o ) = Y ( g, χ, f o ) } ≈ G × V × V ∗ . (3.20)Interestingly, Eq. (3.18) implies that there is an invariant subset of P o given by ¯ P o = { ( g, ˙ g, χ, f o , ˙ χ, ˙ f o ) ∈ P o | g ∈ Diff ω ( P ) } ≈ Diff ω ( P ) × V × V ∗ , (3.21)where Diff ω ( P ) is the set of symplectic diffeomorphisms of P . However, ¯ P o is not in one-to-onecorrespondence with all solutions of the AGVP equations, whereas P o is. Therefore we will regard P o as the phase space for the AGVP equations. P o Because P o is a valid phase space for the AGVP equations, the AGVP dynamics formally define atime-independent flow map F t : P o = G × V × V ∗ → P o = G × V × V ∗ , which is characterizedby the relations F = id P o (3.22) dd t F t ( g, χ, f o ) = Y ( F t ( g, χ, f o )) , (3.23)where Y is the vector field on G × V × V ∗ defined by Eqs. (3.16), (3.17), and (3 . . We cantherefore define a mapping Sol : P o → P ( G × V × V ∗ ) given bySol ( g, χ, f o )( t ) = F t − t ( g, χ, f o ) . (3.24) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS S to the augmentedphase space P o , thereby defining the restricted augmented action S P o = Sol ∗ S . (3.25)By examining the exterior derivative of the restricted augmented action, we can identify a sym-plectic form, and therefore Poisson brackets, on the augmented phase space P o . This can be seenusing the following formal manipulation.Let ( g, χ, f o ) ∈ P o be an arbitrary point in the augmented phase space. Let ( g ( t ) , χ ( t ) , f o ( t )) = F t − t ( g, χ, f o ) . (3.26)Because the AGVP dynamical equations imply f o ( t ) = f o , the restricted augmented action evalu-ated at ( g, χ, f o ) is given by S P o ( g, χ, f o ) = Z t t L f o ( g ( t ) , ˙ g ( t )) dt = Z t t ℓ ( ξ ( t ) , g ( t ) ∗ f o ) , (3.27)where ξ ( t ) = ˙ g ( t ) ◦ g ( t ) − . Using the AGVP equations of motion, we can therefore write d S P o = F ∗ t − t Ξ − Ξ , (3.28)where Ξ is a one-form on P o given by Ξ( g, χ, f o )[ δg, δχ, δf o ] = Z P χ δf o + ϑ ( δg ◦ g − ) g ∗ f o . (3.29) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS t (and remembering that S P o depends on t ), we obtain d ˙ S P o = L Y Ξ ⇒ i Y d Ξ = − d (cid:16) Ξ( Y ) − ˙ S P o (cid:17) , (3.30)where the functional ˙ S P o : P o → R is given by ˙ S P o ( g, χ, f o ) = ℓ (cid:0) X δ H /δf ( g ∗ f o ) , g ∗ f o (cid:1) . (3.31)Equation (3.30) immediately implies that the -form ω P o = − d Ξ is preserved by the AGVP flow, F t . Moreover, because it is not hard to show that ω P o is non-degenerate, the AGVP equations canbe written in Poisson bracket form, i.e. given a functional F : P o → R , ˙ F = [ F , H P o ] P o , (3.32)where H P o ( g, χ, f o ) = Ξ( Y ) − ˙ S P o = H ( g ∗ f o ) (3.33)is the augmented system’s energy functional and [ · , · ] P o is the Poisson bracket obtained by invertingthe two-form ω P o .We will conclude this section by deriving an explicit expression for the bracket [ · , · ] P o . Firstobserve that if F : P o → R is a functional, then the associated Hamiltonian vector field Y F , i.e.the vector field on P o that satisfies i Y F ω P o = d F , (3.34) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS L X F G = [ G , F ] P o . (3.35)Because the Lie derivative L X F G would be easy to calculate if X F were known, we will compute ageneral expression for X F . To this end, it is useful to observe that any integral curve of the vectorfield X F is a critical point of a phase space variational principle. That is, an integral curve of Y F isautomatically a critical point of action functional A F : P ( G × V × V ∗ ) → R given by A F ( g , χ , f o ) = Z t t (cid:16) Ξ ( g ( t ) , χ ( t ) , f o ( t )) [( ˙ g ( t ) , ˙ χ ( t ) , ˙ f o ( t ))] − F ( g ( t ) , χ ( t ) , f o ( t )) (cid:17) dt. (3.36)Therefore we can derive an expression for Y F by varying the action given in Eq. (3.36). For thesake of varying F w.r.t. g , we introduce the convention that if H is a functional on P o , then δH/δg ( g, χ, f o ) is the unique -form on P that satisfies d H ( g, χ, f o )[ δg, ,
0] = Z P (cid:18) δHδg ( g, χ, f o ) ⊗ g ∗ f o (cid:19) · ( δg ◦ g − ) . (3.37)The first (fixed-endpoint) variation of A F is given by δ A F ( g , χ , f o )[ δ g , δ χ , δ f o ] = Z t t Z P (cid:20) (cid:18) ˙ f o ( t ) − δ F δχ (cid:19) δ χ + (cid:18) g ( t ) ∗ ( ϑ ( ξ ( t ))) − δ F δf o − ˙ χ ( t ) (cid:19) δ f o ( t )+ i ξ ( t ) ω − g ( t ) ∗ ˙ f o g ( t ) ∗ f o ϑ − δ F δg ! ⊗ ( g ( t ) ∗ f o ( t )) · η ( t ) (cid:21) dt, (3.38) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ξ ( t ) = ˙ g ( t ) ◦ g ( t ) − (3.39) η ( t ) = δ g ( t ) ◦ g ( t ) − , (3.40)and functional derivatives are evaluated at ( g ( t ) , χ ( t ) , f o ( t )) . It follows that Y F is specified by therelations ˙ g ( t ) = ω − (cid:18) δ F /δg + g ( t ) ∗ δ F /δχ g ( t ) ∗ f o ϑ (cid:19) ◦ g ( t ) (3.41) ˙ χ ( t ) = g ( t ) ∗ (cid:18) ϑ (cid:18) ω − (cid:18) δ F /δg + g ( t ) ∗ δ F /δχ g ( t ) ∗ f o ϑ (cid:19)(cid:19)(cid:19) − δ F /δf o (3.42) ˙ f o ( t ) = δ F /δχ, (3.43)where ω − denotes the inverse of the linear map X → i X ω . This formula for Y F proves that thetwo-form − d Ξ is non-degenerate. Moreover, we can now write down the Poisson bracket [ · , · ] P o using Eq. (3.35). The result is [ F , G ] P o = Z P B (cid:18) δ F δg , δ G δg (cid:19) g ∗ f o + δ F δf o δ G δχ − δ F δχ δ G δf o + Z P g ∗ (cid:18) δ G δχ (cid:19) B (cid:18) δ F δg , ϑ (cid:19) − g ∗ (cid:18) δ F δχ (cid:19) B (cid:18) δ G δg , ϑ (cid:19) , (3.44)where B denotes the Poisson tensor assocaited with the symplectic form ω , i.e. given -forms on P , α and β , B ( α, β ) = ω ( ω − ( α ) , ω − ( β )) . (3.45) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS We have now identified the Hamiltonian, H P o ( g, χ, f o ) = H ( g ∗ f o ) , (3.46)and Poisson bracket for the augmented generalized Vlasov-Poisson system. The Jacobi identity issatisfied because the bracket has been obtained by inverting the symplectic form − d Ξ . The modi-fier “augmented” is appropriate because the dynamical variable χ has no direct physical meaning.On the other hand, the variables g and f o together comprise an element of the physical Lagrangian (as opposed to Eulerian) phase space; g gives the configuration of particles in the single-particlephase space P and f o gives the reference phase space density. The purpose of the additional vari-able χ is to extend the Lagrangian phase space just enough to allow for a non-degenerate Poissonbracket.The appearance of the variable χ perhaps seems awkward at this stage. However, observethe following. The set V is a Lie group under addition that is a symmetry group for the AGVPequations. Specifically, for each δχ ∈ V , we can define a mapping T δχ : P o → P o given by T δχ ( g, χ, f o ) = ( g, χ + δχ, f o ) , (3.47)which clearly satisfies the defining properties of a group action, T δχ + δχ = T δχ ◦ T δχ (3.48) T = id P o . (3.49) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS H P o and the augmented Poisson bracket [ · , · ] P o invariant in the sense that T ∗ δχ H P o = H P o (3.50) T ∗ δχ [ F , G ] P o = [ T ∗ δχ F , T ∗ δχ G ] P o , (3.51)for arbitrary δχ ∈ V and functionals F , G . These properties are quick to verify. Therefore wecan define a Hamiltonian and Poisson bracket on the first reduced phase space P L = P o /V , whichis naturally the physical Lagrangian phase space, G × V ∗ . The Lagrangian Hamiltonian, H P L : P L → R , is simply given by H P L ( g, f o ) = H ( g ∗ f o ) . (3.52)The Lagrangian Poisson bracket, [ · , · ] P L , is defined in terms of the projection map π L : P o → P L given by π L ( g, χ, f o ) = ( g, f o ) . (3.53)We have π ∗ L [ F , G ] P L = [ π ∗ L F , π ∗ L G ] P o , (3.54) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS [ · , · ] P L unambiguously because T δχ leaves the Poisson bracket invariant. In particu-lar, [ F , G ] P L ( g, f o ) = [ π ∗ L F , π ∗ L G ] P o ( g, ˜ χ, f o ) (3.55) = Z P B (cid:18) δ F δg , δ G δg (cid:19) g ∗ f o , (3.56)where ˜ χ is arbitrary and the functional derivatives are evaluated at ( g, f o ) . It does not matter which ˜ χ is chosen because [ π ∗ L F , π ∗ L G ] P o ( g, ˜ χ + δχ, f o ) = T ∗ δχ ([ π ∗ L F , π ∗ L G ] P o )( g, ˜ χ, f o )= [ T ∗ δχ π ∗ L F , T ∗ δχ π ∗ L G ] P o ( g, ˜ χ, f o ) (by Eq. (3.193)) = [ π ∗ L F , π ∗ L G ] P o ( g, ˜ χ, f o ) (by Eq. (3.53)) . (3.57)The relation given in Eq. (3.54) shows that the Lagrangian bracket automatically satisfies the Jacobiidentity. Thus, the awkwardness introduced by the additional variable χ is only apparent; we haveobtained a physical Hamiltonian formulation for the generalized Vlasov-Poisson in Lagrangianlabeling by recognizing that the augmented Hamiltonian and bracket are independent of χ . ThisHamiltonian formulation is “physical” in the sense that all dynamical variables are physicallysignificant.The set G is a symmetry group of the generalized Vlasov-Poisson system in Lagrangian label-ing. Specifically, for each h ∈ G , we can define a mapping R h : P L → P L given by R h ( g, f o ) = ( g ◦ h, h ∗ f o ) , (3.58) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS right group action, namely R h ◦ h = R h ◦ R h (3.59) R id P = id P L . (3.60)The Lagrangian Hamiltonian and Poisson bracket are each invariant under this group action, in thesense that R ∗ h H P L = H P L (3.61) R ∗ h [ F , G ] P L = [ R ∗ h F , R ∗ h G ] P L . (3.62)The invariance of the Lagrangian Hamiltonian is quick to verify. The invariance of the Poissonbracket follows from the identity (cid:18) δδg R ∗ h F (cid:19) ( g, f o ) = δ F δg ( g ◦ h, h ∗ f o ) . (3.63)Therefore, the Lagrangian Hamiltonian and Poisson bracket define corresponding quantities on the Eulerian phase space, P E = P L /G ≈ V ∗ . The Eulerian Hamiltonian is given by H P E ( f ) = H ( f ) . (3.64)The Eulerian Poisson bracket is defined in terms of the Eulerian projection map, π E : P L → P E ,given by π E ( g, f o ) = g ∗ f o , (3.65) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS π E ◦ R h = π E . We have for functionals F , G : P E → R , [ F , G ] P E ( f ) = [ π ∗ E F , π ∗ E G ] P L (˜ g, ˜ f o )= Z P B (cid:18) d δ F δf , d δ G δf (cid:19) f = Z P (cid:26) δ F δf , δ G δf (cid:27) f, (3.66)where ˜ g and ˜ f o are any group element and -form that satisfy f = ˜ g ∗ ˜ f o , and {· , ·} is the Poissonbracket associated with the symplectic form ω . Just as earlier, this bracket automatically satisfiesthe Jacobi identity. However, while this fact for the Lagrangian bracket is perhaps not well-known,here it comes as no surprise; [ · , · ] P E is none other than a Lie-Poisson bracket. We will use the following notation.
Particle configuration space —
Let Q = R be the single-particle configuration space withmetric tensor h· , ·i and associated hodge star ∗ . Typical elements of Q will be denoted q ∈ Q . Thecodifferential on Q will be denoted δ , the Laplace-deRham operator will be denoted ∆ = δ d + d δ ,and the Laplace-deRham Green operator will be denoted G . We will also make use of the trans-verse and longitudinal projection operators Π T = δG d and Π L = d Gδ . Particle phase space —
The set
T Q will serve as the single-particle velocity phase space. Atypical element of
T Q will be denoted v q ∈ T q Q . The map π : T Q → Q will denote the tangentbundle projection. Let F : T Q → T ∗ Q be the diffeomorphism given by v q
7→ h v q , ·i . The symbol HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ϑ will stand for F ∗ θ o , where θ o is the canonical -form on T ∗ Q . Function spaces —
The sets Diff ( T Q ) , C ∞ ( T Q ) , and den ( T Q ) denote the diffeomorphismgroup of T Q , the space of smooth functions on
T Q , and the the space of distributional densitieson
T Q . Typical elements of these spaces will be denoted g ∈ Diff ( T Q ) , χ ∈ C ∞ ( T Q ) , and f ∈ den ( T Q ) . The sets Ω k ( Q ) for integer k ≥ are the k -forms on Q . For our purposes, the -forms, -forms, and N -forms are the most important. Typical elements of the latter will bedenoted φ ∈ Ω ( Q ) , A ∈ Ω ( Q ) , and λ ∈ Ω N ( Q ) . The sets X ( T Q ) and X ( T Q ) ∗ are the vectorfields and -form densities on T Q . If S is any space, P ( S ) will denote the space of paths in S parameterized by the time interval [ t , t ] . If the symbol s is used to denote a typical element of S , we will use a bold version of the same symbol to denote a typical path in S , i.e. s ∈ P ( S ) . Multi-species objects —
Let N s be the number of plasma species. Set G = Diff ( T Q ) N s , V = ( C ∞ ( T Q )) N s , and V ∗ = den ( T Q ) N s . We will denote typical elements of these spaces with ˜ g = ( g , ..., g N s ) ∈ G , ˜ χ = ( χ , ..., χ N s ) ∈ V , and ˜ f o = ( f o, , ..., f o,N s ) ∈ V ∗ . Set g = X ( T Q ) N s .A typical element of g will be denoted ˜ ξ ∈ g .Sugama Sugama et al. (2013) gives the following Euler-Poincar´e formulation of the Vlasov-Darwin system (which he calls the Vlasov-Poisson-Amp`ere system). Define the parameter-dependent Lagrangian, L S ˜ f o : T G × T (Ω ( Q ) × Ω ( Q ) × Ω N ( Q )) → R , given by L S ˜ f o (˜ g, ˙˜ g, φ, A, λ, ˙ φ, ˙ A, ˙ λ ) = N s X s =1 Z T Q g s ∗ f o,s (cid:16) m s ϑ + e s c π ∗ A (cid:17) ( ˙ g s ◦ g − s ) − g s ∗ f o,s ( K s + e s π ∗ φ )+ Z Q π ( d φ ∧ ∗ d φ − d A ∧ ∗ d A ) + 14 πc λ δ A, (3.67) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS K s ( v q ) = m s h v q , v q i / . The Vlasov-Darwin system of equations in Lagrangian labelingthen follow from Hamilton’s principle applied to the action functional S S : P ( G × Ω ( Q ) × Ω ( Q ) × Ω N ( Q )) → R given by S S ˜ f o (˜ g , φ , A , λ ) = Z t t L S ˜ f o (˜ g ( t ) , ˙˜ g ( t ) , φ ( t ) , A ( t ) , λ ( t ) , ˙ φ ( t ) , ˙ A ( t ) , ˙ λ ( t )) dt. (3.68)The Euler-Lagrange equations associated with Sugama’s Lagrangian are given byi ξ s ( t ) d θ s = − ˙ θ s − d H s (3.69) δ d A ( t ) = 4 πc N s X s =1 e s ∗ u ( ξ s ( t ) , f s ( t )) + 1 c d ∗ λ ( t ) (3.70) δ d φ ( t ) = 4 π N s X s =1 e s ∗ n ( f s ( t )) (3.71) δA ( t ) = 0 , (3.72)where ξ s ( t ) = ˙ g s ( t ) ◦ g s ( t ) − , f s ( t ) = g s ( t ) ∗ f o,s , the quantities θ s = m s ϑ + e s c π ∗ A ( t ) (3.73) H s = K s + e s π ∗ φ ( t ) , (3.74)and the operators u : X ( T Q ) × den ( T Q ) → Ω ( Q ) and n : den ( T Q ) → Ω ( Q ) are given by thefiber integrals u ( ξ, f )( q ) = Z π − ( q ) i ξ f (3.75) n ( f )( q ) = Z π − ( q ) f. (3.76)We will refer to u ( ξ, f ) as the particle flux -form and n ( f ) as the particle spatial density -form. HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS L ˜ f o : T G → R . A straightforwardcalculation shows that L ˜ f o has the simple expression L ˜ f o (˜ g, ˙˜ g ) = ℓ ( ˙˜ g ◦ ˜ g − , ˜ g ∗ ˜ f o ) , (3.77)where ℓ ( ˜ ξ, ˜ f ) = N s X s =1 Z T Q f s (cid:16) m s ϑ + e s c π ∗ A ( ˜ ξ, ˜ f ) (cid:17) ( ξ s ) ! − H ( ˜ f ) . (3.78)Here we have introduced the Hamiltonian functional H : V ∗ → R H ( ˜ f ) = N s X s =1 Z T Q f s (cid:18) K s + 12 e s π ∗ Φ( ˜ f ) (cid:19) , (3.79)the potential operators A : g × V ∗ → Ω ( Q ) and Φ : V ∗ → Ω ( Q ) , A ( ˜ ξ, ˜ f ) = 4 πc G Π T J ( ˜ ξ, ˜ f ) (3.80) Φ( ˜ f ) = 4 π G ρ ( ˜ f ) , (3.81)and the charge and current density operators ρ : V ∗ → Ω ( Q ) and J : g × V ∗ → Ω ( Q ) , ρ ( ˜ f ) = N s X s =1 e s ∗ n ( f s ) (3.82) J ( ˜ ξ, ˜ f ) = N s X s =1 e s ∗ u ( ξ s , f s ) . (3.83) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS L ˜ f o reproduces theVlasov-Darwin equations. Because the equations for the potentials are satisfied by construction,we will merely verify that the Euler-Poincar´e equation associated with L ˜ f o reproduces Eq. (3.69).The Euler-Poincar´e equation follows from Hamilton’s principle applied to the action functional S ˜ f o : P ( G ) → R given by S ˜ f o (˜ g ) = Z t t L ˜ f o (˜ g ( t ) , ˙˜ g ( t )) dt. (3.84)Varying this action, we obtain the general Euler-Poincar´e equation given originally by Holm, dd t δℓδξ s + L ξ s ( t ) δℓδξ s = d δℓδf s ⊗ f s ( t ) . (3.85)In order to calculate the functional derivatives appearing in this expression, we will first expressthe reduced Lagrangian ℓ in the form ℓ ( ˜ ξ, ˜ f ) = N s X s =1 Z T Q m s ϑ ( ξ s ) f s + 12 c D A ( ˜ ξ, ˜ f ) , J ( ˜ ξ, ˜ f ) E − H ( ˜ f ) , (3.86)where h· , ·i denotes the natural integration pairing of differential forms h α, β i = Z Q α ∧ ∗ β. (3.87)Next we introduce the linear operators J ˜ f : g → Ω ( Q ) and J ˜ ξ : V ∗ → Ω ( Q ) given by J ˜ f ( ˜ ξ ) = J ( ˜ ξ, ˜ f ) (3.88) J ˜ ξ ( ˜ f ) = J ( ˜ ξ, ˜ f ) . (3.89) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ( J ˜ ξ ) † s : Ω ( Q ) → C ∞ ( T Q ) and ( J ˜ f ) † s : Ω ( Q ) → X ( T Q ) ∗ defined by therelations D α, J ˜ ξ ( δ ˜ f ) E = N s X s =1 Z T Q ( J ˜ ξ ) † s ( α ) δf s (3.90) D α, J ˜ f ( δ ˜ ξ ) E = N s X s =1 Z T Q ( J ˜ f ) † s ( α ) · δξ s , (3.91)are readily found to be given by the formulae ( J ˜ ξ ) † s ( α ) = e s π ∗ α ( ξ s ) (3.92) ( J ˜ f ) † s ( α ) = e s π ∗ α ⊗ f s . (3.93)Finally, we compute the Fr´echet derivative of ℓ , Dℓ ( ˜ ξ, ˜ f )[ δ ˜ ξ, δ ˜ f ] = N s X s =1 Z T Q ( m s ϑ ⊗ f s ) · δξ s + (cid:18) m s ϑ ( ξ s ) − δ H δf (cid:19) δf + 1 c D A ( ˜ ξ, ˜ f ) , J ˜ f ( δ ˜ ξ ) E + 1 c D A ( ˜ ξ, ˜ f ) , J ˜ ξ ( δ ˜ f ) E = N s X s =1 Z T Q (cid:18) m s ϑ ⊗ f s + 1 c ( J ˜ f ) † s ( A ( ˜ ξ, ˜ f )) (cid:19) · δξ s + (cid:18) m s ϑ ( ξ s ) + 1 c ( J ˜ ξ ) † s ( A ( ˜ ξ, ˜ f )) − δ H δf (cid:19) δf = N s X s =1 Z T Q (cid:18) m s ϑ ⊗ f s + e s c π ∗ A ( ˜ ξ, ˜ f ) ⊗ f s (cid:19) · δξ s + (cid:18) m s ϑ ( ξ s ) + e s c π ∗ A ( ˜ ξ, ˜ f )( ξ s ) − δ H δf (cid:19) δf, (3.94)from which the functional derivatives can be quickly extracted, giving δℓδξ s = m s ϑ ⊗ f s + e s c π ∗ A ( ˜ ξ, ˜ f ) ⊗ f s (3.95) δℓδf s = m s ϑ ( ξ s ) + e s c π ∗ A ( ˜ ξ, ˜ f )( ξ s ) − δ H δf . (3.96) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS G Π T is self-adjoint. Equation (3.85) therefore reduces toi ξ s ( t ) d (cid:16) m s ϑ + e s c π ∗ A ( ˜ ξ ( t ) , ˜ f ( t )) (cid:17) + ddt (cid:16) m s ϑ + e s c π ∗ A ( ˜ ξ ( t ) , ˜ f ( t )) (cid:17) + d δ H δf s = 0 , (3.97)which is readily verified to be equivalent to Eq. (3.69). We now define the parameter-independent Lagrangian L : T G × T ( V × V ∗ ) → R by L (˜ g, ˙˜ g, ˜ χ, ˜ f o , ˙˜ χ, ˙˜ f o ) = L ˜ f o (˜ g, ˙˜ g ) + N s X s =1 Z T Q χ s ˙ f o,s . (3.98)When Hamilton’s principle is applied to the augmented action functional , S : P ( G × V × V ∗ ) → R ,given by S (˜ g , ˜ χ , ˜ f o ) = Z t t L (˜ g ( t ) , ˙˜ g ( t ) , ˜ χ ( t ) , ˜ f o ( t ) , ˙˜ χ ( t ) , ˙˜ f o ( t )) dt, (3.99)the resulting Euler-Lagrange equations are given by ˙ f o,s ( t ) = 0 (3.100) ˙ χ s ( t ) = g s ( t ) ∗ δℓδf s = g s ( t ) ∗ (cid:18) θ s ( ξ s ( t )) − δ H δf s (cid:19) (3.101)i ξ s ( t ) d θ s = − ˙ θ s − d δ H δf s , (3.102)where θ s = m s ϑ + e s c π ∗ A ( ˜ ξ ( t ) , ˜ f ( t )) , (3.103) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ˜ g ( t ) ∗ ˜ f o ( t ) . The parameter-independent Lagrangian L therefore succeeds at embedding the Vlasov-Darwin dynamics, including those of the distribu-tion function, into a larger system. We will refer to the system defined by Eqs. (3.100), (3.101),and (3.102) as the augmented Vlasov-Darwin equations (AVD equations, for short). In order to identify a phase space for the AVD equations, we must identify a submanifold of
T G upon which the Euler-Lagrange equations associated with the augmented Lagrangian L define asystem of equations that are first order in time. As the Euler-Lagrange equations are written in theprevious section, the AVD equations are not written as a first order system, and so there is somework to do.The most non-trivial AVD Euler-Lagrange equation is the one that takes the form of a time-dependent Hamilton equation:i ξ s ( t ) d (cid:16) m s ϑ + e s c π ∗ A ( ˜ ξ ( t ) , ˜ f ( t )) (cid:17) + ddt (cid:16) m s ϑ + e s c π ∗ A ( ˜ ξ ( t ) , ˜ f ( t )) (cid:17) + d δ H δf s = 0 . (3.104)This equation appears to implicitly relate the Eulerian velocity field ˜ ξ ( t ) with its time derivative ˙˜ ξ ( t ) and the distribution function ˜ f ( t ) ; the time derivative of ˜ ξ ( t ) appears as a result of the identity ddt A ( ˜ ξ ( t ) , ˜ f ( t )) = A ( ˙˜ ξ ( t ) , ˜ f ( t )) − A ( ˜ ξ ( t ) , L ˜ ξ ( t ) ˜ f ( t )) . (3.105)Fortunately, this implicit relationship is not quite as complicated as it seems. It turns out thatEq. (3.104) implies ξ s ( t ) must be a second-order vector field for each t . Therefore A ( ˜ ξ ( t ) , ˜ f ( t )) and u ( ξ s ( t ) , f s ( t )) can be expressed in terms of the free-streaming vector field X o , i.e. the unique HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS X o d m s ϑ = − d K s . We have u ( ξ s ( t ) , f s ( t )) = u ( X o , f s ( t )) (3.106) A ( ˜ ξ ( t ) , ˜ f ( t )) = A ( ˜ X o , ˜ f ( t )) , (3.107)which implies that the time derivative of the vector potential simplifies to ddt A ( ˜ ξ ( t ) , ˜ f ( t )) = −A ( ˜ X o , L ˜ ξ ( t ) ˜ f ( t )) . (3.108)Thus, the time-dependent Hamilton equation simplifies toi ξ s ( t ) d (cid:16) m s ϑ + e s c π ∗ A ( ˜ X o , ˜ f ( t )) (cid:17) − e s c π ∗ A ( ˜ X o , L ˜ ξ ( t ) ˜ f ( t )) + d δ H δf s = 0 , (3.109)which is merely a relationship between ˜ ξ ( t ) and ˜ f ( t ) .The relationship between ˜ ξ ( t ) and ˜ f ( t ) can be resolved explicitly as follows. By applying fiberintegrals to the Vlasov equation, ˙ f s ( t ) = − L ξ s ( t ) f s ( t ) , we obtain the fluid equation for the particleflux u s ( t ) = u ( ξ s ( t ) , f s ( t )) , ddt ∗ u s ( t ) = − div ( T s ( t )) ♭ − e s m s ( ∗ n s ) 1 c ˙ A ( t ) − e s m s ( ∗ n s ( t )) d φ ( t ) + e s m s ∗ ( ∗ u s ( t ) ∧ ∗ dA ( t )) , (3.110)where n s ( t ) = n ( f s ( t )) (3.111) A ( t ) = A ( ˜ ξ ( t ) , ˜ f ( t )) (3.112) φ ( t ) = Φ( ˜ f ( t )) (3.113) T s ( t ) = T ( f s ( t )) (3.114) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS T is the stress tensor operator. Given a pair of -forms α, β ∈ Ω ( Q ) , the defining relation forthe stress tensor operator is T ( f )( α, β ) = ∗ (cid:18)Z π − π ∗ α ( X o ) π ∗ β ( X o ) f (cid:19) . (3.115)By multiplying the fluid equation for u s ( t ) by e s and then summing over species, we then obtain ddt ∗ J ( ˜ ξ ( t ) , ˜ f ( t )) = − N s X s =1 e s div ( T s ( t )) ♭ − πc ω p ( t ) ˙ A ( t ) − π ω p ( t ) d φ ( t ) + N s X s =1 e s m s ∗ ( ∗ u s ( t ) ∧ ∗ dA ( t )) (3.116)as an equation for the time-derivative of the current density. Here we have introduced the localplasma frequency ω p ( t ) = ω p ( ˜ f ( t )) , where ω p ( ˜ f ) = N s X s =1 πe s m s ∗ n ( f s ) . (3.117)Finally, by applying the operator − πc G Π T to both sides of Eq. (3.116), we obtain a linear operator E I : V ∗ → Ω ( Q ) that gives the inductive electric field in terms of the distribution function, E L ( ˜ f ) = (cid:20) c G Π T ˆ ω p ( ˜ f ) (cid:21) − c G Π T ˆ ω p ( ˜ f )[ O ( ˜ f )] , (3.118)where O : V ∗ → Ω ( Q ) is given by O ( ˜ f ) = d Φ( ˜ f ) + 4 πω p ( ˜ f ) N s X s =1 (cid:18) e s div ( T ( ˜ f )) ♭ − e s m s ∗ [ ∗ u ( X o , f s ) ∧ ∗ d A ( ˜ X o , ˜ f )] (cid:19) . (3.119) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS E L is precisely what is necessary to express the Eulerian phase space velocity ξ s ( t ) in terms of the distribution function. Indeed, we have ξ s ( t ) = ω s ( ˜ f ( t )) − (cid:18) d δ H δf s − e s π ∗ E I ( ˜ f ( t )) (cid:19) , (3.120)where, for each ˜ f ∈ V ∗ , ω s ( ˜ f ) is the symplectic form on T Q given by ω s ( ˜ f ) = − d (cid:16) m s ϑ + e s c π ∗ A ( ˜ X o , ˜ f ) (cid:17) . (3.121)Note that ω s ( ˜ f ) = − d θ s ( ˜ f ) , where θ s ( ˜ f ) = m s ϑ + e s c π ∗ A ( ˜ X o , ˜ f ) . (3.122)With the relationship between ξ s ( t ) and ˜ f ( t ) resolved, we can now substitute it into the AVDEuler-Lagrange equations. When this substitution is performed, the AVD equations become afirst-order ODE in the variables (˜ g ( t ) , ˜ χ ( t ) , ˜ f o ( t )) ∈ G × V × V ∗ . Explicitly, we have ˙ g s ( t ) = ω − s (cid:18) d δ H δf s − e s π ∗ E I (cid:19) ◦ g s ( t ) (3.123) ˙ χ s ( t ) = g s ( t ) ∗ (cid:18) θ s ( X o ) − δ H δf s (cid:19) (3.124) ˙ f o,s ( t ) = 0 , (3.125)where the operators ω s , θ s , E I , and the functional derivatives δ H /δf s are evaluated at ˜ g ( t ) ∗ ˜ f o ( t ) ∈ V ∗ . In writing the equation for ˙ χ s ( t ) , we have made use of the fact that ω − s (cid:16) d δ H δf s − e s π ∗ E I (cid:17) isa second order vector field.As is true of first-order ODEs in general, this first-order ODE for the variables (˜ g ( t ) , ˜ χ ( t ) , ˜ f o ( t )) ∈ G × V × V ∗ is identifiable with a vector field Y on G × V × V ∗ . Setting Z ( t ) = (˜ g ( t ) , ˜ χ ( t ) , ˜ f o ( t )) , HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS Y is defined by the relation ddt Z ( t ) = Y ( Z ( t )) . (3.126)Because the AVD equations can be written in this form, it follows that the submanifold P o ⊂ T ( G × V × V ∗ ) that serves as the AVD phase space is given by P o = { ( Z, ˙ Z ) ∈ T ( G × V × V ∗ ) | ˙ Z = Y ( Z ) } ≈ G × V × V ∗ . (3.127) P o We will denote points in G × V × V ∗ with the letter Z , i.e. Z = (˜ g, ˜ χ, ˜ f o ) . The vector field Y defines a time-independent flow map F t : P o = G × V × V ∗ → P o = G × V × V ∗ , which ischaracterized by the relations F = id P o (3.128) dd t F t ( Z ) = Y ( Z ) . (3.129)We can therefore define a mapping Sol : P o → P ( G × V × V ∗ ) , given bySol ( Z )( t ) = F t − t ( Z ) , (3.130)that sends initial conditions (at t = t ) to their corresponding solution path in P ( G × V × V ∗ ) .The mapping Sol can be used to pull back the augmented action functional S to the augmentedphase space P o , thereby defining the restricted augmented action S P o = Sol ∗ S . (3.131) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS S P o is proportional to the free-endpoint variation of the augmentedaction functional S . Because this variation will be evaluated at a curve in G × V × V ∗ that satisfiesthe AVD Euler-Lagrange equations, only the endpoint contributions to the free-endpoint variationwill appear. Specifically we have d S P o = F ∗ t − t Θ − Θ , (3.132)where Θ is the -form on P o given by Θ( Z )[ δZ ] = N S X s =1 Z T Q (cid:16) θ s (˜ g ∗ ˜ f o ) ⊗ ( g s ) ∗ f o,s (cid:17) · η s + χ s δf o,s . (3.133)Here, η s = δg s ◦ g − s . If we now differentiate Eq. (3.132) in the variable t , keeping in mind that S P o depends on t via the upper limit of time integration, we obtain d F ∗ t − t L P o = F ∗ t − t L Y Θ , (3.134)where L P o is the augmented Lagrangian pulled back to the augmented phase space via Y : P o → T ( G × V × V ∗ ) , i.e. L P o ( Z ) = ( Y ∗ L )( Z )= L ( Y ( Z ))= N s X s =1 Z P K s ( g s ) ∗ f o,s + 12 c D A ( ˜ X o , ˜ g ∗ ˜ f o ) , J ( ˜ X o , ˜ g ∗ ˜ f o ) E − D ρ (˜ g ∗ ˜ f o ) , Φ(˜ g ∗ ˜ f o ) E . (3.135) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS L Y Θ = i Y d Θ + d i Y Θ , Eq. (3.134) can be re-written asi Y d Θ = − d H P o , (3.136)where H P o ( Z ) = (Θ( Y ) − L P o )( Z )= N s X s =1 Z P K s ( g s ) ∗ f o,s + 12 c D A ( ˜ X o , ˜ g ∗ ˜ f o ) , J ( ˜ X o , ˜ g ∗ ˜ f o ) E + 12 D ρ (˜ g ∗ ˜ f o ) , Φ(˜ g ∗ ˜ f o ) E (3.137)is the augmented Hamiltonian functional. Because the -form − d Θ turns out to be non-degenerate,Eq. (3.136) shows that the AVD equations are an infinite-dimensional Hamiltonian system withsymplectic form − d Θ and Hamiltonian H P o .We will now calculate the Poisson bracket defined by the symplectic form − d Θ . We willproceed in two steps. First, we will find an expression for an arbitrary Hamiltonian vector field on P o by solving the equation i Y G d Θ = − d G , (3.138)for Y G given an arbitrary functional G : P o → R . Next we will identify an explicit expression forthe Poisson bracket associated with − d Θ using the formula [ F , G ] P o = L Y G F . (3.139)Here, [ · , · ] P o is the augmented system’s Poisson bracket. Because calculating the Lie derivative inthe last expression is simple, all of the nontrivial work will be done in the first step. HAPTER
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50A helpful tool for solving Eq. (3.138) is the phase space variational principle. This variationalprinciple states that (fixed-endpoint) variations of the action functional S G : P ( P o ) → R given by S G ( Z ) = Z t t (cid:18) Θ[ ˙ Z ( t )] − G ( Z ( t )) (cid:19) dt (3.140)are zero if and only if Z is a solution of the equation ddt Z ( t ) = Y G ( Z ( t )) . (3.141)Thus, we know a priori that the Euler-Lagrange equations associated with the action S G areEq. (3.138). On the other hand, we can compute the fixed-endpoint variation of S G directly, giving δ S G ( Z )[ δ Z ] = N s X s =1 Z t t Z T Q (cid:20) d (cid:16) e s c π ∗ A ( ˜ ξ ( t ) , ˜ f ( t ))( X o ) (cid:17) − ddt θ s − g s ( t ) ∗ ˙ f o,s ( t ) f s ( t ) θ s − i ξ s ( t ) d θ s − δ G δg s (cid:21) ⊗ f s ( t ) · η s ( t ) dt + N s X s =1 Z t t Z T Q (cid:20) g s ( t ) ∗ (cid:16) θ s ( ξ s ( t )) + e s c π ∗ A ( ˜ ξ ( t ) , ˜ f ( t ))( X o ) (cid:17) − ˙ χ s ( t ) − δ G δf o,s (cid:21) δ f o,s ( t ) dt + N s X s =1 Z t t Z T Q (cid:20) ˙ f o,s ( t ) − δ G δχ s (cid:21) δ χ s ( t ) dt, (3.142)where f s ( t ) = g s ( t ) ∗ f o,s ( t ) , η s ( t ) = δ g ( t ) ◦ g ( t ) − , and θ s is evaluated at ˜ f ( t ) . By the phasespace variational principle, if we set this variation equal to zero and then solve for ˙ Z ( t ) in termsof the functional derivatives of G , the result will be the solution to Eq. (3.138). We now turn toperforming this task. HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ˙ Z ( t ) involvesdealing with an implicit (linear) equation for ˜ ξ ( t ) . Indeed, the Euler-Lagrange equation given by d (cid:16) e s c π ∗ A ( ˜ ξ ( t ) , ˜ f ( t ))( X o ) (cid:17) − ddt θ s − g s ( t ) ∗ ˙ f o,s ( t ) f s ( t ) θ s − i ξ s ( t ) d θ s − δ G δg s , (3.143)is an implicit equation for ξ s ( t ) , because ˜ ξ ( t ) appears in the first, second, and fourth terms. Wewill solve this implicit equation by working in the tangent lift of an arbitrary coordinate system q i on Q . First we will express A ( ˜ ξ ( t ) , ˜ f ( t )) , which appears in the first term in Eq. (3.143), in termsof Z ( t ) . Then we will express A ( ˜ X o , L ˜ ξ ( t ) ˜ f ( t )) , which appears in the second term of Eq. (3.143),in terms of Z ( t ) . Solving for A ( ˜ ξ ( t ) , ˜ f ( t )) — Let g ij denote the components of the metric tensor in our cho-sen coordinate system. The determinant of this matrix of components will be denoted | g | .Set ξ s ( t ) = u is ∂∂q i + a is ∂∂ ˙ q i (3.144) f s ( t ) = F s d ϑ ∧ d ϑ ∧ d ϑ (3.145) A ( ˜ ξ ( t ) , ˜ f ( t )) = A i d q i (3.146) δ G δg s = Q i d q i + ˙ Q i d ˙ q i . (3.147)Note that d ϑ ∧ d ϑ ∧ d ϑ = | g | d q ∧ d q ∧ d q ∧ d ˙ q ∧ d ˙ q ∧ d ˙ q . (3.148)In order to compute A ( ˜ ξ ( t ) , ˜ f ( t )) , we will draw upon coordinate expressions for the currentdensity operator J ( ˜ ξ ( t ) , ˜ f ( t )) = P s e s ∗ u ( ξ s ( t ) , f s ( t )) . As is readily verified, the fiber integrals HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS -forms, ∗ u , can be written as ∗ Z π − i ξ s ( t ) f s ( t ) = (cid:18)Z F s u is g ij p | g | d ˙ q (cid:19) d q j , (3.149)where d ˙ q = d ˙ q d ˙ q d ˙ q denotes the “bare” measure on ˙ q space. The current density -form istherefore given by J ( ˜ ξ ( t ) , ˜ f ( t )) = N s X s =1 e s (cid:18)Z F s u is g ij p | g | d ˙ q (cid:19) d q j . (3.150)We will also draw upon the expression for u is that is implied by the d ˙ q i -component of Eq. (3.143),namely g ij u js = − e s m s c A i + 1 m s ˙ Q i . (3.151)By definition, the -form A ( ˜ ξ ( t ) , ˜ f ( t )) is given by A ( ˜ ξ ( t ) , ˜ f ( t )) = 4 πc G Π T (cid:18) N s X s =1 e s (cid:18)Z F s u is g ij p | g | d ˙ q (cid:19) d q j (cid:19) , (3.152)where we have used the fiber integral identity given above. Both sides of this equation depend on ˜ ξ ( t ) . However, upon inserting the expression for u is given above, we obtain A ( ˜ ξ ( t ) , ˜ f ( t )) = − c G Π T (cid:18) ω p A ( ˜ ξ ( t ) , ˜ f ( t )) (cid:19) + 4 πc G Π T (cid:18) N s X s =1 e s ∗ u ( ω − s [ δ G /δg s ] , f s ( t )) (cid:19) (3.153) = − c G Π T (cid:18) ω p A ( ˜ ξ ( t ) , ˜ f ( t )) (cid:19) + A ( ω − [ δ G /δ ˜ g ] , ˜ f ( t )) , (3.154) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ω − [ δ G /δ ˜ g ] is the element of g given by ( ω − [ δ G /δ ˜ g ]) s = ω − s [ δ G /δg s ] , and we have usedthe identity ∗ n ( f s ( t )) = Z F s p | g | d ˙ q. (3.155)We therefore arrive at the simple conclusion A ( ˜ ξ ( t ) , ˜ f ( t )) = (cid:20) c G Π T ˆ ω p (cid:21) − A ( ω − [ δ G /δ ˜ g ] , ˜ f ( t )) (3.156) = n ( ˜ f ( t )) h A ( ω − [ δ G /δ ˜ g ] , ˜ f ( t )) i , (3.157)where we have introduced the squared refractive index operator n ( ˜ f ) = (cid:20) c G Π T ˆ ω p ( ˜ f ) (cid:21) − . (3.158) Solving for A ( ˜ X o , L ˜ ξ ( t ) ˜ f ( t )) — In order to solve for A ( ˜ X o , L ˜ ξ ( t ) ˜ f ( t )) in terms of Z ( t ) , wewill make use of coordinate expressions for the -form d θ s and the -form ∗ u ( X o , L ξ s ( t ) f s ( t )) .We have d θ s = m s g ij d ˙ q i ∧ d q j + (cid:18) m s Γ jki ˙ q i + 12 e s c B kj (cid:19) d q k ∧ d q j (3.159) ∗ u ( X o , L ξ s ( t ) f s ( t )) = p | g | ∂∂q k (cid:16)p | g | T ki (cid:17) − Z a is F s p | g | d ˙ q ! g ij d q j , (3.160)where Γ jki = 12 ( g jk,i + g ji,k − g ki,j ) (3.161) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS B kj d q k ∧ d q j = d A ( ˜ X o , ˜ f ( t )) , (3.162)and T ki = Z u ks ˙ q i F s p | g | d ˙ q. (3.163)We will also employ a coordinate expression for the inverse of the -form ω s = − d θ s . Given a -form on T Q , α = α j d q j + ˙ α j d ˙ q j , ω − s ( α ) = (cid:18) m s ˙ α l (cid:19) ∂∂q l − (cid:18) e s m s c ˙ α k B ∗ kj g jl + 1 m s α l (cid:19) ∂∂ ˙ q l , (3.164)where B ∗ kj = cm s e s ˙ q i [Γ jki − Γ kji ] + B kj . (3.165)Set A ( t ) = A ( ˜ X o , L ˜ ξ ( t ) ˜ f ( t )) . By definition, the -form A ( t ) is given by A ( t ) = 4 πc G Π T N s X s =1 e s ∗ u ( X o , L ξ s ( t ) f s ( t )) ! . (3.166)If we decompose ξ s ( t ) as ξ s ( t ) = ν s ( t ) − e s c ω − s ( A ( t )) , (3.167) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ν s ( t ) is the vector field on phase space defined byi ν s ( t ) ω s = δ G δg s + g s ( t ) ∗ ( δ G /δχ s ) f s ( t ) θ s + e s c π ∗ A ( ˜ X o , ˜ g ( t ) ∗ ( δ G /δ ˜ χ )) − d (cid:16) e s c π ∗ n ( ˜ f ( t )) h A ( ω − [ δ G /δ ˜ g ] , ˜ f ( t )) i ( X o ) (cid:17) , (3.168)then we can decompose A ( ˜ X o , L ˜ ξ ( t ) ˜ f ( t )) into two pieces, A ( t ) = 4 πc G Π T N s X s =1 e s ∗ u ( X o , L ν s ( t ) f s ( t )) ! − G Π T N s X s =1 πe s c ∗ u ( X o , L ω − s ( A ( t )) f s ( t )) ! . (3.169)The first term on the right-hand-side of this expression is given entirely in terms of Z ( t ) becausethe vector field ν s ( t ) only depends on Z ( t ) . The second term on the right-hand-side involves thequantity we are trying to solve for, A ( t ) . Using the coordinate identities given earlier, the -form ∗ u ( X o , L ω − s ( A ( t )) f s ( t )) can be expressed as ∗ u ( X o , L ω − s ( A ( t )) f s ( t )) = 1 m s ∗ n ( f s ( t )) A ( t ) . (3.170)Therefore Eq. (3.169) simplifies to A ( t ) = 4 πc G Π T N s X s =1 e s ∗ u ( X o , L ν s ( t ) f s ( t )) ! − c G Π T (cid:0) ω p A ( t ) (cid:1) , (3.171)which provides us with the expression for A ( t ) = A ( ˜ X o , L ˜ ξ ( t ) ˜ f ( t )) we have sought after: A ( ˜ X o , L ˜ ξ ( t ) ˜ f ( t )) = n ( ˜ f ( t )) h A ( ˜ X o , L ˜ ν ( t ) ˜ f ( t )) i . (3.172) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS A ( ˜ X o , L ˜ ξ ( t ) ˜ f ( t )) and A ( ˜ ξ ( t ) , ˜ f ( t )) in hand, we can finally writedown an explicit expression for Y G . Set Y (˜ g, ˜ χ, ˜ f o ) = (˜ g, ˜ χ, ˜ f o , ˙˜ g G , ˙˜ χ G , ( ˙˜ f o ) G ) . We have ( ˙ g s ) G ◦ g − s = ν s (cid:18) ˜ g ∗ ˜ f o ; δ G δ ˜ g ; ˜ g ∗ δ G δ ˜ χ (cid:19) − e s c ω − s (cid:16) π ∗ n (˜ g ∗ ˜ f o ) h A ( ˜ X o , L ˜ ν ( ˜ g ∗ ˜ f o ; δ G δ ˜ g ;˜ g ∗ δ G δ ˜ χ )˜ g ∗ ˜ f o ) i(cid:17) (3.173) ( ˙ χ s ) G = − δ G δf o,s + g ∗ s (cid:18) θ s (cid:18) ν s (cid:18) ˜ g ∗ ˜ f o ; δ G δ ˜ g ; ˜ g ∗ δ G δ ˜ χ (cid:19)(cid:19) + e s c π ∗ n (˜ g ∗ ˜ f o ) h A ( ω − [ δ G /δ ˜ g ] , ˜ g ∗ ˜ f o ) i ( X o ) (cid:19) (3.174) ( ˙ f o,s ) G = δ G δχ s , (3.175)where the operator ν s : V ∗ × g ∗ × V ∗ → X ( T Q ) is given by ν s (cid:16) ˜ f ; ˜ α ; ˜ h (cid:17) = ω − s (cid:18) α s + h s f s θ s + e s c π ∗ A ( ˜ X o , ˜ h ) − d (cid:16) e s c π ∗ n ( ˜ f ( t )) h A ( ω − [ ˜ α ] , ˜ f ( t )) i ( X o ) (cid:17)(cid:19) . (3.176)It follows from these expressions together with Eq. (3.139) that the Poisson bracket on theaugmented Vlasov-Darwin phase space can be written down immediately. However, the mostobvious form of the bracket is not manifestly antisymmetric. After some toil, an antisymmetric HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS [ F , G ] P o ( Z ) = N s X s =1 Z T Q B s (cid:20) δ F δg s − d L s (cid:18) ω − (cid:18) δ F δ ˜ g (cid:19) , ˜ g ∗ ˜ f o (cid:19) , δ G δg s − d L s (cid:18) ω − (cid:18) δ G δ ˜ g (cid:19) , ˜ g ∗ ˜ f o (cid:19)(cid:21) g s ∗ f os + N s X s =1 Z T Q B s (cid:20) δ F δg s − d L s (cid:18) ω − (cid:18) δ F δ ˜ g (cid:19) , ˜ g ∗ ˜ f o (cid:19) , θ s (cid:21) (cid:18) g s ∗ δ G δχ s (cid:19) − N s X s =1 Z T Q B s (cid:20) δ G δg s − d L s (cid:18) ω − (cid:18) δ G δ ˜ g (cid:19) , ˜ g ∗ ˜ f o (cid:19) , θ s (cid:21) (cid:18) g s ∗ δ F δχ s (cid:19) − N s X s =1 Z T Q L s (cid:18) ω − (cid:18) δ F δ ˜ g (cid:19) , ˜ g ∗ ˜ f o (cid:19) (cid:18) g s ∗ δ G δχ s (cid:19) − L s (cid:18) ω − (cid:18) δ G δ ˜ g (cid:19) , ˜ g ∗ ˜ f o (cid:19) (cid:18) g s ∗ δ F δχ s (cid:19) + N s X s =1 Z T Q δ F δf os δ G δχ s − δ G δf os δ F δχ s , (3.177)where L s : g × V ∗ → C ∞ ( T Q ) is a non-linear operator given by L s ( ˜ ξ, ˜ f ) = e s c π ∗ (cid:18) n ( ˜ f ) h A ( ˜ ξ, ˜ f ) i (cid:19) ( X o ) , (3.178)and ω − (cid:16) δ F δ ˜ g (cid:17) ∈ g is given by (cid:20) ω − (cid:18) δ F δ ˜ g (cid:19)(cid:21) s = ω − s (cid:18) δ F δg s (cid:19) . (3.179)In deriving this expression for the augmented system’s Poisson bracket, we have made use of thefact that the operator n ( ˜ f ) G Π T is self-adjoint. See the next brief subsection for a proof of theself-adjoint property. HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS Properties of the squared refractive index operator
The squared refractive index operator is defined by n ( f ) = (cid:20) c G Π T ˆ ω p ( ˜ f ) (cid:21) − , (3.180)where ˆ ω p ( ˜ f ) is the operator that simply multiplies by the (squared) local plasma frequency. n ( ˜ f ) operates on differential forms over Q . This definition may be perplexing because it is not immedi-ately obvious that A ≡ c G Π T ˆ ω p ( ˜ f ) should be an invertible operator. To see that it is, supposethat A has a non-trivial null eigenvector α o . Then α o would have to satisfy c G Π T ˆ ω p ( ˜ f ) α o = − α o . (3.181)In particular, α o would have to be an eigenvector of the operator c G Π T ˆ ω p ( ˜ f ) with eigenvalue − .This is impossible for the following reason. Define the weighted inner product h α, β i ω = Z Q ( α ∧ ∗ β ) ω p ( ˜ f ) . (3.182)Because the operator G Π T is the product of non-negative definite operators, it is non-negativedefinite itself. Therefore, for each α , (cid:28) α, c G Π T ˆ ω p ( ˜ f ) α (cid:29) ω = D ˆ ω p ( ˜ f ) α, G Π T ˆ ω p ( ˜ f ) α E ≥ . (3.183)But this contradicts our assumption that there is a non-trivial null eigenvector of α o . Indeed, (cid:28) α o , c G Π T ˆ ω p ( ˜ f ) α o (cid:29) ω = − h α o , α o i < . (3.184)It follows that the operator A is invertible, and that n ( ˜ f ) is well-defined. HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS n ( ˜ f ) is that it commutes with E ≡ c G Π T ˆ ω p ( ˜ f ) . To see this, set C = n ( ˜ f ) E − E n ( ˜ f ) . We have (1 + E ) C = E − (1 + E ) E n ( ˜ f ) ⇒ (1 + E ) C (1 + E ) = E (1 + E ) − (1 + E ) E = ˆ0 ⇒ C = ˆ0 . (3.185)While n ( ˜ f ) is not self-adjoint as on operator on the standard L space of differential forms,it is self-adjoint as an operator on the weighted L ω space defined by the weighted inner productintroduced earlier. To see this, let α and β be arbitrary k -forms in L ω . Set α = n ( ˜ f ) α and β = n ( ˜ f ) β . We have D α, n ( ˜ f ) β E ω = (cid:28)(cid:20) c G Π T ˆ ω p ( ˜ f ) (cid:21) α , β (cid:29) ω = D ˆ ω p ( ˜ f ) α , β E + (cid:28) c G Π T ˆ ω p ( ˜ f ) α , ˆ ω p ( ˜ f ) β (cid:29) = D ˆ ω p ( ˜ f ) α , β E + (cid:28) ˆ ω p ( ˜ f ) α , c G Π T ˆ ω p ( ˜ f ) β (cid:29) = (cid:28) ˆ ω p ( ˜ f ) α , (cid:20) c G Π T ˆ ω p ( ˜ f ) (cid:21) β (cid:29) = (cid:28) α , (cid:20) c G Π T ˆ ω p ( ˜ f ) (cid:21) β (cid:29) ω = D n ( ˜ f ) α, β E ω , (3.186)where we have used the fact that G Π T is a self-adjoint operator on L . HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS n ( ˜ f ) G Π T is self-adjoint on L . Indeed, D α, n ( ˜ f ) G Π T β E = * αω p ( ˜ f ) , n ( ˜ f ) G Π T β + ω = * n ( ˜ f ) αω p ( ˜ f ) , G Π T ˆ ω p ( ˜ f ) βω p ( ˜ f ) + ω = * G Π T ˆ ω p ( ˜ f ) n ( ˜ f ) αω p ( ˜ f ) , βω p ( ˜ f ) + ω = D n ( ˜ f ) G Π T α, β E . (3.187) We have now identified the Hamiltonian, H P o (˜ g, ˜ χ, ˜ f o ) = N s X s =1 Z P K s ( g s ) ∗ f o,s + 12 c D A ( ˜ X o , ˜ g ∗ ˜ f o ) , J ( ˜ X o , ˜ g ∗ ˜ f o ) E + 12 D ρ (˜ g ∗ ˜ f o ) , Φ(˜ g ∗ ˜ f o ) E , (3.188)and Poisson bracket for the augmented Vlasov-Darwin system. The Jacobi identity is satisfiedbecause the bracket has been obtained by inverting the symplectic form − d Ξ . The modifier “aug-mented” is appropriate because the dynamical variable ˜ χ has no direct physical meaning. On theother hand, the variables ˜ g and ˜ f o together comprise an element of the physical Lagrangian (asopposed to Eulerian) phase space; ˜ g gives the configuration of particles in the single-particle phasespace T Q and ˜ f o gives the reference phase space density. The purpose of the additional variable ˜ χ is to extend the Lagrangian phase space just enough to allow for a non-degenerate Poisson bracket. HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ˜ χ perhaps seems awkward at this stage. However, observe thefollowing. The set V is a Lie group under addition that is a symmetry group for the AVD equations.Specifically, for each δ ˜ χ ∈ V , we can define a mapping T δ ˜ χ : P o → P o given by T δ ˜ χ (˜ g, ˜ χ, ˜ f o ) = (˜ g, ˜ χ + δ ˜ χ, ˜ f o ) , (3.189)which clearly satisfies the defining properties of a group action, T δ ˜ χ + δ ˜ χ = T δ ˜ χ ◦ T δ ˜ χ (3.190) T = id P o . (3.191)This group action leaves the augmented Hamiltonian H P o and the augmented Poisson bracket [ · , · ] P o invariant in the sense that T ∗ δ ˜ χ H P o = H P o (3.192) T ∗ δ ˜ χ [ F , G ] P o = [ T ∗ δ ˜ χ F , T ∗ δ ˜ χ G ] P o , (3.193)for arbitrary δ ˜ χ ∈ V and functionals F , G on the augmented phase space. These properties arequick to verify. Therefore we can define a Hamiltonian and Poisson bracket on the first reducedphase space P L = P o /V , i.e we can perform Poisson reduction .The details of applying Poisson reduction to pass from the AVD phase space to the Lagrangianphase space P L follow. The Lagrangian Hamiltonian, H P L : P L → R , is uniquely determined byrequiring H P L ( π L (˜ g, ˜ χ, ˜ f o )) = H P o (˜ g, ˜ χ, ˜ f o ) (3.194) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS (˜ g, ˜ χ, ˜ f o ) ∈ P o . Here the projection map π L : P o → P L is given by π L (˜ g, ˜ χ, ˜ f o ) = (˜ g, ˜ f o ) . (3.195)The Lagrangian Poisson bracket, [ · , · ] P L , is defined by requiring π ∗ L [ F , G ] P L = [ π ∗ L F , π ∗ L G ] P o . (3.196)These definitions make sense because the augmented Hamiltonian and Poisson bracket are invari-ant under the the action of V . We find that the Lagrangian Hamiltonian is given by H P L (˜ g, ˜ f o ) = N s X s =1 Z P K s g s ∗ f os + 12 c D A ( ˜ X o , ˜ g ∗ ˜ f o ) , J ( ˜ X o , ˜ g ∗ ˜ f o ) E + 12 D ρ (˜ g ∗ ˜ f o ) , Φ(˜ g ∗ ˜ f o ) E . (3.197)The Lagrangian Poisson bracket is given by [ F , G ] P L = N s X s =1 Z T Q B s (cid:20) δ F δg s − d L s (cid:18) ω − (cid:18) δ F δ ˜ g (cid:19) , ˜ g ∗ ˜ f o (cid:19) , δ G δg s − d L s (cid:18) ω − (cid:18) δ G δ ˜ g (cid:19) , ˜ g ∗ ˜ f o (cid:19)(cid:21) g s ∗ f os . (3.198)This bracket and Hamiltonian give a Hamiltonian formulation of the Vlasov-Darwin equations inLagrangian labeling.Now we will pass from Lagrangian labeling to Eulerian labeling by applying Poisson reductiona second time. The set G is a symmetry group of the Vlasov-Darwin system in Lagrangian labeling.Specifically, for each ˜ h ∈ G , we can define a mapping R ˜ h : P L → P L given by R ˜ h (˜ g, ˜ f o ) = (˜ g ◦ ˜ h, ˜ h ∗ ˜ f o ) , (3.199) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS right group action, namely R ˜ h ◦ ˜ h = R ˜ h ◦ R ˜ h (3.200) R id P = id P L . (3.201)The Lagrangian Hamiltonian and Poisson bracket are each invariant under this group action, in thesense that R ∗ ˜ h H P L = H P L (3.202) R ∗ ˜ h [ F , G ] P L = [ R ∗ ˜ h F , R ∗ ˜ h G ] P L . (3.203)The invariance of the Lagrangian Hamiltonian is quick to verify. The invariance of the Poissonbracket follows from the identity (cid:18) δδ ˜ g R ∗ ˜ h F (cid:19) (˜ g, ˜ f o ) = δ F δ ˜ g (˜ g ◦ ˜ h, ˜ h ∗ ˜ f o ) . (3.204)Therefore, the Lagrangian Hamiltonian and Poisson bracket define corresponding quantities on the Eulerian phase space, P E = P L /G ≈ V ∗ . The Eulerian Hamiltonian is given by H P E ( ˜ f ) = N s X s =1 Z P K s f s + 12 c D A ( ˜ X o , ˜ f ) , J ( ˜ X o , ˜ f ) E + 12 D ρ ( ˜ f ) , Φ( ˜ f ) E . (3.205)The Eulerian Poisson bracket is defined in terms of the Eulerian projection map, π E : P L → P E ,given by π E (˜ g, ˜ f o ) = ˜ g ∗ ˜ f o , (3.206) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS π E ◦ R h = π E . We have for functionals F , G : P E → R , [ F , G ] P E ( ˜ f ) = [ π ∗ E F , π ∗ E G ] P L (˜ g, ˜ f o )= N s X s =1 Z T Q (cid:26) δ F δf s − L s (cid:16) ˜ X δ F /δ ˜ f , ˜ f (cid:17) , δ G δf s − L s (cid:16) ˜ X δ G /δ ˜ f , ˜ f (cid:17)(cid:27) s f s . (3.207)where ˜ g and ˜ f o are any group element and -form that satisfy ˜ f = ˜ g ∗ ˜ f o , {· , ·} s is the Poissonbracket associated with the symplectic form ω s , and ˜ X δ F /δ ˜ f ∈ } is given by (cid:16) ˜ X δ F /δ ˜ f (cid:17) s = X δ F /δf s ≡ ω − s ( d δ F /δf s ) . (3.208) The gyrokinetic Maxwell-Vlasov system is most naturally defined by specifying its action, whichis the sum of the net gyrocenter action and the Maxwell action. If a gauge-invariant form of thegyrocenter transformation is employed, we have S ˜ f o (˜ g , A , φ ) = Z t t (cid:18) L p ( ˜ ξ ( t ) , ˜ f ( t ) , E ( t ) , B ( t )) + L int ( ˜ ξ ( t ) , ˜ f ( t ) , A ( t ) , φ ( t )) + L Max ( E ( t ) , B ( t )) (cid:19) dt, (3.209)where ˜ ξ ( t ) = ˙˜ g ◦ ˜ g − is the multi-species Eulerian phase space fluid velocity, ˜ f ( t ) = ˜ g ( t ) ∗ ˜ f o is themulti-species gyrocenter phase space density (a collection of -forms), E ( t ) = − d φ ( t ) − ˙ A ( t ) /c is the electric field -form, B ( t ) = d A ( t ) is the magnetic field -form and the various Lagrangianfunctions are defined as follows. L p is the “free gyrocenter” Lagrangian (in analogy with the notion HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS L p ( ˜ ξ, ˜ f , E, B ) = N s X s =1 (cid:18) (cid:20)Z T Q f s Ξ s ( ξ s ) (cid:21) − K s ( f s , E, B ) (cid:19) , (3.210)where Ξ s is the guiding center -form and K s is the gyrocenter kinetic energy functional K s ( f, E, B ) = Z T Q f K s ( E, B ) . (3.211)Here K s : Ω ( Q ) × d Ω ( Q ) → C ∞ ( T Q ) is the gyrocenter kinetic energy function, which hasa non-local dependence on the electric and magnetic fields. L int is the “interaction” Lagrangiangiven by L int ( ˜ ξ, ˜ f , A, φ ) = N s X s =1 (cid:18) Z T Q f s h e s c π ∗ A ( ξ s ) − e s π ∗ φ i (cid:19) = 1 c D J ( ˜ ξ, ˜ f ) , A E − D ρ ( ˜ f ) , φ E , (3.212)where the current density -form and charge density operators, J and ρ , were introduced in theprevious section. Finally, L Max is the free electromagnetic field action L Max ( E, B ) = 18 π h E, E i − π h B, B i . (3.213)Explicit expressions for Ξ and K s will not be necessary; it is enough to know they can be found inprinciple. HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS (cid:18) dd t + L ξ s ( t ) (cid:19) (cid:18) δ ( L p + L int ) δξ s (cid:19) = d (cid:18) δ ( L p + L int ) δf s (cid:19) ⊗ f s (3.214) c dd t δ ( L p + L Max ) δE + δ δ ( L p + L Max ) δB = − δ L int δA (3.215) δ δ ( L p + L Max ) δE = δ L int δφ . (3.216)Upon calculating the relevant functional derivatives, these equations reduce toi ξ s ( t ) d Ξ gy s = − ˙Ξ gy s − d H gy s (3.217) c dd t D ( t ) − δH ( t ) = − πc J ( ˜ ξ ( t ) , ˜ f ( t )) (3.218) δD ( t ) = − πρ ( ˜ f ( t )) , (3.219)where we have introduced the gyro center -form and Hamiltonian, Ξ gy s = Ξ s + e s c π ∗ A (3.220) H gy s = K s + e s π ∗ φ, (3.221)the -form D ( t ) = E ( t ) − π δ K δE = ˆ D ( ˜ f ( t ) , E ( t ) , B ( t )) , (3.222)and the -form H ( t ) = B ( t ) + 4 π δ K δB = ˆ H ( ˜ f ( t ) , E ( t ) , B ( t )) . (3.223) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS E ( t ) = − d φ ( t ) − ˙ A ( t ) /c , B ( t ) = d A ( t ) , and K = P s K s ,which implies d E ( t ) = − ˙ B ( t ) /c . Equation (3.217) gives the gyrocenter equations of motion inHamiltonian form. Equation (3.218) is the “macroscopic” Amp`ere equation (written in terms offorms). Finally, Eq. (3.219) is the macroscopic Gauss equation.The reference gyrocenter phase space density ˜ f o is not a dynamical variable in this formulationof gyrokinetics. However, we can elevate ˜ f o to the status of a dynamical variable by embeddinggyrokinetics within a slightly larger system. In particular, if we define the augmented gyrokineticaction S (˜ g , A , φ , ˜ f o , ˜ χ ) = S ˜ f o ( t ) (˜ g , A , φ ) + N s X s =1 Z T Q ˙ f os ( t ) χ s ( t ) , (3.224)the resulting Euler-Lagrange equations are given byi ξ s ( t ) d Ξ gy s = − ˙Ξ gy s − d H gy s (3.225) c dd t D ( t ) − δH ( t ) = − πc J ( ˜ ξ ( t ) , ˜ f ( t )) (3.226) δD ( t ) = − πρ ( ˜ f ( t )) , (3.227) ˙ f os ( t ) = 0 (3.228) ˙ χ s ( t ) = g s ( t ) ∗ (Ξ gy s ( ξ s ( t )) − H gy s ) . (3.229)The first four equations in this set decouple from the fifth and reproduce the gyrokinetic Vlasov-Maxwell dynamics. The fifth equation defines the dynamics of the additional variable ˜ χ ( t ) . Wewill refer to this larger system of equations as the augmented gyrokinetic Vlasov-Maxwell equa-tions, or the AGVM equations for short.It may seem awkward to introduce the additional dynamical variable ˜ χ ( t ) . However, ˜ χ ( t ) will be a help rather than a hinderance as we continue our derivation of the gyrokinetic bracket.Moreover, the evolution equation for ˜ χ ( t ) is very much reminiscent of the evolution equation HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS f os ( t ) depends on time, under the gauge transformation A ( t ) → A ( t ) + ϕ ( t ) , φ ( t ) → φ ( t ) − ˙ ϕ ( t ) /c the augmented gyrokinetic action is not weaklyinvariant (it changes by more than temporal boundary terms). However, if we also change χ s ( t ) according to χ s ( t ) → χ s ( t ) − e s c g s ( t ) ∗ π ∗ ϕ ( t ) , the gyrokinetic action is left weakly invariant.Thus, χ s ( t ) changes in the same way as the quantum phase under a gauge transformation.The AGMV equations determine an evolution equation for the variable Z ( t ) = (˜ g ( t ) , A ( t ) , D ( t ) , ˜ f o ( t ) , ˜ χ ( t )) that is first order in time. To see this, first choose the gauge φ ( t ) = 0 . In thisgauge, B ( t ) = d A ( t ) and E ( t ) = − ˙ A ( t ) /c . Therefore, if we introduce the electric field operator ˆ E , which is defined by the implicit equation D = ˆ D ( ˜ f , ˆ E ( ˜ f , D, B ) , B ) , (3.230)we can write dd t A ( t ) = − c ˆ E ( ˜ f ( t ) , D ( t ) , d A ( t )) , (3.231)where ˜ f ( t ) = ˜ g ( t ) ∗ ˜ f o ( t ) , which gives the time derivative of A ( t ) in terms of Z ( t ) . For the timederivative of ˜ g ( t ) , we note that ˙ g s ( t ) = ξ s ( t ) ◦ g s ( t ) , (3.232)and that ξ s ( t ) is a functional of E and B according to Hamilton’s equations, i.e. ξ s ( t ) = X gy s ( ˆ E ( ˜ f ( t ) , D ( t ) , d A ( t )) , d A ( t )) for a functional X gy s : Ω ( Q ) × Ω ( Q ) → X ( T Q ) . Because ξ s ( t ) can be expressed in terms of Z ( t ) , it follows that the macroscopic Amp`ere equation gives HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS D ( t ) in terms of Z ( t ) . For the same reason, the time derivative of χ s ( t ) canbe written in terms of Z ( t ) . Finally, the time derivative of f os ( t ) is trivially a functional of Z ( t ) .It follows that the variable Z ( t ) obeys an (infinite-dimensional) autonomous first order ODE ˙ Z ( t ) = Y ( Z ( t )) , (3.233)where Y is a vector field on P o = G × Ω ( Q ) × Ω ( Q ) × V ∗ × V . We will refer to the space P o as the AGVM phase space. We will use F t : P o → P o to denote the (formal) flow map associatedwith Y .By the existence and uniqueness of solutions to first order ODEs, for each point Z ∈ P o , thereis a unique path Z such that Z ( t ) = Z and ˙ Z ( t ) = Y ( Z ( t )) . In terms of the flow map F t , thispath is given by Z ( t ) = F t − t ( Z ) ≡ Sol ( Z )( t ) , (3.234)where we have defined the function Sol : P o → P ( P o ) . The augmented gyrokinetic systemtherefore gives us a natural way of mapping points in the augmented phase space into a pathspace. But recall that the augmented gyrokinetic action maps points in a path space into the realnumbers. This suggests that we can construct a special real-valued function on P o by composingthe gyrokinetic action with the function Sol in some sense. Actually, composition doesn’t makeliteral sense because the augmented gyrokinetic action is a functional defined on a path spacethat differs from P ( P o ) ; the argument of the augmented gyrokinetic action is a path of the form (˜ g , A , φ , ˜ f o , ˜ χ ) ∈ P ( G × Ω ( Q ) × Ω ( Q ) × V ∗ × V ) . Nevertheless, there is a simple mapping Π : P ( P o ) → P ( G × Ω ( Q ) × Ω ( Q ) × V ∗ × V ) given by Π(˜ g , A , D , ˜ f o , ˜ χ ) = (˜ g , A , , ˜ f o , ˜ χ ) , (3.235) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS φ ( t ) = 0 . We can still therefore define a specialreal-valued function on P o S P o ( Z ) = S (Π( Sol ( Z ))) . (3.236)We will refer to S P o as the restricted augmented action. Note that S P o implicitly depends on t and t as these appear in the action integral’s limits of integration.It is extremely interesting to study the (infinite-dimensional) exterior derivative of the restrictedaugmented action. A simple direct calculation shows d S P o ( δZ ) = F ∗ t − t Θ P o − Θ P o , (3.237)where Θ P o is a -form on P o given by Θ P o ( δZ ) = N s X s =1 (cid:18) Z T Q (Ξ gy ⊗ g s ∗ f os ) · η s + χ s δf os (cid:19) − πc h D, δA i . (3.238)Here ˜ η = δ ˜ g ◦ ˜ g − . This identity becomes very interesting indeed when both sides are differentiatedwith respect to t . We have F ∗ t − t d ˙ S P o = F ∗ t − t L Y Θ P o = F ∗ t − t ( i Y d Θ P o + d (Θ P o ( Y ))) , (3.239)where ˙ S P o is given by ˙ S P o ( Z ) = L p ( ˜ X gy , ˜ g ∗ ˜ f o , ˆ E, d A ) + L int ( ˜ X gy , A,
0) + L Max ( ˆ E, d A ) . (3.240)Here we are using the short hand notation ˜ X gy = ˜ X gy ( ˆ E (˜ g ∗ ˜ f o , D, d A ) , d A ) and ˆ E =ˆ E ( ˆ E (˜ g ∗ ˜ f o , D, d A ) . Equation (3.239) tells us that the augmented gyrokinetic dynamical vec- HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS Y obeys an infinite-dimensional version of Hamilton’s equations,i Y Ω P o = d E P o , (3.241)where the closed (and non-degenerate, as it turns out) -form Ω P o = − d Θ P o and the energy func-tional E P o = Θ P o ( Y ) − ˙ S P o . It follows that the AGVM equations have a Poisson formulation on P o with a Poisson bracket given by inverting the symplectic form Ω and a Hamiltonian functionalgiven by E P o .In order to derive an explicit expression for the AGVM Poisson bracket [ · , · ] P o , the simplestapproach is to first derive an expression for a general Hamiltonian vector field, X G , using theinfinite-dimensional phase space variational principle δ Z t t (cid:18) Θ P o ( ˙ Z ( t )) − G ( Z ( t )) (cid:19) dt = 0 , (3.242)and then calculate the Poisson bracket using the formula [ F , G ] P o = L X G F . (3.243)We find that X G = ( ˜ ξ G ◦ ˜ g, ˙ A G , ˙ D G , ( ˙˜ f o ) G , ˙˜ χ G ) , where ( ξ s ) G = ( ω gy s ) − (cid:18) δ G δg s − πe S π ∗ δ G δD + g s ∗ δ G /δχ s g s ∗ f os Ξ gy (cid:19) (3.244) ˙ A G = − πc δ G δD (3.245) ˙ D G = 4 πc δ G δA − πJ ( ˜ ξ G , ˜ g ∗ ˜ f o ) (3.246) ( ˙ f os ) G = δ G δχ s (3.247) ( ˙ χ s ) G = g ∗ s (Ξ gy s [( ξ s ) G ]) − δ G δf os . (3.248) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS [ F , G ] P o = N s X s =1 Z T Q B gy s (cid:18) δ F δg s − πe s π ∗ δ F δD , δ G δg s − πe s π ∗ δ G δD (cid:19) g s ∗ f os + N s X s =1 Z T Q (cid:18) g s ∗ δ F δχ s (cid:19) B gy s (cid:18) Ξ gy , δ G δg s − πe s π ∗ δ G δD (cid:19) − (cid:18) g s ∗ δ G δχ s (cid:19) B gy s (cid:18) Ξ gy , δ F δg s − πe s π ∗ δ F δD (cid:19) + N s X s =1 Z T Q δ F δf os δ G δχ s − δ G δf os δ F δχ s ! + 4 πc (cid:18)(cid:28) δ F δD , δ G δA (cid:29) − (cid:28) δ G δD , δ F δA (cid:29)(cid:19) , (3.249)where B gy s is the gyrocenter Poisson tensor.The AGVM Poisson bracket in conjunction with the energy functional E P o ( Z ) = (Θ P o ( Y ) − ˙ S P o )( Z )= K (˜ g ∗ ˜ f o , ˆ E, d A ) + D ˆ P , ˆ E E + 18 π D ˆ E, ˆ E E + 18 π h d A, d A i , (3.250)cast the AGVM equations as an infinite dimensional Poisson dynamical system. Here we haveintroduced the polarization operator ˆ P ( ˜ f , D, B ) = 14 π ( D − ˆ E ( ˜ f , D, B )) . (3.251)Note that the AGVM Poisson bracket is derived by inverting a symplectic form, and so it does not have casimirs. The main drawback of this Poisson formulation of gyrokinetics is that it containsa superfluous dynamical variable ˜ χ . A secondary drawback is that it uses the gauge-dependentvector potential A as a dynamical variable. A tertiary drawback is that it is expressed in Lagrangianlabeling.All of the drawbacks of the augmented Poisson formulation of gyrokinetics can be removedusing Poisson reduction. To refresh the reader’s memory, the basic premise of Poisson reductionis that if there is a Lie group H that acts freely on a Poisson manifold P in a manner that leaves HAPTER
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P/H . The bracket on
P/H is known as the reduced bracketand the Hamiltonian on
P/H is known as the reduced Hamiltonian. In our case, the poissonmanifold will be P = P o equipped with the AGVM bracket, and the Hamiltonian function will be E P o . The Lie group will be H = Ω ( Q ) × ( G ⋊ V ) . The first factor in H corresponds to gaugesymmetry, the second to particle-relabeling symmetry, and the third to symmetry with respectto translations in the augmented variable ˜ χ . The quotient P/H will be the gyrokinetic Eulerianphase space, i.e. the space of triples ( ˜ f , D, B ) . The reduced bracket and reduced Hamiltonian onthe Eulerian phase space will provide us with a Poisson formulation of the gyrokinetic Vlasov-Maxwell system without any of the drawbacks of the AGVM Poisson formulation.Instead of applying Poisson reduction to the entire symmetry group of the AGVM system H = Ω ( Q ) × ( G ⋊ V ) , we will apply the Poisson reduction procedure three times, eliminatingone factor of H in each step. The general theory behind breaking a symmetry group into subgroupsand then applying step-wise Poisson reduction is known as Poisson reduction by stages . We willnot need to draw upon any of the general results from this theory. First we will factor out thetranslations in ˜ χ , which are embodied by the additive Lie group V . This will give us a Poissonformulation of gyrokinetics on the space P LA = P o /V , which consists of tuples (˜ g, A, D, ˜ f o ) . P LA is the gauge-dependent Lagrangian phase space for gyrokinetics. In the second reduction step, wewill factor out the gauge symmetry, which is generated by a second additive Lie group Ω ( Q ) . Theresult will be a Poisson formulation of gyrokinetics on the gauge-independent Lagrangian phasespace P L = P LA / Ω ( Q ) . Finally, we will quotient by the particle relabeling symmetry group, G , which will provide us with a Poisson formulation of gyrokinetics on the Eulerian phase space P E = P L /G . HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS P LA = P o /V , is given by [ F , G ] P LA (˜ g, A, D, ˜ f o ) = N s X s =1 Z T Q B gy s (cid:18) δ F δg s − πe s π ∗ δ F δD , δ G δg s − πe s π ∗ δ G δD (cid:19) g s ∗ f os + 4 πc (cid:18)(cid:28) δ F δD , δ G δA (cid:29) − (cid:28) δ G δD , δ F δA (cid:29)(cid:19) . (3.252)The reduced Hamiltonian on P LA is given by E P LA (˜ g, A, D, ˜ f o ) = K (˜ g ∗ ˜ f o , ˆ E, d A ) + D ˆ P , ˆ E E + 18 π D ˆ E, ˆ E E + 18 π h d A, d A i . (3.253)Note that this reduced bracket and Hamiltonian are both invariant under time-independent gaugetransformations A → A + d ϕ , where ϕ ∈ Ω ( Q ) .The Poisson bracket on the second reduced phase space, P L = P LA / Ω ( Q ) , which is thegauge-independent Lagrangian phase space, is given by [ F , G ] P L (˜ g, D, B, ˜ f o ) = N s X s =1 Z T Q B gy s (cid:18) δ F δg s − πe s π ∗ δ F δD , δ G δg s − πe s π ∗ δ G δD (cid:19) g s ∗ f os + 4 πc (cid:18)(cid:28) δ F δD , δ δ G δB (cid:29) − (cid:28) δ G δD , δ δ F δB (cid:29)(cid:19) . (3.254)The Hamiltonian on P L is given by E P o (˜ g, D, B, ˜ f o ) = K (˜ g ∗ ˜ f o , ˆ E, B ) + D ˆ P , ˆ E E + 18 π D ˆ E, ˆ E E + 18 π h B, B i . (3.255)Note that the -form B is required to be exact (ignoring possible Homological complications). HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS P E = P L /G is given by [ F , G ] P E ( ˜ f , D, B ) = N s X s =1 Z T Q B gy s (cid:18) d δ F δf s − πe s π ∗ δ F δD , d δ G δf s − πe s π ∗ δ G δD (cid:19) f s + 4 πc (cid:18)(cid:28) δ F δD , δ δ G δB (cid:29) − (cid:28) δ G δD , δ δ F δB (cid:29)(cid:19) , (3.256)and the Eulerian Hamiltonian is given by E P o ( ˜ f , D, B ) = K ( ˜ f , ˆ E, B ) + D ˆ P , ˆ E E + 18 π D ˆ E, ˆ E E + 18 π h B, B i . (3.257)Note that this bracket, and all of the previous brackets satisfy the Jacobi identity by construction. The basic symmetry group of the AGVM equations is H = Ω ( Q ) × ( G ⋊ V ) , which as a set issimply Ω ( Q ) × G × V . For technical reasons, we will regard Ω ( Q ) as the functions on Q thatvanish at infinity. Some basic properties of H are:• The group identity e = (0 , ˜ id T Q , . The group product of s = ( ϕ , ˜ h , ˜ τ ) and s =( ϕ , ˜ h , ˜ τ ) is given by s ∗ s = ( ϕ + ϕ , ˜ h ◦ ˜ h , h ∗ τ + τ ) . (3.258)The inverse of s = ( ϕ, ˜ h, ˜ τ ) is given by s − = ( − ϕ, ˜ h − , − ˜ h ∗ ˜ τ ) . (3.259) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS H ’s Lie algebra h = Ω ( Q ) × g × V . We will denote typical elements of h with the symbol x = ( δϕ, ˜ ζ, δ ˜ χ ) . The adjoint action of H on h is given byAd s x = ( δϕ, ˜ h ∗ ˜ ζ, ˜ h ∗ [ L ˜ ζ ˜ τ + δ ˜ τ ]) . (3.260)The Lie bracket is therefore [ x , x ] = (0 , − [ ˜ ζ , ˜ ζ ] , L ˜ ζ δ ˜ τ − L ˜ ζ δ ˜ τ ) . (3.261)• The dual to H ’s Lie algebra h ∗ = Ω ( Q ) × g ∗ × V ∗ . We will denote typical elements of h ∗ with the symbol µ = ( ρ, ˜ α, ˜ f ) . The coadjoint action of H on h ∗ is given byAd ∗ s µ = ( ρ, ˜ h ∗ [ ˜ α − d ˜ τ ⊗ ˜ f ] , ˜ h ∗ ˜ f ) . (3.262)The conditions for a functional A : h ∗ → R to be invariant under the coadjoint action aretherefore L δA/δ ˜ α f = 0 (3.263) L δA/δ ˜ α ˜ α + d δAδ ˜ f ⊗ ˜ f = 0 . (3.264)Note that these conditions can be regarded as first-order functional partial differential equa-tions for the Casimirs of the Lie Poisson bracket on h ∗ .• There is a right H -action on P o given by R s (˜ g, D, A, ˜ f o , ˜ χ ) = (˜ g ◦ ˜ h, D, A + d ϕ, ˜ h ∗ ˜ f o , ˜ h ∗ ( ˜ χ + ( e/c )˜ g ∗ π ∗ ϕ ) + τ ) . (3.265) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS right infinitesimal generator is given by x R ( Z ) = (˜ g ∗ ˜ ζ ◦ ˜ g, , d δϕ, L ˜ ζ ˜ f o , L ˜ ζ ˜ χ + ( e/c )˜ g ∗ π ∗ δϕ + δ ˜ τ ) . (3.266)Note that the right infinitesimal generator satisfies R ∗ s x R = ( Ad s x ) R for each s ∈ H .In order to identify the conservation laws associated with this symmetry group, we will nowconsider the invariance properties of the -form Θ P o . A straightforward calculation shows R ∗ s Θ P o = Θ P o + d c s , (3.267)where c s ( Z ) = N s X s =1 Z T Q e s c π ∗ ϕ g s ∗ f os + h s ∗ τ s f os . (3.268)Differentiating this equation in s at e ∈ H , we obtaini x R d Θ = − d (Θ( x R ) − δc ( x )) , (3.269)for each x ∈ h . Here we have introduced the mapping δc : h → C ∞ ( P o ) given by δc ( x )( Z ) = dd ǫ (cid:12)(cid:12)(cid:12)(cid:12) c exp( ǫx ) ( Z ) (3.270) = N s X s =1 Z T Q δτ s f os + 1 c D ρ (˜ g ∗ ˜ f o ) , δϕ E . (3.271) Θ( x R ) is given by Θ( x R )( Z ) = N s X s =1 Z T Q ( g ∗ s Ξ gy s ⊗ f os − d χ ⊗ f os ) · ζ s − πc h δ D, δϕ i . (3.272) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS m : P o → h ∗ associated with the action R s is given by m (˜ g, D, A, ˜ f o , ˜ χ ) = (cid:18) − πc δ D − c ρ (˜ g ∗ ˜ f o ) , [˜ g ∗ ˜Ξ gy − d ˜ χ ] ⊗ ˜ f o , − ˜ f o (cid:19) . (3.273)It is straightforward to verify that m satisfies the following equivariance property m ◦ R s = Ad ∗ s − ◦ m, (3.274)i.e. translating in P o along the H -action only changes the value of m by shifting it along a coadjointorbit in h ∗ . Thus, m : P o → h ∗ is a Poisson map when h ∗ is equipped with its Lie-Poisson bracket.We can now formally write down many of the Casimirs of the gyrokinetic Vlasov-Maxwellbracket in Eulerian labeling. Because m is a constant of motion for the AGVM system, any func-tion of m is also a constant of motion. In particular, if A : h ∗ → R is an Ad ∗ -invariant functionalon g ∗ , A ◦ m is a constant of motion. The constant of motion A ◦ m satisfies R ∗ s ( A ◦ m ) = A ◦ m by the equivariance of m and the Ad ∗ -invariance of A . Therefore A ◦ m descends to the quotient P o /H = P E , i.e. there is a functional C A : P E → R uniquely characterized by the formula π ∗ E C A = A ◦ m, (3.275)where π E : P o → P o /H = P E is the quotient map. C A is a Casimir of the Eulerian gyrokineticVlasov-Maxwell bracket [ · , · ] P E because π ∗ E [ C A , F ] P E = [ π ∗ E C A , π ∗ E F ] P o (3.276) = [ A ◦ m, π ∗ E F ] P o (3.277) = 0 . (3.278) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS The neutral Vlasov model was introduced by Tronci and Camporeal in Tronci and Camporeale(2015). Its defining equations are given by ∂ t f s + v · ∇ f s + e s m s ( E + v × B ) · ∇ v f s = 0 (3.279) ∂ t B = −∇ × E (3.280) ∇ × B = µ o X s e s Z v f s d v , X s e s Z f s d v = 0 . (3.281)We will first formulate the quasineutral model as an initial value problem without assuming ∇ · E = 0 , which is not obviously implied by the Euler-Lagrange equations. We will work interms of potentials in the temporal gauge ϕ = 0 . I will also make use of a hodge decomposition ofthe vector potential A = ∇ × α + ∇ λ, (3.282)where α satisfies ∇ · α = 0 (3.283)and λ is defined modulo the addition of constants (i.e. λ is really an equivalence class of functionswhose elements differ from one another by constant functions). Given an A , there is a unique pair α , λ that satisfy the previous three equations. An equation for α — According to the variational formulation of the neutral Vlasov modelgiven in Tronci and Camporeale (2015), one of the Euler-Lagrange equations is ρ = 0 , which HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ∇ · J = 0 . This is consistent with another one of the Euler-Lagrange equations, µ o J = ∇ × B = −∇ × ∆ α . (3.284)Thus, we come to the conclusion that α is uniquely determined by J (and therefore f ) via α = µ o G [ G [ ∇ × J ]] , (3.285)where G is the inverse of the Laplacian ( G stands for Green operator). Note that λ is not yetdetermined. An equation for ∂ t λ — If we take the time derivative of the Amp`ere equation, we obtain µ o ˙ J = −∇ × ( ∇ × E ) . (3.286)The time derivative of the current density can also be written in terms of the electromagnetic fieldusing the momentum equation ∂ t ( n s u s ) + ∇ · T s = e s n s m s ( E + u s × B ) , (3.287)where T is the stress tensor, T s = Z vv f s d v . (3.288)We have ˙ J = −∇ · Q + ǫ o ω p E + ǫ o ω p h u i × B , (3.289) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS Q = X s e s T s (3.290) ω p = X s ω ps = X s n s e s ǫ o m s (3.291) h u i = P s ω ps u s P s ω ps . (3.292)If we now equate our two expressions for the time derivative of the current density, we obtain anequation that functionally relates the electric field to the distribution function and the magneticfield, c ω p ∇ × ( ∇ × E ) + E = −h u i × B + 1 ǫ o ω p ∇ · Q . (3.293)We will now use this expression to derive an expression for ∂ t λ .Note that the ( f -dependent) linear operator E
7→ D [ E ] , where D [ E ] = c ω p ∇ × ( ∇ × E ) + E , (3.294)does not have non-zero null eigenvectors. Indeed, if E o is a null eigenvector then we have ≤ Z ω p c | E o | d x = − Z E o · ∇ × ( ∇ × E o ) d x = − Z |∇ × E o | d x ≤ , (3.295)which implies that | E o | = 0 everywhere. Thus, we can use the inverse of D to define an electricfield functional E = E ( f, B ) , where E = D − (cid:20) −h u i × B + 1 ǫ o ω p ∇ · Q (cid:21) . (3.296) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS E = − ∂ t A , thefunctional E is related to ∂ t λ according to ∇ · E ( f, B ) = − ∆ ∂ t λ. (3.297)Because B can be regarded as a functional of f according to Eq. (3.285), we can now express ∂ t λ entirely in terms of f : ∂ t λ = − G [ ∇ · E ( f, B )] . (3.298) The quasineutral model’s initial value formulation —
We can now formulate the initial valueproblem for the quasineutral model. The state of the system is determined by the pair ( f, λ ) , where f is the multi-species distribution function and λ appears in the Hodge decomposition of the vectorpotential as above. The time evolution equation for ( f, λ ) is given by ∂ t f s = − v · ∇ f s − e s m s ( E + v × B ) · ∇ v f s (3.299) ∂ t λ = − G [ ∇ · E ( f, B )] , (3.300)where B = − µ o G [ ∇ × J ] (3.301)is expressed in terms of f . Note that the dynamics of f decouple from the dynamics of λ . Remarks —
This system of evolution equations admits solutions that do not satisfy the Amp`ereequation and the neutrality condition. However, using the definition of E , it is straightforward toverify that ∇ · J is a constant of motion. Thus, if we choose initial conditions with ∇ · J = 0 , thenthis identity will hold for all subsequent times, and, as a consequence, the Amp`ere equation will be HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ∇ · J = 0 , we demand that our initialcondition satisfies ρ = 0 , then ρ will be zero for all times. Indeed, ∂ t ρ = −∇ · J , (3.302)which implies that ρ is constant along solutions that initially satisfy ∇ · J = 0 . So far we have deduced that the phase space for the neutral Vlasov model is given by the space ofpairs ( f, λ ) subject to the holonomic constraint ∇ · J = 0 . Because we will need a slightly moreprecise notation in the discussion that follows, we will write the constraint as ∇ · J ( X o , f ) = 0 ,where the functional J ( V, f ) depends on the multi-species velocity field V and the multi-speciesdistribution function f according to J ( V, f ) = X s e s Z ˙ x s f s d v , (3.303)where ˙ x s = ˙ x s ( x , v ) is the spatial component of the phase space velocity V s = ( ˙ x s , ˙ v s ) . Thevector field X o is given by ˙ x s = v and ˙ v s = 0 , which implies J ( X o , f ) = X s e s Z v f s d v , (3.304)as expected. HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS L H ( g, F o , χ, λ, ˙ g, ˙ F o , ˙ χ, ˙ λ ) = Θ( V, ˙ F o , ˙ χ, ˙ λ ) − H ( g, F o , χ, λ ) (3.305)Here g is a multi-species phase space fluid configuration, i.e. an element of Diff ( T Q ) ; F o is amulti-species reference phase space density related to the reference distribution function f o by F os = f os Ω , where Ω = d x ∧ d y ∧ d z ∧ d v x ∧ d v y ∧ d v z ; χ is a multi-species function onphase space; and λ is a function on Q modulo constant functions. The multi-species phase spacefluid velocity is related to ˙ g by V = ˙ g ◦ g − . The quantities g, F o are constrained to satisfy ∇ · J ( X o , F ) = 0 , where F s = g s ∗ F os . The Hamiltonian functional is given by H ( g, F o , χ, λ ) = X s h K s , F s i + 12 µ o h∇ × A T ( J o ) , ∇ × A T ( J o ) i (3.306)where, J o = J ( X o , F ) , A T ( J o ) is the transverse vector potential given by A T ( J o ) = µ o G [ G [ ∇ × ( ∇ × J o )]] , (3.307)and K s = m s v · v is the single-particle kinetic energy. The Lagrange -form is given by Θ( V, ˙ F o , ˙ χ, ˙ λ ) = X s h ϑ s ( V s ) , F s i + X s h ˙ χ s , F os i . (3.308)Here the singe-particle Lagrange -form ϑ s is given by ϑ s = ( m s v + e s [ A T ( J o ) + ∇ λ ]) · d x . (3.309) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS X H with Hamiltonian H is determined by varying the action S H = R t t L H dt . More generally, the Hamiltonian vector field X F with Hamiltonian F is deter-mined by varying the action S F = R t t L F dt . The Poisson bracket [ F , G ] on the augmented phasespace (i.e. ( g, F o , χ, λ ) -space) is given by [ F , G ] = d F ( X G ) . (3.310)In order to derive the bracket, we will first find an expression for the general Hamiltonian vectorfield X F and then deduce the bracket from the previous formula.We will find X F by manipulating the Euler-Lagrange equations associated with the Lagrangian L F . In order to derive the Euler-Lagrange equations associated with the phase space Lagrangian L F , it is necessary to account for the constraint ∇· J ( X o , F ) = 0 . This holonomic constraint placesa an awkward constraint on the variations of the phase space variables ( g, F o , χ, λ ) . I have found itvery convenient to work with unconstrained variations and a Lagrange multiplier. That is, insteadof varying the action S F = R t t L F dt while respecting the divergence-free current constraint, it iseasier to vary S ′F = Z t t ( L F + h ψ, ∇ · J o i ) dt (3.311)using unconstrained variations. Here the Lagrange multiplier ψ is a time-dependent function on Q modulo constant functions.The Euler-Lagrange equations associated with the action S ′F are given as follows. Varying g s — Varying the multi-species phase space fluid configuration gives ddt ϑ s + i V s d ϑ s + e s d ( v · ∇ ψ − v · A T ( J )) + g s ∗ ˙ F os F os ! ϑ s + δ g s F = 0 . (3.312) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS J = J ( V, F ) (note that J = J o !) and δ g s F is a -form on the single-particle phase spacethat can be thought of as the functional derivative of F with respect to g s . To be precise, ddǫ (cid:12)(cid:12)(cid:12)(cid:12) F ( g sǫ ) = h δ g s F ( ξ s ) , F s i , (3.313)where ξ s = δg s ◦ g − s . In deriving the expression (3.312), it is useful to make use of the self-adjointproperty of the transverse vector potential, h J , A T ( J ) i = h A T ( J ) , J i , (3.314)where J , J are arbitrary vector fields on Q . It is also useful to note that 3.312 implies that thevelocity V s can be decomposed as V s = ν F s + P s ( e s ˙ A F · d x ) , (3.315)where P s is the Poisson tensor associated with the symplectic form ω s = − d ϑ s and ν F s = P s ( δ g s F + e s d [ v · A o F ] − e s A T ( J F χ ) − g s ∗ (cid:18) δ F /δχ s F os (cid:19) ϑ s ) (3.316) Varying λ — Varying the scalar λ gives ∇ · J + δ F δλ = 0 . (3.317)Note that δ F /δλ lives in the dual to the space of scalar functions modulo constant functions,which is precisely the space of functions on Q with vanishing integral. HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS Varying F os — Varying the reference phase space density F os gives ˙ χ s + g ∗ s [ ϑ s ( V s ) + e s v · ( A T ( J ) − ∇ ψ )] − δ F δF os = 0 . (3.318)This equation implies that the phase space function χ s behaves very much in the same way as thephase of a quantum wave in the WKB approximation. Varying χ s — Varying the phase-like function χ s gives ˙ F os + δ F δχ s = 0 . (3.319) Varying ψ — As expected, varying the Lagrange multiplier ψ gives ∇ · J o = 0 . (3.320)Taken together, these Euler-Lagrange equations comprise a puzzle that must be solved in or-der to find the components of the Hamiltonian vector field X F . Some parts of the puzzle aresimple. For instance, the time derivative of F os is given immediately by Eq. (3.319). However,Eq. (3.312) represents a much more serious challenge. There the velocity field V s is woveninto various terms in a rather intricate manner. Complicating matters further is the fact that theLagrange multiplier ψ must somehow be eliminated from the equations. HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ˙ A F = ∇ ˙ λ − A T [ J ( X o , L V F )] (3.321) A o F = ∇ ψ − A T [ J ( V, F )] , (3.322)in terms of functional derivatives of F . In principle, this task is not so difficult. However, obtainingthe desired expressions in a useful form is a subtle enterprise. It turns out that a particularly usefulway of writing the expressions is ˙ A F = [ n − (cid:20) µ o J F ν + µ o µ [ J F χ ] ω p /c (cid:21) − µ o G Π T J F χ (3.323) A o F = [ n − (cid:20) µ o J F + µ o ∇ G (cid:2) δ F δλ (cid:3) ω p /c (cid:21) . (3.324)Here the various current densities are given by J F = J ( P ( δ g F ) , F ) (3.325) J F ν = J ( ν F , F ) (3.326) J F χ = J ( X o , g ∗ ( δ F /δχ )) (3.327)and the f -dependent linear operators n, µ are given by n = −D − (cid:18) c ω p ∆Π T (cid:19) (3.328) µ = 1 − ω p c G Π T . (3.329) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS n ( c /ω p ) is self-adjoint. The self-adjoint property can be verified by first noting D † = ω p c D c ω p , which implies (cid:18) n c ω p (cid:19) † = − (cid:18) c ω p ∆Π T (cid:19) D − c ω p . (3.330)Then one uses the fact that c ω p ∆Π T commutes with the operator D = 1 − c ω p ∆Π T to conclude that ( n c ω p ) † = n c ω p . It is also useful to be aware of the identity n − −D − (cid:18) c ω p ∆Π T + D (cid:19) = −D − . (3.331)With equations (3.323) and (3.324) in hand, we can now express the components of the Hamil-tonian vector field X F as V s = ν F s + P s ( e s ˙ A F · d x ) (3.332) ˙ F os = − δ F δχ s (3.333) ˙ χ s = δ F δF os − g ∗ s (cid:0) ϑ s ( ν F s ) − e s v · A o F (cid:1) (3.334) ˙ λ = G [ ∇ · ˙ A F ] . (3.335) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS [ G , F ] = L X F G , we find the following ex-pression for the Poisson bracket, [ G , F ] = X s h ω s ( ν G s , ν F s ) , F s i + X s h δ G δχ s , δ F δF os i − h δ G δF os , δ F δχ s i− (cid:28) µ o µ ( J G χ ) (cid:12)(cid:12)(cid:12)(cid:12) D − c ω p (cid:12)(cid:12)(cid:12)(cid:12) J F + ∇ G [ δ F /δλ ] (cid:29) + (cid:28) µ o µ ( J F χ ) (cid:12)(cid:12)(cid:12)(cid:12) D − c ω p (cid:12)(cid:12)(cid:12)(cid:12) J G + ∇ G [ δ G /δλ ] (cid:29) . (3.336)This Poisson bracket is defined on a space larger than the phase space we are actually interestedin. The “unimportant” variables are the χ s and the g s . We can find the bracket on ( F, λ ) -spaceusing Poisson reduction. First we will reduce by the abelian group P s C ∞ ( T Q ) , which acts bytranslation, χ s χ s + δχ s . It is simple to verify that both the bracket and Hamiltonian givenearlier are invariant under this group action. The first reduced bracket is therefore [ G , F ] r = X s h ω s ( ν G s , ν F s ) , F s i , (3.337)where F , G are functionals of ( g, F o , λ ) and ν F s = P s ( δ g s F + e s d [ v · A o F ]) . (3.338)Next we will reduce by the product of diffeomorphism groups Π s Diff ( T Q ) which acts (on theright) according to ( g s , F os ) ( g s ◦ h s , h ∗ s F os ) . Again, the bracket and Hamiltonian given earlierare invariant under this group action. The second reduced bracket is [ G , F ] r = X s h{G F s + e s v · A o G , F F s + e s v · A o F } , F s i , (3.339) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS A o F = −D − c ω p (cid:18) µ o J ( X F F , F ) + µ o ∇ G [ δ F /δλ ] (cid:19) , (3.340)and F F = δ F /δF . Here, F , G are functionals on ( F, λ ) -space. Force-free electrodynamics is a single-fluid model sometimes used in astrophysics to describeextremely-magnetized relativistic plasmas. As discussed in Gralla and Jacobsen (2014), the defin-ing equations (which are Lorentz invariant) are given by ∂ t B = − c ∇ × E (3.341) ∂ t E = c ∇ × B − π J (3.342) J = c πB (cid:20) ( ∇ · E ) E × B + ( B · ∇ × B − E · ∇ × E ) B (cid:21) (3.343) E · B = 0 . (3.344) For the sake of deriving the FFE Poisson bracket by reduction of a non-degenerate bracket, it isuseful to formulate a phase space variational principle on an augmented phase space P + . The space P + is the space of all ordered lists ( E, A, g, ρ o , χ ) where• E is the electric field -form• A is the vector potential -form• g is a diffeomorphism of R that represents the fluid configuration HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ρ o is the reference charge density -form• χ is an auxiliary scalar field,and E and A are constrained to satisfy E ∧ d A = 0 , (3.345)which guarantees that the electric and magnetic fields are perpendicular. The Lagrange -form onthis space is given by Θ( ˙ E, ˙ A, ˙ g, ˙ ρ o , ˙ χ ) = h A ⊗ ρ, u/c i − π ( E, ˙ A/c ) + h ρ o , ˙ χ i . (3.346)Here u = ˙ g ◦ g − , angle brackets denote a “natural pairing”, ρ = g ∗ ρ o , and parentheses denote thestandard inner product of differential forms on R . The Hamiltonian functional is given by H ( E, A, g, ρ o , χ ) = 18 π ( E, E ) + 18 π ( d A, d A ) . (3.347)The phase space Lagrangian is given by L H = Θ( ˙ E, ˙ A, ˙ g, ˙ ρ o , ˙ χ ) − H ( E, A, g, ρ o , χ ) . (3.348) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS L F , which is given by δL F = (cid:18) ∗ [ ι u/c ρ ] + 14 π ˙ E/c + 14 π δ ∗ ( b · ˙ A/c ) E | B | − B ! , δA (cid:19) + (cid:18) − π ˙ A/c − π E, δE ⊥ (cid:19) + (cid:28) [ − ˙ A/c − ι u/c B − ( A/c ) g ∗ ( ˙ ρ o /ρ o )] ⊗ ρ, ξ (cid:29) + h g ∗ ( A ( u/c )) + ˙ χ, δρ o i + h− ˙ ρ o , δχ i + ddt Θ( δE, δA, δg, δρ o , δχ ) . (3.349)Here B = d A , | B | = p ∗ ( B ∧ ∗ B ) , b = ∗ B/ | B | , the dot product between two k -forms α, β isgiven by α · β = ∗ ( α ∧ ∗ β ) , (3.350)and if α is a -form, α ⊥ = α − ( α · b ) b . Note that δE is constrained to satisfy δE · b = − E · ∗ δB | B | , (3.351) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS E is completely determined by the variation of B . By setting thevariation of L H equal to zero, the following equations of motion emerge: ˙ ρ o = 0 (3.352) ˙ χ = − g ∗ ( A ( u/c )) (3.353) ˙ A/c = − E (3.354) δ B = ∗ [ ι u/c ρ ] + 14 π ˙ E/c (3.355) E = ι u/c B, (3.356)which are equivalent to the equations of force free electrodynamics provided π ∗ ρ = − δ E at t = 0 (note that π ∗ ρ + δ E is a constant of motion). Symmetry under translations in χ Consider the abelian group G o = C ∞ ( R ) , whose typical element we will denote τ . G o acts onthe augmented FFE phase space according to χ χ + τ . The phase space Lagrangian is invariantunder this symmetry, which implies that Θ(0 , , , , τ ) = h ρ o , τ i is a constant of motion for each τ ∈ G o . Because τ is an arbitrary smooth function, ρ o must be independent of time. HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS Consider again the abelian group G o . Let s ∈ G o be a typical element. This group acts on P + in asecond way, namely A A + d s (3.357) χ χ − c g ∗ s. (3.358)Note that this transformation preserves the constraint E ∧ d A = 0 . The phase space Lagrangian isinvariant under this symmetry which implies that µ s = Θ(0 , d s, , , − g ∗ s/c ) = − πc ( δ E + 4 π ∗ ρ, s ) (3.359)is a constant of motion for each s ∈ G o . Thus, δ E + 4 π ∗ ρ is a constant of motion. Symmetry under particle relabling
Let G = Diff ( R ) be the non-abelian group of diffeomorphisms of R with typical element h ∈ G . There is a (right) G action on P + given by g g ◦ h (3.360) ρ o h ∗ ρ o (3.361) χ h ∗ χ. (3.362)The phase space Lagrangian is invariant this symmetry. Thus, the contraction of the Lagrange -form with the infinitesimal generator of this symmetry is a constant of motion. The infinitesimal HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ξ P + = ( δE, δA, δg, δρ, δχ ) = (0 , , T g ◦ ξ, L ξ ρ, L ξ χ ) , (3.363)where ξ is an arbitrary vector field. This implies that µ ξ = Θ( ξ P + ) = h ( g ∗ A/c + d χ ) ⊗ ρ o , ξ ) i (3.364)is a constant of motion for each ξ . Because ξ is arbitrary and ρ o is a constant of motion, the quantity A o = g ∗ A + d cχ is a constant of motion. To understand this conservation law, consider the exteriorderivative of A o , d A o = g ∗ B. (3.365)By applying the pushforward by g to each side of this expression, and noting that A o is a constantof motion, we conclude that the magnetic field -form is advected by the fluid velocity u , B = g ∗ d A o ⇒ ˙ B = − L u B. (3.366)But this implies that the vector potential A is advected modulo an exact -form, A = g ∗ A o + d ψ. (3.367)Thus, a second conclusion we can draw about the conservation of µ ξ is that d cχ can be interpretedas the difference between A in the temporal gauge and A in the “advection gauge”, wherein A isadvected as a -form. HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS Symmetry under isometries of configuration space
Let G be the non-abelian group of isometries of R with typical element R . G is naturally amatrix group, and so it is equipped with a natural (left) action on R , x R x . (3.368)Given an element ν ∈ g we can therefore define an infinitesimal generator on R , ν Q = ddǫ (cid:12)(cid:12)(cid:12)(cid:12) exp( ǫν ) , (3.369)where exp denotes the matrix exponential. The left action of G on R lifts to a left action on P + given by A R ∗ A (3.370) E R ∗ E (3.371) g R ◦ g. (3.372)The infinitesimal generator of this action is given by ν P + = ( δE, δA, δg, δρ, δχ ) = ( − L ν Q E, − L ν Q A, ν Q ◦ g, , . (3.373)The phase space Lagrangian is invariant under this symmetry. Therefore µ ν = Θ( ν P + ) = 14 π ( A ( ν Q /c ) , π ∗ ρ + δ E ) + 14 π ( E, ι ν Q B ) (3.374) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ν ∈ g . This conservation law is equivalent to the global conser-vation of linear and angular momentum. Provided initial conditions are chosen so that the Gaussequation is satisfied, µ ν is equal to the total momentum of the electromagnetic field. Combining the symmetry groups G o , G o , and G . There is a “big group” that encodes three of the symmetries discussed so far: phase translation,gauge transformation, and particle relabling. The big group will be denoted G = G o × ( G o ⋊ G ) ,which is suggestive of the fact that H is a direct product of the abelian gauge group G o with thenon-ablian semidirect product G o ⋊ G . The identity element is e = (0 , , id ) ∈ G o × G o × G .The group product is given by ( s , τ , h ) ∗ ( s , τ , h ) = ( s + s , h ∗ τ + τ , h ◦ h ) , (3.375)which can be seen to be associative by direct calculation. The group inverse is given by ( s, τ, h ) − = ( − s, − h ∗ τ, h − ) . (3.376)While the structure of this group may appear a bit mysterious, it can be uncovered in a straight-forward way as follows. By an abuse of notation, denote the right actions of G o , G o , and G by R s , R τ , and R h , respectively. We have R s ( E, A, g, ρ o , χ ) = ( E, A + d s, g, ρ o , χ − g ∗ s/c ) (3.377) R τ ( E, A, g, ρ o , χ ) = ( E, A, g, ρ o , χ + τ ) (3.378) R h ( E, A, g, ρ o , χ ) = ( E, A, g ◦ h, h ∗ ρ o , h ∗ χ ) . (3.379) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS R s ◦ R τ = R τ ◦ R s (3.380) R s ◦ R h = R h ◦ R s (3.381) R h ◦ R τ = R h ∗ τ ◦ R h . (3.382)Therefore we have the identity ( R s ◦ R τ ◦ R h ) ◦ ( R s ◦ R τ ◦ R h ) = R s + s ◦ R h ∗ τ + τ ◦ R h ◦ h . (3.383)This identity strongly suggests defining the group product given above. Morevoer, by combiningthis identity with the fact that the product (3.375) does indeed satisfy the group axioms, we canquickly deduce the appropriate right action of G on P + , namely R ( s,τ,h ) = R s ◦ R τ ◦ R h . (3.384)The identity (3.383) can be rewritten as R ( s ,τ ,h ) ◦ R ( s ,τ ,h ) = R ( s ,τ ,h ) ∗ ( s ,τ ,h ) , (3.385)which is the most non-trivial property satisfied by a right G action. The other properties of a rightaction follow immediately from the definition (3.384) and the commutation relations. The quotient space P = P + / G The Poisson bracket, {· , ·} P + , on P + given by inverting the symplectic form − d Θ will automat-ically be symmetric under the group G because Θ has the same symmetry. Therefore there is a reduced Poisson bracket on P = P + / G that is given formally as follows. Let π : P + → P HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS P + to its orbit under the action of G . Given a pairof functionals F, G : P → R , we can pull them back to P + , thereby obtaining the functionals π ∗ F, π ∗ G : P + → R . Each of these functionals on P + is automatically G -invariant, which impliesthat the functional { π ∗ F, π ∗ G } P + is also G -invariant. Because π is surjective, there is therefore aunique functional, { F, G } P : P → R , that satisfies π ∗ { F, G } P = { π ∗ F, π ∗ G } P + . (3.386)The formula (3.386) defines the Poisson bracket on P . The Jacobi and Leibniz identities arestraightforward to check.Because P represents a “physical” phase space for FFE, the bracket on P is in some ways moredesireable than the bracket on P + . It is therefore useful to have a concrete model of the space P .The purpose of this subsection is to supply this model.It will turn out that P can be represented as the space of triples ( E, B, ρ ) , where E is theelectric field -form, B is the (exact) magnetic field -form, ρ is the charge density -form, and theelectric and magnetic fields are constrained to satisfy E ∧ B = 0 . To see this, first note that thereis a surjective map π : P + → P given by π ( E, A, g, ρ o , χ ) = ( E, d A, g ∗ ρ o ) . (3.387)Next suppose ( E, A, g, ρ o , χ ) and ( E ′ , A ′ , g ′ , ρ ′ o , χ ′ ) each map to ( E, B, ρ ) under π . Immediatelywe see that E ′ = E and d A ′ = d A , which implies that there is some s ∈ G o such that A ′ = A + d s . If we now set h = g − ◦ g ′ and τ = χ ′ − h ∗ χ + h ∗ g ∗ s/c , a simple calculation shows that ( E ′ , A ′ , g ′ , ρ ′ o , χ ′ ) = R ( s,τ,h ) ( E, A, g, ρ o , χ ) . This shows that points in π − ( E, B, ρ ) are all on thesame G -orbit. Because it is also true that π ◦ R ( s,τ,h ) = π for each ( s, τ, h ) ∈ G , we must therefore HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS P along π are precisely the G -orbits. This proves that P + / G is diffeomorphic to P . P In order to compute the Poisson bracket on P , we will compute { π ∗ F, π ∗ G } P + , where F, G : P → R are functionals on P . By the definition of a Hamiltonian vector field, we have { π ∗ F, π ∗ G } P + = ( d π ∗ F )( X π ∗ G ) . (3.388)In order to evaluate the right hand side of Eq. (3.388), we must compute the components of theHamiltonian vector field X π ∗ G = ( ˙ E G , ˙ A G , ˙ g G , ˙ ρ oG , ˙ χ G ) . This computation is most easily done byanalyzing the Euler-Lagrange equations associated with the phase space Lagrangian L G ( E, A, g, ρ o , χ, ˙ E, ˙ A, ˙ g, ˙ ρ o , ˙ χ ) = Θ( ˙ A, ˙ g, ˙ ρ o , ˙ χ ) − π ∗ G ( E, A, g, ρ o , χ ) . (3.389)The first variation of L G is given by δL G = (cid:18) ∗ [ ι u/c ρ ] + 14 π ˙ E/c + 14 π δ ∗ ( b · [ ˙ A/c + 4 πδG/δE ]) E | B | − π δGδB ! , δA (cid:19) + (cid:18) − π ˙ A/c − δGδE , δE ⊥ (cid:19) + (cid:28) [ − ˙ A/c − ι u/c B − ( A/c ) g ∗ ( ˙ ρ o /ρ o ) − d δG/δρ ] ⊗ ρ, ξ (cid:29) + h g ∗ ( A ( u/c )) + ˙ χ − g ∗ δG/δρ, δρ o i + h− ˙ ρ o , δχ i + ddt Θ( δE, δA, δg, δρ o , δχ ) . (3.390) HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS X π ∗ G are given by ˙ ρ o = 0 (3.391) c ˙ A G = − π α G ⊥ − π d δGδρ (3.392) c j G = 4 π ∗ ρ | B | ∗ ( α G ∧ b ) − (cid:18) E | B | · δ ( ∗ α G ⊥ ) + b · δ (cid:18) α G k ∗ E | B | − δGδB (cid:19)(cid:19) b (3.393) c ˙ E = − π ∗ ρ | B | ∗ (4 π α G ∧ b ) − π δ (cid:18) α G k ∗ E | B | − δGδB (cid:19) ⊥ + (cid:18) E | B | · δ ( ∗ α G ⊥ ) (cid:19) b (3.394) ˙ χ G = g ∗ (cid:18) δGδρ − A ( u G /c ) (cid:19) , (3.395)where α G = δG/δE − (1 / π ) d δG/δρ and j G = ∗ ι u G ρ is the current density -form.After some tedious but straightforward calculations involving substituting the previous expres-sions into the identity (3.388), we find that the Poisson bracket on P is given by { F, G } P = − πc (cid:18) (4 π ∗ ρ ) B | B | , α F ∧ α G (cid:19) + 4 πc (cid:18) α F ⊥ , δ (cid:18) δGδB − α G k ∗ e (cid:19)(cid:19) − πc (cid:18) α G ⊥ , δ (cid:18) δFδB − α F k ∗ e (cid:19)(cid:19) , (3.396)where e = E/ | B | .It is not difficult to show that any functional of δ E + 4 π ∗ ρ is a Casimir of the bracket {· , ·} P .Therefore the submanifold of P defined by δ E + 4 π ∗ ρ = 0 is a Poisson submanifold that canbe parameterized by the space P o of pairs ( E, B ) that satisfy E ∧ B = 0 . Being a Poissonsubmanifold, P o has a bracket {· , ·} P o that is naturally induced by {· , ·} P . The expression for this HAPTER
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OUNDARY TERMS AND P OISSON BRACKETS ( E, B ) -space is given by { F, G } P o =4 πc (cid:18) ( δ E ) B | B | , δFδE ∧ δGδE (cid:19) +4 πc (cid:18) δFδE ⊥ , δ (cid:18) δGδB − δGδE k ∗ e (cid:19)(cid:19) − πc (cid:18) δGδE ⊥ , δ (cid:18) δFδB − δFδE k ∗ e (cid:19)(cid:19) . (3.397) hapter 4Energetically-consistent gyrokineticcollision operator One of the greatest unsolved problems in the theory of magnetically-confined plasmas is un-derstanding and controlling the turbulent flux of particles and heat into a fusion reactor’swall Kikuchi and Azumi (2012). It is believed that the predominant cause of these fluxes islow-frequency fluctuating electromagnetic fields with wavelengths on the order of the gyroradius.While a collisionless gyrokinetic model of these fluctuating fields has been developed that is fullyconsistent with the First Law of Thermodynamics (for a recent review see Brizard and Hahm(2007)), this energetically-consistent model has the serious flaw of ignoring collisions altogether.In order to accurately describe irreversible plasma transport processes, the effects of colli-sions must be incorporated into gyrokinetic theory. Previous work on linear gyrokinetic collisionoperators Abel et al. (2008); Li and Ernst (2011); Madsen (2013a) assumed a strict two-scale sepa-ration between a large-scale equilibrium distribution function F o and a small-scale fluctuating part δF = F − F o . Conservation properties of the collision operator in Abel et al. (2008), for example,104 HAPTER
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NERGETICALLY - CONSISTENT GYROKINETIC COLLISION OPERATOR F approach that do not make this split, and that can thus investigate morecompletely the possible effects of finite ǫ = ρ i /L in experiments, such as corrections to gyroBohmscaling and non-local turbulence spreading (see footnote 5 on p. 427 in Brizard and Hahm (2007).)When finite- ǫ effects are accounted for, preserving exact conservation properties, and thereforeensuring consistency with the First Law of Thermodynamics, is a nontrivial unsolved problem.The collision operators in Abel et al. (2008); Li and Ernst (2011), for example, were obtained bytransforming a particle-space collision operator with exact conservation properties into the lowest-order guiding center coordinates. While this approach guarantees the existence of energy andmomentum-like quantities that annihilate the collision operator, these same quantities are not con-served by the full- F collisionless gyrokinetic system, and therefore fail to be conserved by thefull- F collisional system. More generally, existing gyrokinetic collision operators are not ener-getically consistent in a full- F formalism because: (a) the gyrocenter coordinate transformation,and therefore any collision operator transformed into gyrocenter coordinates, is only known as anasymptotic expansion in the gyrokinetic ordering parameter ǫ ; and (b) replacing the asymptoticexpansion of such an operator with a truncated power series destroys exact conservation laws. Thepurpose of this Chapter is to present the first collisional formulation of global full- F gyrokineticswith exact conservation laws. For the sake of simplicity, our discussion will focus on quasi-neutral electrostatic gyrokinetics (forinstance, see Parra and Calvo (2011)). However, the ideas behind our discussion apply equally-well to electromagnetic gyrokinetics (for example, see Sugama (2000).) Our primary result con-sists of an expression for the non-linear Landau operator in gyrocenter coordinates that is cor-rected by small terms to ensure exact energy and momentum conservation [see Eq. (4.24).] These
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NERGETICALLY - CONSISTENT GYROKINETIC COLLISION OPERATOR B ∗k -denominators in the Hamiltonian guiding center theoryintroduced by Littlejohn Littlejohn (1981); they do not increase the theory’s order of accuracy, butthey are essential to include for the sake of ensuring exact energy and momentum conservation.As a first step, we review how the energy conservation law is discussed in collisionless kinetictheory. The governing equations of collisionless electrostatic kinetic theory are the Vlasov-Poissonequations, ∂ t f s + { f s , H s } = 0 (4.1) ∆ ϕ = − πρ ( f ) , (4.2)where f s is the species- s distribution function, ϕ is the electrostatic potential, ρ ( f ) is the chargedensity, H s = p / m s + e s ϕ , and {· , ·} is the standard canonical Poisson bracket. Equations(4.1)-(4.2) conserve the total energy E = X s Z p m s f s dz + (cid:28) ϕ, ρ ( f ) + 18 π ∆ ϕ (cid:29) , (4.3)where h· , ·i denotes the standard L -pairing of functions on configuration space and dz = d x d p .Because binary collisions conserve energy, Eq. (4.3) must also be conserved in collisional kinetictheory. In particular, if the Vlasov-Poisson equations are modified by the addition of a bilinearcollision operator, ∂ t f s + { f s , H s } = X ¯ s C s ¯ s ( f s , f ¯ s ) (4.4) ∆ ϕ = − πρ ( f ) , (4.5) HAPTER
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NERGETICALLY - CONSISTENT GYROKINETIC COLLISION OPERATOR C s ¯ s must be chosen to satisfy the condition d E dt = X s Z H s ∂ t f s dz + (cid:28) ∂ t ϕ, ρ ( f ) + 14 π ∆ ϕ (cid:29) = X s, ¯ s Z H s C s ¯ s ( f s , f ¯ s ) dz. (4.6)Because this identity must hold for an arbitrary multi-species distribution function, the collisionoperator therefore has to satisfy the well-known identities Z H s C s ¯ s ( f s , f ¯ s ) dz + Z H ¯ s C ¯ ss ( f ¯ s , f s ) d ¯ z = 0 , (4.7)which express the fact that the energy gained by species s due to collisions with species ¯ s isprecisely the energy lost by species ¯ s due to collisions with species s . The non-linear Landauoperator (summation rule is implied), C s ¯ s ( f s , f ¯ s ) = − Γ s ¯ s { x i , γ s ¯ si } , (4.8)satisfies the identities (4.7), and therefore defines an energetically-consistent collisional kinetictheory. Here Γ s ¯ s = 4 πe s e s ln Λ ; the -component vector γ s ¯ s is γ s ¯ si ( z ) = Z δ ( x − ¯ x ) Q s ¯ s ( z, ¯ z ) A s ¯ s ( z, ¯ z ) d ¯ z ; (4.9)the × matrix Q s ¯ s is given by Q s ¯ s ( z, ¯ z ) = 1 W s ¯ s ( z, ¯ z ) P [ W s ¯ s ( z, ¯ z )] , (4.10) HAPTER
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NERGETICALLY - CONSISTENT GYROKINETIC COLLISION OPERATOR P ( ξ ) ≡ I − ˆ ξ ˆ ξ is the orthogonal projection onto the plane perpendicular to the vector ξ ; thevelocity difference W s ¯ s is given by W s ¯ s ( z, ¯ z ) = { x , H s } ( z ) − { x , H ¯ s } (¯ z ); (4.11)and the vector A s ¯ s ( z, ¯ z ) = f s ( z ) { x , f ¯ s } (¯ z ) − { x , f s } ( z ) f ¯ s (¯ z ) . (4.12)When comparing this form of the Landau operator to more conventional expressions, it is usefulto note that { x , g } = ∂ p g , where g is any function on phase space, so that the collision operator(4.8) describes collisions in momentum space. Moreover, the identities (4.7) follow immediatelyfrom the fact that the velocity difference W s ¯ s is a null-eigenvector of the matrix Q s ¯ s . In order to apply this same argument to gyrokinetic theory, we start with the gyrokinetic Vlasov-Poisson system ∂ t F s + { F s , H gy s } gc s = 0 (4.13) ∇ · P = ρ ( F ) . (4.14)Here, F s is the gyrocenter distribution function; ϕ is the electrostatic potential; {· , ·} gc s is the guid-ing center Poisson bracket; H gy s = H gc s + e s h ψ i + e s h{ ˜ ψ, ˜Ψ } gc s i ≡ K s ( E ) + e s ϕ (4.15) HAPTER
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NERGETICALLY - CONSISTENT GYROKINETIC COLLISION OPERATOR ψ ( z ) = ϕ ( X + ρ os ) , where ρ os is the lowest-order guiding-centergyroradius; h·i denotes the gyroaverage; ˜Ψ denotes the gyroangle antiderivative of ˜ ψ ≡ ψ − h ψ i ; K s ( E ) is the gyrocenter kinetic energy; P = − δ K /δ E is the gyrocenter polarization den-sity; K = P s R F s K s ( E ) dz gc s ; and dz gc s denotes the guiding center Liouville volume element.These equations govern collisionless quasineutral electrostatic gyrokinetic theory in the “high-flow” regime (see Krommes and Hammett (2013) and references therein) and they conserve thetotal energy, E gy = X s Z F s H gy s dz gc s , (4.16)exactly. Note that the quasineutrality equation (4.14) implies that this system governs plasma dy-namics on time scales long compared to the period of plasma oscillations.The equations governing collisional gyrokinetic theory are given by adding a bilinear collisionoperator to the gyrokinetic Vlasov-Poisson equations, ∂ t F s + { F s , H gy s } gc s = X ¯ s C gy s ¯ s ( F s , F ¯ s ) (4.17) ∇ · P = ρ ( F ) . (4.18)Because the conservation laws of ordinary collisional kinetic theory are consistent with those ofcollisionless kinetic theory, the gyrokinetic collision operator C gy s ¯ s must not alter the conservationof E gy . Thus, d E gy dt = X s Z H gy s ∂ t F s dz gc s + D ρ ( F ) − ∇ · P , ∂ t ϕ E = X s, ¯ s Z H gy s C gy s ¯ s ( F s , F ¯ s ) dz gc s . (4.19) HAPTER
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NERGETICALLY - CONSISTENT GYROKINETIC COLLISION OPERATOR Z H gy s C gy s ¯ s ( F s , F ¯ s ) dz gc s + Z H gy ¯ s C gy ¯ ss ( F ¯ s , F s ) d ¯ z gc ¯ s = 0 , (4.20)which is the gyrokinetic version of Eq. (4.7). The identities (4.20) must be satisfied exactly by anyenergetically-consistent gyrokinetic collision operator. While Eq. (4.20) imposes important qualitative constraints, they cannot determine the form of thegyrokinetic collision operator by themselves. A quantitative constraint is necessary as well. Tothis end, it is important that the gyrokinetic collision operator agrees with the the transformation ofthe particle-space Landau operator into gyrocenter coordinates, at least up to some desired orderin the gyrokinetic ordering parameter ǫ . Is it possible to satisfy these qualitative and quantitativeconstraints simultaneously? The answer is “yes”.We have discovered an accurate gyrokinetic collision operator that is consistent with the con-servation laws of collisionless gyrokinetic theory, and therefore the first law of thermodynamics.The form of the operator is suggested by the somewhat-peculiar presentation of the particle-spaceLandau operator given earlier. Let y s = X + ρ os and define the gyrocenter velocity difference W gy s ¯ s ( z, ¯ z ) = { y s , H gy s } gc s ( z ) − { y ¯ s , H gy ¯ s } gc ¯ s (¯ z ) , (4.21) Necessary conditions for the use of the Landau operator are ω c < ω p and ( ∂ t F ) / ( ω p F ) < . When theseconditions are not satisfied, our discussion must be modified. HAPTER
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NERGETICALLY - CONSISTENT GYROKINETIC COLLISION OPERATOR × matrix Q s ¯ s gy ( z, ¯ z ) = 1 W gy s ¯ s ( z, ¯ z ) P [ W gy s ¯ s ( z, ¯ z )] , (4.22)and the vector A gy s ¯ s ( z, ¯ z ) = F s ( z ) { y ¯ s , F ¯ s } gc ¯ s (¯ z ) − { y s , F s } gc s ( z ) F ¯ s (¯ z ) . (4.23)The energetically-consistent gyrokinetic Landau operator is given by C gy s ¯ s ( F s , F ¯ s ) = − Γ s ¯ s { y s i , γ s ¯ s gy i } gc s , (4.24)where γ s ¯ s gy ( z ) = Z δ gy s ¯ s ( z, ¯ z ) Q s ¯ s gy ( z, ¯ z ) A gy s ¯ s ( z, ¯ z ) d ¯ z gc ¯ s , (4.25)and δ gy s ¯ s ( z, ¯ z ) = δ ( y s ( z ) − y ¯ s (¯ z )) . Note that this operator depends explicitly on the electric fieldthrough the gyrocenter Hamiltonians that appear in Eq. (4.21). Using a straightforward, but tediousargument that is not reproduced here, we have shown that this operator agrees with the Landauoperator transformed into gyrocenter coordinates with leading-order accuracy.Because the proof is simple, we will now show explicitly that the gyrokinetic Landau-Poissonsystem (4.17) defined in terms of the collision operator (4.24) has exact conservation laws for en-ergy and momentum. We hope to convey the similarity of this demonstration with the analogousdemonstration for the ordinary Landau-Poisson system (4.4)-(4.5). However, a word of cautionis in order here. It is essential that the guiding center Poisson brackets that appear in Eq. (4.24)be genuine Poisson brackets (i.e., the brackets must satisfy the Leibniz and Jacobi identities). HAPTER
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Proving that the gyrokinetic Landau operator (4.24) satisfies the identities (4.20) is very similarto proving that the particle-space Landau operator satisfies the identities (4.7). Setting ˙ E s ¯ s = R H gy s C gy s ¯ s ( F s , F ¯ s ) dz gc s , it is simple to verify that ˙ E s ¯ s + ˙ E ¯ ss = Γ s ¯ s Z Z ( W gy s ¯ s ) † Q s ¯ s gy A gy s ¯ s δ gy s ¯ s d ¯ z gc ¯ s dz gc s , (4.26)where all two-point quantities in the integrand are evaluated at ( z, ¯ z ) and · † denotes the ordinarymatrix transpose. Because Q s ¯ s gy is a symmetric matrix with null eigenvector W gy s ¯ s , the right-hand-side of this equation vanishes exactly. Thus the gyrokinetic Landau operator (4.24) satisfies theidentities (4.20) exactly, and the gyrokinetic Landau-Poisson system (4.17) has an exact energyconservation law, d E gy /dt = 0 . We will prove that if the background magnetic field is axisymmetric, then the gyrokinetic Landau-Poisson system conserves the total toroidal momentum P φ = X s Z p φs F s dz gc s , (4.27) HAPTER
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NERGETICALLY - CONSISTENT GYROKINETIC COLLISION OPERATOR p φs is the guiding center canonical toroidal momentum . If the background magnetic fieldhas additional symmetries, a similar proof of the conservation of the corresponding total momen-tum can easily be constructed. The time derivative of Eq. (4.27) yields dP φ dt = X s, ¯ s Z p φs C gy s ¯ s ( F s , F ¯ s ) dz gc s = X s, ¯ s ˙ P φs ¯ s , (4.28)where P φ is conserved exactly by the gyrokinetic Vlasov-Poisson system. Here, we find ˙ P φs ¯ s + ˙ P φ ¯ ss =Γ s ¯ s Z Z ( { y s , p φs } gc s − { y ¯ s , p φ ¯ s } gc ¯ s ) † Q s ¯ s gy A gy s ¯ s δ gy s ¯ s d ¯ z gc ¯ s dz gc s . (4.29)Now using the fact that p φs is the generator of infinitesimal toroidal rotations, we can see that { y s , p φs } gc s = e z × y s , where e z is the unit vector along the axis of rotation. Therefore the vectorquantity ( { y s , p φs } gc s − { y ¯ s , p φ ¯ s } gc ¯ s ) δ gy s ¯ s = e z × ( y s − y ¯ s ) δ gy s ¯ s = 0 , which follows from standard δ -function properties. This shows that ˙ P φs ¯ s + ˙ P φ ¯ ss = 0 , which in turn implies total toroidalmomentum conservation dP φ /dt = 0 . As we have discussed, these conservation laws ensure that the gyrokinetic Landau-Poisson systemis consistent with the the First Law of Thermodynamics. On the other hand, they do not directlyimply that the gyrokinetic Landau-Poisson system is consistent with the Second Law of Thermo-dynamics. To verify that entropy is indeed a non-decreasing function of time, we have computed Rather than give an explicit expression for p φs , which will depend on ones choice of guiding center representation,it is better to define it operationally via the guiding center Poisson bracket: for each phase space function f , thecanonical toroidal momentum satisfies { f, p ϕs } gc s = ∂ φ f , where ∂ φ is the toroidal angle derivative. HAPTER
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NERGETICALLY - CONSISTENT GYROKINETIC COLLISION OPERATOR S = − P s R F s ln F s dz gc s and found dSdt = Γ s ¯ s Z Z F s F ¯ s ( A gy s ¯ s ) † Q s ¯ s gy A gy s ¯ s δ gy s ¯ s d ¯ z gc ¯ s dz gc s . (4.30)Because Q s ¯ s gy is a positive semi-definite matrix and the distribution function is positive , the right-side of Eq. (4.30) is non-negative, which is the desired result.Note that this proves one “half” of a gyrokinetic version of Boltzmann’s H -theorem. Themissing ingredient is a complete characterization of the distributions that satisfy dS/dt = 0 , i.e.the gyrokinetic Maxwellians. Because the guiding center Poisson bracket is rather complicated,we have not yet found a complete characterization. However, we have verified that the distribution F Ms = 1 Z s exp (cid:18) − H gy s T (cid:19) , (4.31)where Z s = R exp( − H gy s /T ) dz gc s is the partition function, maximizes the entropy. We leave thecharacterization of the most general gyrokinetic Maxwellian, which would be useful for the sakeof deriving dissipative gyrofluid models with exact conservation laws Madsen (2013b), as a topicfor future study. When the collision frequency is much smaller than the gyrofrequency Brizard (2004), the fullgyrokinetic Landau operator (4.24) can be replaced with that operator’s gyroaverage, h C gy s ¯ s i . Whenthis is done, the gyrokinetic Landau-Poisson system becomes the gyroaveraged Landau-Poisson Positivity of the distribution function is also guaranteed by the positive semi-definiteness of Q s ¯ s gy . HAPTER
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NERGETICALLY - CONSISTENT GYROKINETIC COLLISION OPERATOR ∂ t F s + { F s , H gy s } gc s = X s h C gy s ¯ s ( F s , F ¯ s ) i (4.32) ∇ · P = ρ ( F ) , (4.33)where F s is now interpreted as the gyroaveraged part of the distribution function. Because thefunctions H gy s and p φs are independent of the gyrophase, the proofs of energy and momentumconservation given earlier work with C gy s ¯ s replaced by h C gy s ¯ s i . Thus, the gyroaveraged Landau-Poisson system has exact energy and momentum conservation laws. Closely related to the gyroaveraged Landau-Poisson system is the collisionally-linear gyroaver-aged Landau-Poisson system, ∂ t F s + { F s , H gy s } gc s = X ¯ s (cid:16) δC test s ¯ s + δC field s ¯ s (cid:17) , (4.34) ∇ · P = ρ ( F ) , (4.35)where the linearized test-particle and field-particle collision operators are δC test s ¯ s ( F s ) = h C gy s ¯ s ( F s , F M ¯ s ) i , (4.36) δC field s ¯ s ( F ¯ s ) = h C gy s ¯ s ( F Ms , F ¯ s ) i . (4.37)This system of equations is obtained from the gyroaveraged Landau-Poisson system by assuming F s = F Ms + δF s and then dropping the non-linear term in the collision operator, h C gy s ¯ s ( δF s , δF ¯ s ) i . HAPTER
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NERGETICALLY - CONSISTENT GYROKINETIC COLLISION OPERATOR h C gy s ¯ s ( F Ms , F M ¯ s ) i = 0 . Because the gyrokinetic Landau operator satisfies the identi-ties (4.20), it is straightforward to prove that these equations have the same conservation laws forenergy and momentum as the gyroaveraged Landau-Poisson system. The key to deriving an energetically-consistent formulation of collisional gyrokinetics was firstexpressing the particle-space Landau operator in terms of Poisson brackets “as much as possible,”which was an idea first championed by Brizard in Brizard (2004). In particular, the identity v − ¯ v = { x , H s } ( z ) − { x , H ¯ s } (¯ z ) (4.38)suggests that the appropriate definition of the gyrocenter velocity difference is given by Eq. (4.21).This idea, together with the procedure given earlier for determining the energetic consistency con-straints, appears to be appropriate for deriving energetically-consistent collision operators for otherreduced plasma models as well. In future work, we will report on the energy-conserving collisionalformulations of electromagnetic gyrokinetics and oscillation center theory.We note that, although the gyrokinetic Landau operator (4.24) and its linearized forms (4.36)-(4.37) may prove difficult to implement numerically, they identify the proper formalism for theinclusion of collisional transport in gyrokinetic theory. Hence, these gyrokinetic collision operatorsform the basis from which approximations can be implemented for practical applications.Lastly, by setting ϕ = 0 in the above formulas, our results reduce to an energy-momentum-conserving guiding center collision operator. This operator would be ideally suited to incorporatingcollisions into orbit-following codes such as ORBIT White and Chance (1984); see Hirvijoki et al.(2013) for recent work on the Monte Carlo implementation of a 5D guiding center Fokker- Note that this identity does not contradict the message presented in Madsen (2013a). In that reference, the gyroki-netic Maxwellian is defined using only the lowest-order gyrocenter Hamiltonian.
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NERGETICALLY - CONSISTENT GYROKINETIC COLLISION OPERATOR ad hoc methods to ensure exact conservation lawsBoozer and Kuo-Petravic (1981), or else do not fully account for inhomogeneities in the magneticfield Tessarotto et al. (1994).
The results presented in this chapter were obtained in collaboration with Professor Alain Brizard.They can also be found on the arXiv at arXiv:1503.07185. hapter 5Hamiltonian mechanics of stochasticacceleration
The term “stochastic acceleration” refers to the chaotic motion of particles subjected to a prescribedrandom force. Such motion occurs in myriad contexts; the turbulent electromagnetic fields presentin the interstellar medium and the RF wave fields found in magnetic fusion devices are just twoexamples. In the astrophysical context, it is thought to be partially responsible for the presence ofcosmic rays in our solar system Fermi (1949). In the magnetic fusion context, it might explain thepresence of certain high-energy tails observed in the National Spherical Torus Experiment whenneutral beams are fired into RF-heated plasmas Liu et al. (2009).Robust modeling of stochastic acceleration requires statistical approaches. The dominantapproach is to employ the Fokker-Planck equation Sturrock (1966); Hall and Sturrock (1967);Barbosa (1979); Petrosian and Liu (2004); Hamilton and Petrosian (1992) for the one-particle118
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Mathematically, this type of problem can be described as follows. Each particle moves througha n -dimensional single-particle phase space M according to a dynamical law given by a time-dependent vector field X t ; if z t ∈ M denotes the trajectory of a particle in M , then ˙ z t = X t ( z t ) . (5.1)Because the only forces present are Hamiltonian, X t must be Hamiltonian in the sense that thereis some Poisson bracket {· , ·} and some time-dependent Hamiltonian, H t , such that ˙ z i = { z i , H t } ,where z i denotes an arbitrary coordinate system on M Grebogi et al. (1979). By standard mathe-matical convention, this is written X t = X H t Abraham and Marsden (2008). The presumed formof the force then implies H t = H + ǫh t , where ǫ ≪ , H describes the mean time-independentbackground, and h t describes the small-amplitude random perturbation. Moreover, X h t evaluatedon a particle trajectory must have a correlation time τ ac much shorter than some constant τ , which,in turn, is much shorter than any bounce time associated with the perturbation τ b , τ ac ≪ τ ≪ τ b .Our goal in this Chapter is to find the correct coarse-grained version of the microscopic equa-tions of motion, X H t . Specifically, we seek a Langevin equation in the form δz t = X ( z t ) d t + X k ≥ X k ( z t ) δW kt (5.2)whose solutions correctly reproduce the late-time statistical behavior of solutions to the micro-scopic equations of motion. Here X k are vector fields on M that must be determined, W k are in-dependent ordinary Wiener processes, and δ denotes the Stratonovich differential Gardiner (2009) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION ◦ d ). We will identify the X k by demanding that Eq. (5.2) possess twoproperties: it must generate the Fokker-Planck equations for the one- and two-particle distributionfunctions, f t ( z ) and g t ( z , z ) . The two-particle distribution function is defined such that the proba-bility particle is in the region U ⊂ M and particle 2 is in the region U ⊂ M at time t is given by R U R U g t d z d z , where d z denotes the Liouville measure Abraham and Marsden (2008). Baxen-dale Baxendale (1984) has proven that a Langevin equation is uniquely determined by its one- andtwo- particle Fokker-Planck equations. Therefore, these conditions uniquely specify the Langevinequation we seek. In particular, the requirement that two-particle statistics be accurately repro-duced is critical; Baxendale’s work implies that constraining the Langevin equation only to beconsistent with the one-particle Fokker-Planck equation would not identify it uniquely.Physically, the reason that the two-particle Fokker-Planck equation contains more informationthan the one-particle Fokker-Planck equation can be understood as follows. After a short amountof time ∆ t , the displacement of a particle initially located at z at time t is given approximatelyby ∆ t X t ( z ) . Similarly, the displacement of a particle initially located at z is nearly ∆ t X t ( z ) .Because the random force field generally has spatial correlations, X t ( z ) and X t ( z ) are not statis-tically independent. Thus, the probability distribution of ( z ′ , z ′ ) , where z ′ i ≈ z i + ∆ t X t ( z i ) , willnot be given by the product of the distribution of z ′ with that of z ′ . This failure-to-factor precludesdetermining the two-particle distribution function from the mere knowledge of the one-particledistribution function. Note that this is true in spite of the fact that these particles do not interact ;because the random force is assumed to be prescribed, the time-evolution of z is decoupled fromthe time-evolution of z . HAPTER
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The one-particle Fokker-Planck equation associated with Eq. (5.2) is given by Gardiner (2009);Baxendale (1984) ∂f t ∂t = − div ( f t X ) + 12 X k ≥ div ( div ( f t X k ) X k )= A f t , (5.3)while the two-particle Fokker-Planck equation Baxendale (1984); Schmalfuss (2001); Kunita(1987) is given by ∂g t ∂t = A (1)1 g t + A (2)1 g t + X k ≥ div (1) div (2) : g t X k ( z ) ⊗ X k ( z ) . (5.4)The divergence operators in these expressions are defined relative to the Liouville volume form andthe colon indicates the full contraction of second-rank tensors, a : b ≡ a ij b ij . Because these equa-tions follow from Eq. (5.2) via rigorous mathematics, we will refer to them as the mathematicalFokker-Planck equations.On the other hand, under our assumption that the correlation time of the perturbed force ismuch shorter than a bounce time, standard coarse-graining procedures Risken (1996); Bazant(2006) together with a decomposition theorem for time-ordered exponentials Lam (1998) leadto the late-time evolution laws for the one- and two-particle distribution functions associated withthe microscopic equations of motion, Eq. (5.1). The physical one-particle Fokker-Planck equation HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION ∂f t ∂t = − (cid:26) f t , H + ǫ τ E [ s ] (cid:27) + ǫ τ E [ {{ f t , s } , s } ]= A f t , (5.5)while the physical two-particle Fokker-Planck equation (see the supplementary material for aderivation) is given by ∂g t ∂t = A (1)1 g t + A (2)1 g t + ǫ τ E [ α : d (1) d (2) g t ] . (5.6)The notation introduced in these two equations is defined as follows: E denotes an expectationvalue; the functions s , s are defined by s = Z τ exp( λX H ) ∗ h τ − λ d λ (5.7a) s = 12 Z τ Z a { exp( bX H ) ∗ h τ − b , exp( aX H ) ∗ h τ − a } d b d a ; (5.7b) exp( Y ) : M → M denotes the time-one advance map of the dynamical system defined by thevector field Y ; (exp( Y ) ∗ h )( z ) ≡ h (exp( − Y )( z )) ; the superscripts indicate which argument of g t that A and the exterior derivative d should be applied to; and α ( z , z ) ≡ E [ X s ( z ) ⊗ X s ( z )] is the two-point covariance tensor.The X k must be chosen so that the mathematical Fokker-Planck equations, Eqs. (5.3) and (5.4),are equivalent to the physical Fokker-Planck equations, Eqs. (5.5) and (5.6). However, a directcomparison of these two pairs of equations is difficult with Eqs. (5.5) and (5.6) in their currentform. To eliminate this issue, we will obtain a special decomposition of the two-point covariancetensor α ( z , z ) . HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION ξ ∈ T ∗ z M , then we can define a vector field Y ξ on M by contracting ξ with α on the left according to Y ξ ( z ) = α ( z , z )( ξ, · )= E [ ξ ( X s ( z )) X s ( z )] . (5.8)By forming all possible linear combinations of vector fields of this form, we can construct a (po-tentially infinite dimensional) linear space of vector fields Aronszajn (1950); Baxendale (1976),which we will denote H , H = { linear combinations of Y ξ , ξ ∈ T ∗ M } . (5.9)Because each Y ξ is of the form Y ξ ( z ) = X ¯ H ( z ) with ¯ H ( z ) = E [ ξ ( X s ( z o )) s ( z )] , and the sum ofHamiltonian vector fields is again Hamiltonian, H consists entirely of Hamiltonian vector fields.Moreover, following Baxendale Baxendale (1984, 1976), we see that H is a real Hilbert spacewhose inner product is defined by the formula h Y ξ , Y η i H = α ( z , z )( ξ, η )= E [ ξ ( X s ( z )) η ( X s ( z ))] , (5.10)where ξ ∈ T ∗ z M and η ∈ T ∗ z M . Therefore we may choose an orthonormal basis { e k } k ≥ for H , where each e k must be of the form e k = X H k . A simple calculation then leads to the desireddecomposition of α : α ( z , z ) = X k ≥ X H k ( z ) ⊗ X H k ( z ) . (5.11) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION X k . Indeed, we have found that the physical Langevin equation is given by δz t = X ˜ H ( z t ) d t + X k ≥ X ˜ H k ( z t ) δW kt , (5.12)where ˜ H = H + ǫ τ E [ s ] , ˜ H k = ǫ √ τ H k (5.13)Recall that the X H k are defined to be an orthonormal basis of the Hilbert space H defined inEq. (5.9). Also recall that all of the above manipulations have been performed under the assumptionthat the correlation time of the perturbed force felt by a particle is much shorter than any bouncetime associated with the perturbation.Because the coefficients in the Langevin equation for stochastic acceleration, Eq. (5.12), areall Hamiltonian vector fields, this equation is an example of a stochastic Hamiltonian system , thefoundations of which are developed in L´azaro-Cam´ı and Ortega (2008). It is in this sense that theLangevin equation for stochastic acceleration inherits the Hamiltonian structure of the microscopicequations. In particular, SDEs of this type are known to arise from a stochastic variational principlefor which Noether’s theorem applies. Thus, even at the dissipative macroscopic level, symmetriesimply the presence of conservation laws. We will find the physical Langevin equation for two example stochastic acceleration problems.Generally speaking, finding the coefficients of the physical Langevin equation involves finding anorthonormal basis for the space H , a task which may be analytically intractable. But, by Mercer’s HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION τ -second intervals. Assume that the pulses are uniform in space and constant in magnitude,but uniformly and independently distributed in direction. Thus, the k ’th pulse is generated by apotential of the form φ k ( x , t ) = ( z k · x ) φ o u ( t − kτ ) , where z k is a random vector uniformlydistributed over the unit sphere and u ( t ) is a temporal windowing function localized at t = τ / .In order to find the Langevin equation governing the plasma dynamics at times much longerthan τ , we must (a) calculate s and s using Eqs. (5.7a) and (5.7b), (b) find an orthonormal basis { X H k } k ≥ for the space H defined in Eq. (5.9), and (c) write down Eq. (5.12) with ˜ H and ˜ H k calculated using Eq. (5.13). The results of these three steps are as follows.(a) A quick calculation shows that s = m o z · x − m z · v (5.14a) s = const (5.14b)where m o = ( q/m ) φ o R τ u ( s )d s , m = ( q/m ) φ o R τ ( τ − s ) u ( s )d s , and q/m is the charge-to-massratio.(b) Each Y ξ must be of the form Y ξ = X g βγ , where g βγ ( x , v ) = 13 ( m β + m o γ ) · ( m v − m o x ) , (5.15)and β , γ are arbitrary constant 3-component vectors. Using this expression, it is simple to find anorthonormal basis for H . One is given by { X ¯ H k } k =1 .. , with H i ( x , v ) = 1 √ e i · ( m v − m o x ) , (5.16) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION { e i } i =1 .. is the standard basis for R .(c) Finally, the physical Langevin equation is given by δx i = v i d t + 1 √ τ m δW i (5.17a) δv i = 1 √ τ m o δW i , (5.17b)where i = 1 , , .As is readily verified, the one-particle Fokker-Planck equation for this SDE is given by ∂f t ∂t + v · ∇ f t = 16 τ ( m ∇ f t + m o m ∇ · ∇ v f t + m o m ∇ v · ∇ f t + m o ∇ v f t ) . (5.18)On the other hand, given an arbitrary function φ ( x , v ) , the SDE δx i = v i dt + m √ τ (cid:0) cos( φ ) δW ,i − sin( φ ) δW ,i (cid:1) (5.19a) δv i = m o √ τ (cid:0) cos( φ ) δW ,i − sin( φ ) δW ,i (cid:1) , (5.19b)where the W ,i , W ,j are six independent ordinary Wiener processes, will also generate Eq. (5.18).However, when φ is not constant, the two-particle Fokker-Planck equation generated by Eq. (5.19)will differ from the two-point Fokker-Planck equation generated by Eq. (5.17). This can be verifiedusing Eq. (5.6). The procedure identified here selects φ = 0 as the physical choice. In particular,it shows that a Langevin equation with the correct one-particle Fokker-Planck equation may stillincorrectly reproduce the two-particle distribution function.The inadequacy of Eq. (5.19) can also be understood intuitively as follows. Chaotic motionsof any two particles experiencing the electrostatic pulses are “synchronized” since the pulses areindependent of x and v . The Langevin equation (5.19), on the other hand, desynchronizes particle HAPTER
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Next, consider a minority population of magnetized fast ions moving through a plane lower-hybridwave that propagates perpendicular to the magnetic field. Assume the wave has a high harmonicnumber and a wavelength small compared to a typical ion gyroradius. Karney Karney (1979) hasshown that the dynamics of the perpendicular velocity of these ions are governed by a canonicaltime-dependent Hamiltonian system with Hamiltonian H t = I − ǫ sin( √ I sin θ − νt ) , (5.20)where I is the normalized magnetic moment, t the time normalized by the gyroperiod, θ the gy-rophase, ν the harmonic number, and ǫ the normalized wave amplitude. Moreover, when ǫ exceedsa threshold value, an ion’s motion becomes chaotic. This chaotic motion comes as the result ofthe effective randomization of the wave phase felt by an ion after a gyroperiod. Thus, above thethreshold for chaos, we can model the wave phase as being randomized every gyroperiod by arandom variable η . That is, we can replace the exact chaotic ion motion with a stochastic ap-proximation; see Chirikov (1979) for Chirikov’s application of the same modeling approach to thestandard map. This allows us to apply the formalism developed in this Chapter to find the physi-cal Langevin equation describing the stochastic particle trajectories at times much longer than thegyroperiod. HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION s and s . Set τ = 2 π and adopt therough approximation ∞ X n = −∞ J n ν − n exp( inθ ) ≈ J n o δ exp( in o θ ) , (5.21)where ν = n o + δ , | δ | < , and J n = J n ( √ I ) denotes the Bessel function of the first kindAbramowitz and Stegun (1964). This approximation amounts to selecting the most slowly varyingterm in the sum in Eq. (5.21). Then, upon directly evaluating the integrals in Eqs. (5.7a) and (5.7b),the resulting expressions for s and E [ s ] are s = 2 π sinc ( πδ ) J n o sin( n o θ + η ) (5.22a) E [ s ] = π ∞ X m = −∞ J m +1 − J m − m − ν + π sinc (2 πδ ) J n o +1 − J n o − δ , (5.22b)where η is a random variable uniformly distributed over the interval [0 , π ] and sinc ( x ) = sin( x ) /x .Next, the space H can be constructed using the above expression for s . In this case, H istwo-dimensional and has a basis { X H , X H } , where H ( I, θ ) = √ π sinc ( πδ ) J n o ( √ I ) cos( n o θ ) (5.23a) H ( I, θ ) = √ π sinc ( πδ ) J n o ( √ I ) sin( n o θ ) . (5.23b) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION δI = ǫ √ π sinc ( πδ ) n o J n o ( √ I ) × (cid:0) sin( n o θ ) δW − cos( n o θ ) δW (cid:1) (5.24a) δθ = (cid:18) ǫ π ∂∂I E [ s ] (cid:19) d t + (cid:18) ǫ r π I sinc ( πδ ) J ′ n o ( √ I ) × (cid:0) cos( n o θ ) δW + sin( n o θ ) δW (cid:1) (cid:19) . (5.24b)The diffusion of the magnetic moment I predicted by Eq. (5.24) has already been studied by KarneyKarney (1979). However, Eq. (5.24) extends and compliments Karney’s results by predicting theappropriate diffusion in gyrophase, as well as the correct two-particle statistics. We have shown how to derive the physical Langevin equation for particle trajectories undergoingstochastic acceleration. This SDE correctly generates the correct one- and two-particle Fokker-Planck equations and inherits the Hamiltonian structure of the microscopic equations of motion.This inheritance is theoretically satisfying because it is a direct consequence of demanding con-sistency with the physical one- and two-particle Fokker-Planck equations. It also implies thatsymmetries of the macroscopic physical laws governing stochastic acceleration imply the presenceof conservation laws. While this relationship is well known at the microscopic level, it is a pleasantsurprise that it remains intact upon passing to dissipative macroscopic equations.A Hamiltonian Langevin equation L´azaro-Cam´ı and Ortega (2008) is a Stratonovich SDE ofthe form given in Eq. (5.12). If a loop of initial conditions for this SDE evolves under a givenrealization of the noise, then the action of that loop is constant in time. In addition, these equa-
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This supplement to the article “The Hamiltonian mechanics of stochastic acceleration” consists ofa derivation of the physical two-particle Fokker-Planck equation. In the main text, the two-particleFokker-Planck equation is given in Eq. (6). The derivation will freely draw upon notation definedin the article. The essential idea behind this derivation is not novel; the same idea is presented inBazant (2006) in the simpler context of a one-dimensional random walker.Let F t,s be the time advance map Abraham and Marsden (2008) associated with the dynamicalvector field X H t (Eq. (1) in our manuscript); F t,s ( z ) gives the time t phase space location of aparticle located at z ∈ M at time s . Because this time advance map satisfies the identity F t,r ◦ F r,s = F t,s , where ◦ denotes the composition of functions, we have F Nτ, = F Nτ, ( N − τ ◦ F ( N − τ, ( N − τ ◦ ... ◦ F τ, . This decomposition of the time advance map provides a stroboscopic description ofparticle dynamics; as the integer N increases, it tells us the phase space location of a particle at thetimes t = 0 , t = τ, t = 2 τ, ... . HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION τ ≪ τ b , the results in Lam (1998) may be used to write F ( k +1) τ,kτ = exp( X s ( k +1) τ ) ◦ exp( τ X H ) , where s t = ǫs ,t + ǫ s ,t + ... is given to O ( ǫ ) by s ,t = Z τ exp( λX H ) ∗ h t − λ dλ (5.25) s ,t = 12 Z τ Z a { exp( bX H ) ∗ h t − b , exp( aX H ) ∗ h t − a } db da. Here exp( Y ) : M → M is the time-one advance map of the dynamical system defined by thevector field Y ; {· , ·} denotes the Poisson bracket; and (exp( Y ) ∗ h )( z ) = h (exp( − Y )( z )) . Thismeans that each τ -second step in the discrete-time dynamics is a deterministic drift, exp( τ X H ) ,followed by a small random kick, δ k ≡ exp( X s ( k +1) τ ) . Moreover, the statistical assumptions onthe perturbed force imply that the various δ k are independent identically distributed maps. There-fore, each discrete-time particle trajectory, z Nτ = F Nτ, ( z ) , is a temporally-homogeneous Markovprocess.In terms of F t,s , the trajectory of a pair of particles at z = ( z , z ) ∈ M × M at time s is givenby z t = ( F t,s ( z ) , F t,s ( z )) . (5.26)This motivates introducing the two-particle time-advance map, F s,t : M × M → M × M , whichis defined by the formula F t,s ( z ) = ( F t,s ( z ) , F t,s ( z )) . (5.27)The two-particle time-advance map inherits many of the qualitative features of the one-particletime-advance map. In particular, F t,r ◦ F r,s = F t,s , which implies that F Nτ, = F Nτ, ( N − τ ◦ F ( N − τ, ( N − τ ◦ ... ◦ F τ, . Moreover, F ( k +1) τ,kτ admits the decomposition F ( k +1) τ,kτ = δ k ◦ HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION exp ( τ ¯ X H ) , where δ k ( z ) = ( δ k ( z ) , δ k ( z )) , (5.28)and exp ( τ ¯ X H ) is the time- τ advance map associated with the vector field on M × M given bythe formula ¯ X H ( z ) = X H ( z ) ⊕ X H ( z ) ∈ T z M ⊕ T z M. (5.29)The time-homogeneous Markov property implies that the linear operator P N defined on two-particle observables Q : M × M → R by the formula ( P N Q )( z ) = E [ Q ( F Nτ, ( z ))] satisfies thesemigroup property P N + M = P N P M . Therefore, if we define the time evolution of a two-particleobservable as Q Nτ = P N Q , then τ ( Q ( N +1) τ − Q Nτ ) = τ ( P − Q Nτ . For times sufficientlylarge compared with τ , intuition suggests that the left-hand side of this identity approaches thepartial time derivative ∂ Q t /∂t and that τ ( P − may be approximated by a differential operator.This intuition can be made precise through the use of a Kramers-Moyal expansion Risken (1996);Bazant (2006), which we will describe now. Following Bazant (2006), we will obtain this limitingpartial differential equation by scaling the time variable by the appropriate power of ǫ and lookingfor a dominant balance of the equation τ ( Q ( N +1) τ − Q Nτ ) = 1 τ ( P − Q Nτ (5.30)as ǫ → . HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION ¯ Q t ( z ) = (cid:18) exp ( − t ¯ X H o ) ∗ Q t (cid:19) ( z ) ≡ Q t (cid:18) exp ( − t ¯ X H o )( z ) (cid:19) . (5.31)instead of Q t . In terms of ¯ Q t and t = N τ , Eq. (5.30) becomes ¯ Q t − exp ( − τ ¯ X H o ) ∗ ¯ Q t + ∞ X k =1 ∂ k ¯ Q t ∂t k τ k = (5.32) ¯ Q t − exp ( − τ ¯ X H o ) ∗ ¯ Q t + E (cid:20) ǫ [ ¯ Q t , ¯ s ] + ǫ ([ ¯ Q t , ¯ s ] + [[ ¯ Q t , ¯ s ] , ¯ s ] / (cid:21) + O ( ǫ ) , where ¯ s = exp ( − t ¯ X H o ) ∗ ( π ∗ s ,τ + π ∗ s ,τ ) (5.33) ¯ s = exp ( − t ¯ X H o ) ∗ ( π ∗ s ,τ + π ∗ s ,τ ) ; (5.34)the bracket [ · , · ] is the Poisson bracket on M × M defined by the formula [ f, g ]( z , z ) = { f ( z , · ) , g ( z , · ) } ( z ) + { f ( · , z ) , g ( · , z ) } ( z ); (5.35)and π , π : M × M → M are the projection maps onto the first and second factor respectively. TheTaylor expansion in time is the key step here. It is justified by the fact that we will be consideringlate times when the evolution of ¯ Q t has had time to slow down as a result of diffusion. Notice thatbecause E [ s ,t ] = 0 , E [¯ s ] = 0 as well. HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION t = ( τ /ǫ ) λ , where λ is a renormalized dimensionless time. As ǫ → , thedominant balance of Eq. (5.32) is given by ∂ ¯ Q λ ∂λ = E (cid:20) [ ¯ Q λ , ¯ s ] + [[ ¯ Q λ , ¯ s ] , ¯ s ] / (cid:21) . (5.36)Or, in terms of t and Q t = exp ( t ¯ X H o ) ∗ ¯ Q t , ∂ Q t ∂t = [ Q t , π ∗ H o + ǫ E [ π ∗ s ,τ ] /τ ] + ǫ τ E (cid:2) [ Q t , π ∗ s ,τ ] , π ∗ s ,τ ] (cid:3) + [ Q t , π ∗ H o + ǫ E [ π ∗ s ,τ ] /τ ] + ǫ τ E (cid:2) [ Q t , π ∗ s ,τ ] , π ∗ s ,τ ] (cid:3) + ǫ τ E (cid:2) [[ Q t , π ∗ s ,τ ] , π ∗ s ,τ ] (cid:3) = L Q t . (5.37)Thus, for late times Q t is given formally by Q t ( z ) = h exp( tL ) Q , δ z i = hQ , g t, z i , (5.38)where h· , ·i denotes the L pairing of functions on M × M relative to the two-particle Liouvillemeasure dz dz , δ z is a delta function concentrated at z , and g t, z is the distribution function of apair of particles that begin at z ∈ M × M when t = 0 . Because this identity holds for arbitraryfunctions Q , it implies that g t, z evolves according to g t, z = exp( tA ) δ z , (5.39)where A = L ∗ is the L adjoint of the operator L . Differentiating this last identity in timeand integrating against the initial two-particle distribution function finally leads to the two-particle HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION ∂g t ∂t = A g t , (5.40)where A is given by ( A g t )( z ) = A (1)1 g t + A (2)1 g t + ǫ τ E [ α : d (1) d (2) g t ] . (5.41)The quantities A and α are defined in the main text. The Lorentz plasma consists of a noninteracting gas of electrons moving through a neutralizing ran-dom collection of fixed, infinitely-massive, Debye-screened ions. The typical approach to studyingthe dynamics of the Lorentz plasma is to derive a Fokker-Planck equation governing the single-electron distribution function on time scales long compared with the plasma period. This approacheliminates the need to resolve the complicated structure of the ionic potential, and thereby greatlyreduces the analytical and computational resources required to understand the plasma’s behavior.The structure of the Fokker-Planck equation, which takes the form of a Vlasov equation cor-rected by a collision operator, is ultimately determined by the microscopic Hamiltonian equationsof motion for a single electron. Therefore qualitative features of the electronic equations of motionought to have counterparts at the level of the collision operator. For instance, because an elec-tron that passes through the screened potential of an ion suffers no change in its kinetic energy, itwould be surprising if the Fokker-Planck equation didn’t have a kinetic energy conservation law.Likewise, because there is no mechanism for electron absorption, the collision operator should be
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION ∂ t f + v · ∇ f = C L ( f ) , (5.42)where C L is the Lorentz collision operator. We will find that the collision operator, C HL , in theHamiltonian Fokker-Planck equation is not identical to C L , but agrees with the latter asymptoti-cally in the limit ǫ o , ǫ → , where ǫ o = τ ac /τ and ǫ = v th τ /L , τ is the coarse-graining timestep, τ ac ∼ ω − p is the Lagrangian autocorrelation time, and L is the length scale of the electrondistribution function. We will also show that there is tension between the stochastic Hamiltonianproperty possessed by C HL and the energy-conserving property of C L in the following sense. Theoperator C HL slowly produces energy for all non-zero ǫ o , ǫ , while the limiting energy-conservingoperator C L is provably not Hamiltonian in the sense of L´azaro-Cam´ı and Ortega (2008). Finally,we will prove that any “reasonable” stochastic Hamiltonian collision operator that is associatedwith a path-wise energy-conserving stochastic differential equation must be signficantly differentfrom the Lorentz collision operator C L .Altogether, these results might give the impression that the stochastic Hamiltonian formalismis not appropriate as an underlying mathematical structure for the pitch angle scattering process. HAPTER
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The set Q = R will serve as the configuration space for the non-interacting electrons. The velocityphase space for a single electron is therefore M = T Q ≈ Q × R . If φ is the potential producedby the background ions, then the dynamics of each electron are governed by the Hamiltonian H ( x, v ) = 12 v + q e m e φ ( x ) , (5.43)where q e and m e are the electron charge and mass, respectively. The relationship between thisHamiltonian and the electron dynamical vector field X is given by Hamilton’s equations,i X ω o = d H , (5.44)where ω o = d x i ∧ d v i . Structure of the ionic potential
Let λ D and b o be the Debye length and the electron distance of closest approach, respectively.The plasma parameter Λ = λ D /b o . The structure of the electrostatic potential produced by each HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION λ D and b in the following manner. The charge density producedby an ion centered at x j is given by ρ x j ( x ) = q i δ ( x − x j ) . The electrostatic potential, φ x j , producedby such a Debye-shielded ion satisfies the differential equation −∇ φ x j + 1 λ D φ x j = 4 πρ x j . (5.45)The only spherically-symmetric solution of this equation that decays as | x | → ∞ is given by φ x j ( x ) = q i | x − x j | exp ( −| x − x j | /λ D ) . (5.46)This “raw” potential, φ x j , is not the mathematically-appropriate potential to subject to electrons inthe Lorentz plasma. When an electron passes within a distance b o from the j ’th ion, it experiencesa large angle scattering event. Because such scattering events are exceedingly-rare, and becausewe would like to avoid infinities in our analysis, we will regularize the raw potential. We willaccomplish this regularization by assuming that the potential produced by the j ’th ion is given by φ x j ( x ) = g ( | x − x j | ) ≡ q i λ D g Λ ( | x − x j | /λ D ) , where g Λ ( r ) = g − ( r ) if r < r if < r < g + ( r ) if r > , (5.47)and g − , g + are chosen so that (i) g Λ ( r ) = 0 for r > δ for some small δ > and (ii) thederivative of g Λ vanishes in a neighborhood of r = 0 . The total electrostatic potential produced by N ions with centers x j is then given by φ = N X j =1 φ x j . (5.48) HAPTER
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Statistical properties of the ionic potential
We will assume that the x j are i.i.d. Q -valued random variables with PDF p . We will also workin the “thermodynamic limit.” The precise meaning of this statement is as follows. We allow theionic PDF, p , to depend on the number of particles parametrically, i.e. p ( x ) = p N ( x ) . Then wemake two assumptions:(i) N ≫ (ii) The limit lim N →∞ N p N ≡ n i (5.49)exists (pointwise) and is equal to the constant n i = Λ /λ D .The mean value of the ionic potential in the thermodynamic limit is given by h φ i = lim N →∞ E [ φ ]( x )= lim N →∞ N Z g ( | x − x ′ | ) p N ( x ′ ) dx | prime. = n i Z g ( | x − x ′ | ) dx ′ = 4 πn i ∞ Z g ( r ) r d r , (5.50)where dz denotes the standard volume form on R . It follows that the mean electrostatic force onan electron is zero. HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION C φφ ( x, y ) = lim N →∞ E [( φ − E [ φ ])( x )( φ − E [ φ ])( y )]= n i Z g ( | x − x ′ | ) g ( | y − x ′ | ) dx ′ . (5.51)where we have used the fact that the x i are independent and E [ φ x ] ∝ N − . Notice that C φφ ( x, y ) = C ( | x − y | ) , (5.52)where C ( d ) = Z π T e λ D C Λ ( d/λ D ) , (5.53)and C Λ ( d ) =2 π ∞ Z −∞ ∞ Z g Λ ( q R + ( Z − d ) ) g Λ ( q R + Z ) R dR dZ (5.54)is a dimensionless covariance function. The most important properties of C φφ and C are thefollowing.(P1) C φφ is manifestly positive semi-definite, i.e. for any finite collection of points x j ∈ Q and corresponding real numbers a j , X i X j a i C φφ ( x i , x j ) a j ≥ . (5.55) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION C φφ is an isotropic covariance function , and C is a radial basis function .(P2) C ( d ) = 0 when d > λ D + . This follows from the fact that g ( r ) is compactly sup-ported in the interval [0 , λ D + ) .(P3) C ( d ) = C ( − d ) Using (P3), we can also see that all odd powers of d vanish in C ’s Maclaurin series, i.e. C ( d ) = C (0) + 12 C ′′ (0) d + 124 C ′′′′ (0) d + O ( d ) , (5.56)as d → . Thus, C ′ ( d ) d − C ′′ ( d ) = O ( d ) , (5.57)and C ′ ( d ) = C ′′ (0) d + O ( d ) , (5.58)as d → .The covariance tensor of the electrostatic field produced by the ions is given by C ∇ φ ∇ φ ( x, y ) = lim N →∞ E [ ∇ ( φ − E [ φ ])( x ) ∇ ( φ − E [ φ ])( y )]= lim N →∞ E [ ∇ φ ( x ) ∇ φ ( y )]= C ( x − y ) , (5.59) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION C ( d ) = Z π T e λ D C Λ ( d /λ D ) , (5.60)and C Λ ( d ) = − C ′ Λ ( | d | ) | d | (cid:18) id − d | d | d | d | (cid:19) − C ′′ Λ ( | d | ) (cid:18) d | d | d | d | (cid:19) . (5.61)The most important properties of C ∇ φ ∇ φ and C are the following.(F1) C ∇ φ ∇ φ is positive semi-definite, i.e. given any finite collection of points x j and corre-sponding vectors v j , X j X k v j · C ∇ φ ∇ φ ( x j , x k ) · v k ≥ . (5.62)(F2) C (∆) = 0 when | ∆ | > λ D + .(F3) C (∆) = C ( − ∆) .Using (F2) and integration by parts, we can simplify the following type of definite integralsinvolving C . Let L > λ D + and choose a unit vector e . Set I n ( e ) = L Z − L | λ | n C ( λe ) dλ. (5.63) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION n ≥ , we have I n ( e ) = − ∞ Z λ n − C ′ ( λ ) dλ ( id − ( n + 1) ee ) . (5.64) The purpose of this section is to give a qualitative motivational picture of the ideas that go intoderiving a Hamiltonian Fokker-Planck equation for the Lorentz plasma, as well as to describe pre-cisely what a Hamiltonian Fokker-Planck equation is. As a point of departure, we remind the readerof the justification and derivation of a Fokker-Planck model in terms of so-called jump moments.We then reformulate the same discussion in terms of the symplectomorphism group, Diff ω o ( T Q ) .This reformulation offers a particularly suggestive description of how the Hamiltonian nature ofthe electronic equations of motion influences the structure of the Fokker-Planck equation. In terms of jump moments
The problem of finding a formula for the trajectory of an electron in the Lorentz plasma is ex-tremely complicated. There is not just a single ion; if there were, we would only have to solve thetwo-body problem. There are not just two ions either; if this were true, we would be faced with(an analogue of) the soluble Euler three-body problem. It is better to assume there are ∼ ions,and therefore electron dynamics are surely chaotic. As such, when studying the dynamics of theLorentz plasma, we must be satisfied with less detailed information than exact electron orbits.One way to give a less detailed (and therefore simpler) description of the Lorentz plasma thatstill retains a great deal of dynamical information is to find the evolution equation for the single-electron PDF. This is a much more manageable task than finding the precise electron trajectoriesfor the following reason. Whereas the large number of ions mangles the electron trajectories, it ac- HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION τ ac ∼ ω − p . Thus, ifwe chop the time-axis into intervals of length τ ≫ τ ac and restrict attention to an electron’s phasespace location at the ends of these intervals, we obtain a discrete-time Markov process in phasespace, z n , where n is the discrete time index. If we could estimate the transition probabilitiesof this Markov process, we would be able to derive an equation describing the evolution of theelectron PDF on time scales much longer than τ . This derivation could proceed by analogy withthe derivation of the late-time evolution equation for the PDF of a random walker given in Bazant(2006). Indeed, our electron Markov process is nothing more than a random walk in phase spacewhere the PDF of the walker’s step depends on where the walker is standing.Provided that we make the further restriction τ ≪ τ bounce , where τ bounce is the characteristictime for an electron trajectory to deviate significantly from free streaming, the transition probabil-ities can be calculated using ordinary perturbation theory . It is straightforward to show that theresulting late-time evolution equation for the single-electron PDF takes the form ∂ t f + div ( f u ) = div ( D · df ) , (5.65)where div denotes the divergence relative to the Liouville volume form dx dv , u is a vector fieldon phase space, and D is a rank- tensor on phase space with components D ij . The drift vector u is given by u = u o + h ∆ z i /τ − h div (∆ z )∆ z i /τ, (5.66) Our ability to require that τ satisfies both τ ≪ τ bounce and τ ≫ τ ac follows from the fact that small-angle scatteringevents dominate over large-angle scattering events. HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION ∆ z, ∆ z are vector fields on phase space and u o = v · ∂ x is the free-streaming vector field.The diffusion tensor is given by D = 12 h ∆ z ⊗ ∆ z i /τ. (5.67)We will refer to ∆ z and ∆ z as the jump vectors. They are defined so that a particle starting at z o = ( x, v ) when t = 0 ends at z τ ≈ exp(∆ z + ∆ z )( x + vτ, v ) (5.68)when t = τ , with second order accuracy . The components of D are known as the jump moments.Equation (5.65) is known as the Fokker-Planck equation. In terms of the symplectomorphism group
The previous argument justifying the use of a Fokker-Planck equation for the late-time single-electron PDF is appealing because it draws upon only elementary facts about Markov processes.However, it has the disadvantage of obscuring a striking geometric picture that underlies the wholediscussion. In order to illuminate the geometric picture, we will now give a second justification forusing a Fokker-Planck equation.Let F be the t = τ time-advance map associated with the full single-electron equations ofmotion. Given an initial condition z ∈ T Q , the approximately-Markov process that gives anelectron’s phase space location at the ends of τ -second time intervals is given by z n = F n ( z ) , (5.69) The amplitude of the fluctuating electric field can be regarded as the expansion parameter, which is essentially p / Λ HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION n ∈ Z is the discrete time variable and F n denotes the n -fold composition of F with itself(e.g. when n = 2 , F n ( z ) = F ( F ( z )) .) Note that the mapping F is random because it depends onthe configuration of the ions. Because z n is a Markov process, the operator U n given by ( U n Q )( z ) = h Q ( F n ( z )) i , (5.70)where Q : T Q → R is an arbitrary observable, must satisfy the semi-group property U n + n = U n U n . (5.71)However, by noting U = hF ∗ i , where F ∗ denotes the pullback operator along F , we also have U = hF ∗ F ∗ i 6 = hF ∗ ihF ∗ i = U U . (5.72)Note that F and F are not independent random mappings – they are identical! This contradictiontells us that the process Eq. (5.69) is not precisely Markov.In spite of this contradiction, we know that z n is approximately Markov because τ ≫ τ ac .Therefore, for the sake of modeling it is sensible to replace Eq. (5.69) with z n = F n ◦ F n − ◦ · · · ◦ F ( z ) , (5.73)where ◦ denotes functional composition and the F i are i.i.d. random mappings each with the samePDF as F . The effect of this replacement is that the locations of all of the ions are scrambledafter each time step. While this scrambling effect is, strictly speaking, unphysical, it ought to bestatistically harmless; electrons forget about the orientations of the ions after τ ac seconds anyway.It is easy to check that this redefined z n is rigorously Markov. This formula shows that U is a mean propagator. HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION g n = F n ◦ F n − ◦ · · · ◦ F . (5.74)Indeed, the expected value of Q ( z n ) , for any observable Q , is given by h Q ( z n ) i = h g ∗ n Q i ( z ) = ( h g ∗ n i Q )( z ) , (5.75)which shows that the single-electron PDF is completely determined if we know h g ∗ n i . But whywould we want to do this? Whereas the process z n evolves in a finite-dimensional space, theprocess g n evolves in an infinite-dimensional space of mappings, which suggests that g n is a muchmore complicated object than z n . The answer is that g n is simpler than z n when viewed in the rightway.First notice that g n is a diffeomorphism for all n , i.e. g n is smooth and invertible with a smoothinverse. This follows from the fact that F i is a diffeomorphism for each i (being a time-advancemap for an ODE on phase space) and g n is a composition of the F i ’s. This means that the mapping-valued process g n takes place in a very special space of mappings known as the phase space dif-feomorphism group Diff ( T Q ) . Diff ( T Q ) , which is the set of all diffeomorphisms of the velocityphase space T Q , is a group under functional composition. In a sense that we will not discuss here,Diff ( T Q ) is also a smooth (infinite-dimensional) manifold. Thus, the process g n evolves in a spacewith a very rich structure.Next notice that the increments δg n ,n = g n ◦ g − n , for n ≤ n , have the following simplestatistical properties:(RW1) If n ≤ n ≤ n ≤ n , δg n ,n and δg n ,n are statistically independent. HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION of δg n ,n depends on n , n only through the difference n − n .If we were to replace Diff ( T Q ) with the additive group R , these properties would imply that g n is an ordinary random walk . More generally, if Diff ( T Q ) is replaced by any group G , adiscrete-time process on G that satisfies properties (RW1) and (RW2) is known as a random walkon G . Thus, the process g n is a Diff ( T Q ) -valued random walk.Once nice feature of thinking about the dynamics of the Lorentz plasma as a random walkon the diffeomorphism group is that we have a good intuitive understanding of the long-timebehavior of random walks. In particular, we know that, under an appropriate scaling limit, anordinary random walk is well-approximated by a Brownian motion. Therefore we can reasonablyexpect that the long-time behavior of the random walk g n is described by a Brownian motionon Diff ( T Q ) Baxendale (1984), i.e. a continuous-time process g t ∈ Diff ( T Q ) that satisfies theproperties(BM1) If t ≤ t ≤ t ≤ t , δg t ,t and δg t ,t are statistically-independent.(BM2) The PDF of δg t ,t only depends on t , t through the difference t − t .(BM3) The sample paths of g t are almost surely continuous functions of t .Of course, properties (BM1) and (BM2) are the obvious analogues of the properties (RW1)and (RW2) that we already know g n satisfies. Property (BM3) is motivated by the dominance Here PDF stands for probability distribution functional . The PDF of a step taken by such a walker would be arbitrary.
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION ( T Q ) always arise asthe stochastic time-advance maps of stochastic differential equations. If g t is such a stochastictime-advance map, then a basic fact from the theory of stochastic differential equations statesthat the PDF of the random variable g t ( z ) , where z ∈ T Q is a fixed initial condition, satisfies aFokker-Planck equation of the form given in Eq. (5.65). We have therefore arrived at an alternativejustification for the use of a Fokker-Planck equation to model the Lorentz plasma.Another nice feature of thinking in terms of g n instead of z n is that the Hamiltonian nature ofthe electron dynamical equations manifests itself in a very simple way at the level of Diff ( T Q ) .Indeed, because the electron dynamical vector field X given by Eq. (5.44) is Hamiltonian, the t = τ time-advance map F must preserve the symplectic form, F ∗ ω o = ω o , where ω o = dx i ∧ dv i .Likewise, because the F i have the same PDF as F , we also have F ∗ i ω o = ω o . Therefore, by theidentity ( F i ◦ F j ) ∗ = F ∗ j F ∗ i , the process g n satisfies g ∗ n ω o = ω o (5.76)for all n . In other words, g n is not free to wander everywhere in Diff ( T Q ) , but only along theconstraint set defined by g ∗ ω o = ω o . Actually, the latter constraint set is a subgroup of Diff ( T Q ) known as the symplectomorphism group, Diff ω o ( T Q ) ⊂ Diff ( T Q ) . Because g n does not leave thesymplectomorphism group, the limiting Brownian motion g t also must satisfy the same constraint.It can be shown that the latter requirement constrains the Fokker-Planck equation (5.65) to satisfy u = X h o (5.77) D = X k =1 X h k ⊗ X h k , (5.78) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION X f denotes the Hamiltonian vector field with Hamiltonian f , the h k are arbitrary functionson phase space, and the sum over k is possibly infinite. Conversely, given a Fokker-Planck equationwhose drift vector u and diffusion tensor D take the above form, it is always possible to find aDiff ω o ( T Q ) -valued Brownian motion that generates it.When the drift vector and diffusion tensor of a Fokker-Planck equation are in the form pre-scribed by Eqs. (5.77) and (5.78), we will say that the Fokker-Planck equation is Hamiltonian.Thus, an important consequence of the fact that the single-electron equations of motion are Hamil-tonian in nature is that the Fokker-Planck equation for the single-electron PDF ought to be Hamilto-nian. By making this observation, we can see that in passing from the microscopic single-electronequations of motion to the macroscopic Fokker-Planck equation, we move from the world ofHamiltonian mechanics into the world of stochastic Hamiltonian mechanics. Where Hamiltonianmechanics is concerned with smooth one-parameter subgroups of Diff ω o ( T Q ) , stochastic Hamil-tonian mechanics is concerned with Brownian motion on Diff ω o ( T Q ) . The stochastic Hamiltoniannature of the Fokker-Planck equation is the moral counterpart to the Hamiltonian nature of themicroscopic equations of motion alluded to in the introduction. We will now apply the technique described in Burby et al. (2013b), which we will refer to here-after as BZQ, to derive a Hamiltonian Fokker-Planck equation for the Lorentz plasma. In BZQ’snotation, we have H o = 12 v (5.79) h = q e m e φ, (5.80) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION φ is the ionic potential described earlier. The first- and second-order jump vectors are givenby ∆ z = X s and ∆ z = X s , where s = Z τ F t ∗ h dt (5.81) s = 12 Z τ Z t { F t ∗ h, F t ∗ h } dt dt . (5.82)The unperturbed flow map, F t , is given by F t ( x, v ) = ( x + vt, v ) . (5.83)Using these formulae, the drift vector u HL and the diffusion tensor D HL can be computed in termsof the covariance tensor C given in Eq. (5.60).First we compute u HL = u o + X h s i /τ , which amounts to computing h s i . It is straightforwardto verify that the mean of the Poisson bracket appearing in the definition of s reduces to h{ F t ∗ h, F t ∗ h }i = − (cid:18) q e m e (cid:19) [ t − t ] tr ( C ([ t − t ] v )) , (5.84)where tr denotes the trace. Upon substituting this expression into the double integral, changingintegration variables, and applying Fubini’s theorem, we then arrive at the following expressionfor h s i : h s i = − (cid:18) q e m e (cid:19) Z τ t ( τ − t ) tr ( C ( vt )) dt. (5.85)Note that because h s i only depends on ( x, v ) through v , the drift vector u HL = u o + h ∆ z i /τ = u o + X h s i /τ only has an x -component.Next we derive an expression for the diffusion tensor D HL . For this purpose, we introduce auseful notation for contravariant second rank tensors on phase space. If T is a dyad like C , then HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION T xv is the second rank contravariant tensor on phase space given by dh · T xv · dh = ∇ h · T · ∇ v h , (5.86)where h , h are arbitrary functions on phase space. The tensors T xx , T vx , T vv are similarly defined.In terms of this notation, h X s ⊗ X s i is given by h X s ⊗ X s i = (cid:18) q e m e (cid:19) (cid:18) τ Z τ Z C vv ([ t − t ] v ) dt dt + τ Z τ Z t C vx ([ t − t ] v ) dt dt + τ Z τ Z t C vx ([ t − t ] v ) dt dt + τ Z τ Z t t C xx ([ t − t ] v ) dt dt (cid:19) . (5.87)After simplifying the double integrals and dividing by τ , the diffusion tensor D is then given by D HL = 1 τ (cid:18) q e m e (cid:19) (cid:18) Z τ ( τ − t ) C vv ( vt ) dt + τ Z τ ( τ − t )( C xv ( vt ) + C vx ( vt )) dt + τ Z τ ( τ − t / + t / τ ) C xx ( vt ) dt (cid:19) . (5.88)As explained in Burby et al. (2013b), because D HL = h X s ⊗ X s i / (2 τ ) , there exists an expansionof D in the form D HL = X k =1 X h k ⊗ X h k , (5.89)where the h k form an orthonormal basis for the reproducing kernel Hilbert space associated withthe phase space covariance kernel α ( z , z ) = h X s ( z ) ⊗ X s ( z ) i / (2 τ ) . We will not find the h k HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION h k is equivalent to the well-known problemof finding the so-called Karhunen-Lo`eve Lord et al. (2014) expansion of a Gaussian random fieldwith covariance α .Because u HL and D HL just calculated can be written in the form given in Eq. (5.77), the Fokker-Planck equation ∂ t f + div ( f u HL ) = div ( D HL · df ) (5.90)is Hamiltonian. In the following section, we will compare and contrast this Fokker-Planck equationwith the classical result given in Eq. (5.42). In particular, we will compare the Lorentz collisionoperator C L ( f ) = div ( D L · df ) , (5.91)where D L = ν ( v ) U vv ( v ) , (5.92) U ( v ) = | v | ( id − ˆ v ˆ v ) , and ν ( v ) = ω p π ln ΛΛ v th | v | , with the Hamiltonian collision operator C HL ( f ) = div ( D HL · df ) , (5.93)where D HL is given by Eq. (5.88). HAPTER
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Asymptotic equivalence
In order to demonstrate the asymptotic equivalence of Eqs. (5.42) and (5.90), we introduce thedimensionless variables x, v, t . These normalized position, velocity, and time variables are relatedto their unnormalized counterparts by x = Lx (5.94) v = v th v (5.95) t = T t, (5.96)where L is the length scale of the electron distribution function, v th = λ D ω p is the thermal velocity,and T is the temporal scale of the electron distribution function. We will set T = Λ /ω p , whichis consistent with measuring time in units of the electron-ion collision period. The HamiltonianFokker-Planck equation expressed in these dimensionless variables is ∂ t f + div ( f u ) = div ( D · df ) , (5.97)where u = T u HL and D = T D HL . We will now present expressions for u and D that exhibit theirdependence on the small parameters ǫ o = 1 / ( τ ω p ) , ǫ = v th τ /L , and / Λ .The normalized drift vector is given by u = Λ ǫ o ǫ v · ∂ ¯ x − π ǫ ∇ v χ · ∂ x , (5.98) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION χ is given by χ ( v ) = Z /ǫ o ǫ o λ (1 − ǫ o λ ) tr C Λ ( vλ ) dλ. (5.99)Note that χ depends on ǫ o and Λ , but not ǫ .The normalized diffusion tensor is given by D = 116 π (cid:18) η vv + ǫ η vx + η xv ) + ǫ ζ xx (cid:19) , (5.100)where the dimensionless dyads η , ζ are given by η = Z /ǫ o (1 − ǫ o λ ) C Λ ( vλ ) dλ (5.101) ζ = Z /ǫ o ( / − ǫ o λ / + ǫ o λ / ) C Λ ( vλ ) dλ. (5.102)Note that η , ζ , like χ , depend on ǫ o and Λ , but not ǫ .These expressions for u and D can now be used to study the asymptotic behavior of the Hamil-tonian Fokker-Planck equation as ǫ o , ǫ and / Λ tend to zero. For concreteness, we will study thislimit under the assumption ǫ o = / √ Λ (5.103) ǫ = / √ Λ . (5.104)Effectively, this assumption chooses a specific path to zero through ( ǫ o , ǫ , / Λ) -space along whichour asymptotic limit is taken. HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION χ, η , ζ . For this purpose, it is enoughto consider the limiting behavior of the integrals I n = Z / ǫ o ( ǫ o λ ) n C Λ ( vλ ) dλ = ǫ no | v | n +1 Z | v | / ǫ o s n C Λ (ˆ vs ) ds, (5.105)for non-negative integer n . It is not difficult to show that I n ∝ ln Λ if n = 0 ǫ n if n > . (5.106)Therefore the asymptotic limits of χ, η , ζ are given by χ → (5.107) η → I ≡ | v | Z ∞ C Λ ( s ˆ v ) ds (5.108) ζ → I . (5.109)The limiting drift vector and diffusion tensor are now simple to obtain. For u we have u → v · ∂ x . (5.110)Similarly, D is given by D → π I vv . (5.111)The dyad I can be simplified further by using the fact that when Λ is large, C Λ ( d ) ≈ πe −| d | for | d | > Λ − . In fact, if the Debye screened potential was not regularized, this would not be an HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION I ≈ − | v | (cid:18) Z ∞ Λ − C Λ ( s ) s ds (cid:19) ( id − ˆ v ˆ v )= 2 π | v | (cid:18) Z ∞ Λ − e − s s ds (cid:19) ( id − ˆ v ˆ v ) → π ln Λ | v | ( id − ˆ v ˆ v ) . (5.112)After restoring units, we can therefore write the limiting drift vector and diffusion tensor as u = v · ∂ x (5.113) D = ω p ln Λ8 π Λ U vv = D L , (5.114)where the dyad U = | v | ( id − ˆ v ˆ v ) . Comparing these expressions with Eq. (5.91) reveals that wehave indeed recovered the classical result for the Lorentz plasma Fokker-Planck equation.This result assumes the scalings given in Eqs. (5.103) and (5.104), but many other choices seemallowable. Therefore a natural question is whether or not the form of the limiting Fokker-Planckequation depends on how we send the small parameters to zero. It turns out that there are onlytwo possible limiting equations, and these differ only in the limiting drift vector; either the freestreaming term survives the limit or it doesn’t. We chose our scaling so that the free streamingterm survives because this seems to be the most interesting possible scenario. The Hamiltonian collision operator slowly produces energy
While the Fokker-Planck equation (5.90) is manifestly Hamiltonian, it does not conserve kineticenergy. This can be seen by direct calculation. The total electron kinetic energy is given by E = Z H o f Ω , (5.115) HAPTER
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Ω = dx dv is the Liouville volume form. The rate of change of the plasma kinetic energy istherefore ddt E = Z H o ∂ t f Ω= Z H o div ( − f u HL + D HL · df ) Ω= − Z dH o · D HL · df Ω= Z div ( D HL · dH o ) f Ω , (5.116)where we have used the fact that h s i only depends on v and D HL is symmetric. This expressionshows that the only way kinetic energy will be conserved regardless of initial conditions is ifdiv ( D HL · dH o ) = 0 . However,div ( D HL · dH o ) = (cid:18) q e m e (cid:19) ∇ v · (cid:18) Z τ (1 − t/τ ) v · C ( vt ) dt (cid:19) = − (cid:18) q e m e (cid:19) ∇ v · (cid:18) Z τ (1 − t/τ ) vC ′′ ( | v | t ) dt (cid:19) = (cid:18) q e m e (cid:19) | v | τ | v | ∂∂ | v | (cid:18) | v | [ C (0) − C ( | v | τ )] (cid:19) ≈ (cid:18) q e m e (cid:19) C (0) | v | τ , (5.117)where the last line is valid when | v | τ > λ D + . It follows that d E dt = 0 and that the characteristictime for energy change is τ e = |E || d E /dt | ≈ τ Λ . (5.118)We will have more to say about why C HL does not conserve energy in the final section. HAPTER
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The Lorentz operator is not Hamiltonian
While C HL does not conserve energy, the limiting collision operator C L certainly does. Thereforeit is tempting to hope that the limiting procedure that transforms C HL into C L preserves the Hamil-tonian nature of C HL . In this section we will prove definitively that this is not the case. Specificallywe will show that there is no sequence of functions h k such that D L = P k X h k ⊗ X h k .The level of mathematical discourse increases substantially in this section for two reasons.First, a basic knowledge of vector bundles and operations on vector bundles is assumed. A partic-ularly readable account of this material is given in Bott and Tu (1982), starting on p. 53. Second,we assume a working knowledge of the theory of reproducing kernels. Reproducing kernels arereally nothing more than two-point covariance functions, but much can be said about them with-out referring to ideas from probability theory, and this (somewhat) justifies introducing a secondname for them. The standard reference for learning about the basics of reproducing kernel theoryis Aronszajn (1950).Suppose that M is a smooth manifold and κ : M × M → R is a smooth reproducing kernel.Associated to κ is a smooth section, α κ , of the symmetric tensor product T ∗ M ⊙ T ∗ M . The section α κ is defined as follows. Let v , v ∈ T x M be tangent vectors at x ∈ M . Choose smooth curves c , c : I → M , where I is an open interval of R containing , such that c (0) = c (0) = x , c ′ (0) = v and c ′ (0) = v . We set α κ ( v , v ) = ddǫ (cid:12)(cid:12)(cid:12)(cid:12) ddǫ (cid:12)(cid:12)(cid:12)(cid:12) κ ( c ( ǫ ) , c ( ǫ )) . (5.119)A section α of T ∗ M ⊙ T ∗ M has a reproducing kernel primitive if there is some reproducingkernel κ such that α = α κ . The purpose of this section is to prove a theorem that characterizesthe set of α ’s with reproducing kernel primitives. We will only consider sections α with locallyconstant rank . The rank of a section α at x ∈ M is defined as the codimension of the kernel of α x . HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION T αx M be the collection of vectors v x such that ∀ w x ∈ T x M, α ( v x , w x ) = 0 , (5.120)then the rank of α x is rank ( α x ) = dim( T x M ) − dim( T αx M ) . A section α has locally constantrank if for each x ∈ M there is some open neighborhood of x on which the function u ∈ M rank ( α u ) ∈ Z is constant.First we will prove that if α κ has locally-constant rank, then L X α = 0 for any vector field X that takes values in α κ ’s characteristic distribution . Given an α with locally-constant rank, itscharacteristic distribution is the subbundle of T M whose fiber at x ∈ M is given by T αx M . α ’scharacteristic distribution will be denoted T α M . A general α with locally-constant rank need notsatisfy L X α = 0 . Thus, the following theorem provides a non-trivial necessary condition for asection α with locally-constant rank to have a reproducing kernel primitive. Theorem 1.
Let κ be a smooth reproducing kernel on M . α κ is positive semi-definite as a bilinearform. Moreover, if α κ has locally constant rank, then L X α κ = 0 for each vector field X that takesvalues in T α κ M . In particular T α M is integrable in the sense of Frobenius.Proof. Let { φ j } be an orthonormal basis for the reproducing kernel Hilbert space associated with κ . The kernel κ can be expressed in terms of these basis elements as κ ( x, y ) = P j φ j ( x ) φ j ( y ) ,which implies that α κ = P j ( dφ j ) . This immediately implies that α κ is positive semi-definite asa bilinear form.Select an x o ∈ M and restrict attention to an open neighborhood U of x o where u rank ( α κu ) takes the constant value r . Suppose n is the largest integer with the property that there exists n distinct basis elements f ≡ φ j , . . . , f n ≡ φ j n that satisfy ( dφ j ) x o ∧ · · · ∧ ( dφ j n ) x o = 0 . (5.121) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION n -forms with n > m = dim( M ) vanishes, n must satisfy n ≤ m . Thereforewe must be able to find m − n additional functions f n +1 , . . . , f m so that the f i comprise a coor-dinate chart on U (it may be the case that U must be shrunk). In this special coordinate system on U , Eq. (5.121) implies that each φ j must be independent of the coordinates f n +1 , . . . , f m . Thuseach of the m − n vector fields ∂ m +1 ≡ ∂∂f m +1 , . . . , ∂ n ≡ ∂∂f n take values in the characteristicdistribution of α κ . It follows that the dimension of the characteristic distribution, m − r , must begreater than or equal to m − n , i.e. n ≥ r . In fact, n cannot be greater than r . To see this, note that α κ = α o + δα , where α o = n X i =1 ( df i ) (5.122) δα = X j j ,...,j n } ( dφ j ) . (5.123)It is straightforward to show that rank ( α o ) = n . Moreover, because α o and δα are each positivesemi-definite bilinear forms (being sums of squared -forms), r = rank ( α κ ) ≥ rank ( α o ) = n .Therefore we must have n = r . This shows that ∂ r +1 , . . . , ∂ m in fact span α κ ’s characteristicdistribution. Moreover, being pushforwards of some of the standard basis vectors in R m , these m − r vector fields commute. We have therefore succeeded in proving that the commutator ofany pair of vector fields that take values in T α κ M also takes values in T α κ M ; this is preciselyintegrability in the sense of Frobenius. Actually, we have nearly proved more than this. If X is anyvector field that takes values in T α κ M , then it must be a C ∞ ( U ) -linear combination of the vectors ∂ r +1 , . . . , ∂ m . Therefore, L X α κ = 2 X j ( L X dφ j )( dφ j )= 2 X j ( dL X φ j )( dφ j ) = 0 , (5.124) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION φ j is independent of f n +1 , . . . , f m . In otherwords, for each X that takes values in T α κ M , L X α κ = 0 . It is not hard to show that this lastproperty actually imples T α M is integrable in the sense of Frobenius.Next we will show that the necessary conditions for a constant-rank α to have a reproducingkernel primitive given in Theorem 1 are, in a particular sense, locally sufficient. Theorem 2.
Suppose α ∈ Γ( T ∗ M ⊙ T ∗ M ) is positive-semidefinite, has locally-constant rank, andthat L X α = 0 for each vector field X that takes values in T α M . Then for each x o ∈ M , there is anopen set U containing x o such that α | U = α κ U , where κ U : U × U → R is a smooth reproducingkernel on the open submanifold U .Proof. Let r be the rank of α near x o . By the Frobenius theorem we can choose coordinates f i ona neighborhood U of x o with the following property. The vector fields ∂∂f , . . . , ∂∂f r do not lie in T α M , while the vector fields ∂ r +1 ≡ ∂∂f r +1 , . . . , ∂ m ≡ ∂∂f m do. In this coordinate system, α mustbe of the form α = r X i,j =1 ( df i ) A ij ( df j ) , (5.125)where A ij is an r × r symmetric, positive definite matrix of functions on U (note that the upper limitof the double sum in Eq. (5.125) is r ≤ m ). By assumption, L ∂ l α = 0 for each l ∈ { r + 1 , . . . , m } ,which implies L ∂ l α = r X i,j =1 ( df i )( df j ) ∂A ij ∂f l = 0 . (5.126) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION ( df i )( df j ) with i ≤ j are linearly independent and the matrix A ij is symmetric, theprevious equation shows that ∂A ij ∂f l = 0 , (5.127)for each i, j ∈ { , . . . , r } and l ∈ { r +1 , . . . , m } . i.e. the A ij only depend on the first r coordinatesin this coordinate system, A ij = A ij ( f , . . . , f r ) .Let V ⊂ R r be the image of the submersion π : u ∈ U ( f ( u ) , . . . , f r ( u )) ∈ R r . Withoutloss of generality, we can assume that V is open and connected. Because the A ij only depend onthe first r coordinates on U , they define a positive-definite bilinear form g on V given by g ( x , . . . , x r ) = r X i,j =1 A ij ( x , . . . , x r ) dx i dx j . (5.128)In other words, ( V, g ) is a Riemannian manifold with metric tensor g . By Nash’s embeddingtheorem, there is therefore an isometric embedding I : ( V, g ) → ( R M , g o ) where M is someinteger and g o = P Mi =1 ( dx i ) is the standard metric tensor on R M . In particular, g = I ∗ g o = M X i =1 ( ds i ) , (5.129)where s i = I ∗ x i .Now, it is simple to verify that α = π ∗ g , and therefore α = π ∗ M X i =1 ( ds i ) = M X i =1 ( dφ i ) , (5.130)where φ i = π ∗ I ∗ x i . We have therefore proved that α | U = α κ , where κ ( x, y ) = P Mi =1 φ i ( x ) φ i ( y ) is a reproducing kernel. HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION α ’s characteristic distribution is amanifold, the previous theorem can also be globalized in the following manner. Theorem 3.
Suppose α ∈ Γ( T ∗ M ⊙ T ∗ M ) is positive semi-definite, has locally-constant rank,and that L X α = 0 for each vector field X that takes values in T α M . Also assume that the leafspace of the foliation tangent to T α M is a smooth manifold with the quotient topology. Then thereis a reproducing kernel κ : M × M → R such that α = α κ .Proof. Let M α be the leaf space of the foliation tangent to α ’s characteristic distribution. Let π : M → M α be the map that sends a point in M to its corresponding leaf. Because L X α = 0 foreach X taking values in T α M , there is a unique g ∈ Γ( T ( M α ) ⊙ T ( M α )) such that α = π ∗ g .To see that g is uniquely determined by the formula α = π ∗ g , consider the following. Let w , w ∈ T x M α be a pair of vectors tangent to the leaf space at x ∈ M α . Let ˜ w , ˜ w ∈ T ˜ x M be anypair of vectors tangent to M at ˜ x ∈ M that satisfy T π ( ˜ w i ) = w i . Suppose ˜ w ′ , ˜ w ′ ∈ T ˜ x ′ M are alsolifts of the vectors w , w . Then there must be a diffeomorphism Φ : M → M that preserves theleaves of the foliation tangent to T α M (i.e. π ◦ Φ = π ) such that ˜ x ′ = Φ(˜ x ) . This diffeomorphismallows us to compare the vectors ˜ w , ˜ w and ˜ w ′ , ˜ w ′ . In particular, we can consider the differences δ = ˜ w ′ − T Φ( ˜ w ) and δ = ˜ w ′ − T Φ( ˜ w ) . We have T π ( δ i ) = T π [ ˜ w ′ i − T Φ( ˜ w i )]= w i − T ( π ◦ Φ)( ˜ w i )= w i − w i = 0 . (5.131) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION α ( ˜ w ′ , ˜ w ′ ) and α ( ˜ w , ˜ w ) . Indeed, α ( ˜ w ′ , ˜ w ′ ) = α ( δ + T Φ( ˜ w ) , δ + T Φ( ˜ w ))= α ( T Φ( ˜ w ) , T Φ( ˜ w ))= Φ ∗ α ( ˜ w , ˜ w )= α ( ˜ w , ˜ w ) , (5.132)where the last line follows from L X α = 0 . It follows that α ( ˜ w , ˜ w ) depends only on α , w , and w . Thus, g ( w , w ) = α ( ˜ w , ˜ w ) is well-defined.The bilinear form g is positive definite for if w ∈ T M α is tangent to the leaf space, g ( w, w ) = α ( ˜ w, ˜ w ) , and the right-hand-side is zero only when ˜ w is tangent to T α M , i.e. when w = 0 .The pair ( M α , g ) is therefore a Riemannian manifold. Nash’s theorem then implies that thereexists an isometric embedding I : ( M α , g ) → ( R N o , g o ) where g o is the standard euclidean metric g o = P N o i =1 ( dx i ) . We have therefore proved that α = π ∗ g = π ∗ I ∗ g o = N o X i =1 ( dφ j ) , (5.133)where φ j = π ∗ I ∗ x j . Equivalently, α = α κ where the kernel κ ( x, y ) = P N o i =1 φ j ( x ) φ j ( y ) . We will now use Theorem 1 to prove that the Lorentz diffusion tensor D L does not admit adecomposition of the form D L = P k X h k ⊗ X h k . This will constitute a proof that the classicalFokker-Planck equation for the Lorentz plasma is not Hamiltonian. Suppose that D L does admitsuch a decomposition. Then the symmetric covariant tensor on phase space α L given by α L ( X, Y ) = ( i X ω o ) · D L · ( i Y ω o ) , = X k ( X · dh k )( dh k · Y ) (5.134) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION κ ( z , z ) = P k h k ( z ) h k ( z ) . Moreover, because α L canalso be expressed as α L = ν ( | v | ) U ij dx i dx j (5.135)we also see that α L is positive semi-definite and has the constant rank whenever | v | 6 = 0 . There-fore by Theorem 1, we must have L X α L = 0 for any vector field X that takes values in the kernelof α L . On the other hand, one such X is given by X = X H o = v · ∂ x , and ( L X Ho α L )( Y , Y )= L X Ho ( α L ( Y , Y )) − α L ( Y , L X Ho Y ) − α L ( L X Ho Y , Y ) , (5.136)for arbitrary vector fields Y , Y on phase space. In particular, when Y = Y = w · ∂ x + w · ∂ v where w is a constant -component vector, ( L X Ho α L )( Y , Y ) = 2 w · U · w, (5.137)which is never zero everywhere in phase space. This contradiction implies that α L does not admita reproducing kernel primitive, which in turn implies that D L does not admit a decomposition ofthe form D L = P k X h k ⊗ X h k . Thus, the classical Fokker-Planck equation for the Lorentz plasmais not Hamiltonian. On energy-conserving Hamiltonian collision operators
While C L is not Hamiltonian and C HL does not conserve energy, perhaps there is some othercollision operator ˜ C ( f ) = div ( ˜ D · df ) that approximates C L , satisfies the Hamiltonian property, and conserves energy. The method introduced in BZQ would not be enough to find such an operator, HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION ˜ C is unlikely in the following sense. We will saythat the collision operator, div ( D · df ) in a Fokker-Planck equation conserves energy path-wiseif the corresponding diffusion tensor satisfies D · dH o = 0 , i.e. dH o is a null eigenvector of D .Note that C L conserves energy path-wise. We will show that any Hamiltonian collision operator ˜ C that conserves energy path-wise is necessarily quite different from C L , i.e. ˜ C must be a poorapproximation to C L .Let ˜ D = P k X h k ⊗ X h k be the diffusion tensor associated with the Hamiltonian collisionoperator ˜ C that conserves energy path-wise. Associated with ˜ D is the symmetric covariant tensor ˜ α = P k ( dh k ) . Because ˜ D · dH o = 0 , ˜ α · X H o = 0 . In particular, α ( X H o , X H o ) = X k { h k , H o } , (5.138)which implies that each of the h k Poisson commute with H o . Therefore, L X Ho ˜ α = 2 X k d ( { h k , H o } ) dh k = 0 . (5.139)A reading of the proof of Theorem 3 now shows that there must be a symmetric covariant tensor ˜ a defined on the space of free streaming trajectories that pulls back to give ˜ α . A free streamingtrajectory is a subset of T Q of the form γ ( x o ,v o ) = { ( x, v ) ∈ T Q | v = v o and ∃ t ∈ R s.t. x = v o t + x o } , (5.140)where ( x o , v o ) is an arbitrary point in T Q with v = 0 . The space of free streaming trajectories, F S , is simply the union of all free streaming trajectories. Because the free streaming trajectorieswith a given velocity v can be identified with points in the plane perpendicular to v , F S has thestructure of a rank- vector bundle over the -dimensional velocity space with the zero velocity HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION R o = R − { } . To be precise, F S is diffeomorphic to the subbundle of R o × R givenby F S = { ( v, x ) ∈ R o × R | x · v = 0 } . (5.141)There is also a natural projection map π : T Q → F S given by π ( x, v ) = ( v, x ⊥ ) , where x ⊥ = x − x · ˆ v ˆ v . We must have ˜ α = π ∗ ˜ a , where ˜ a is some symmetric covariant tensor on F S .We will now analyze the form of ˜ α given that it must be the pullback of ˜ a along π . Let e ( v ) , e ( v ) be orthogonal unit vectors that are everywhere perpendicular to v , i.e. e ( v ) · v = e ( v ) · v = 0 . Using these unit vectors, we can introduce local coordinates on F S , ( v, x , x ) ,given by v = v (5.142) x = x · e ( v ) (5.143) x = x · e ( v ) , (5.144)where ( v, x ) ∈ F S . If X = X x · ∂ x + X v · ∂ v is a vector on the velocity phase space, its pushforwardalong π is given by T π ( X ) = (cid:18) X x · e + X v | v | · ( x k e + | v | R e · x ) (cid:19) ∂∂x + (cid:18) X x · e + X v | v | · ( x k e − | v | R e · x ) (cid:19) ∂∂x + X v · ∂∂v , (5.145)where R = ( ∇ v e ) · e and x k = x · ˆ v . Now because ˜ α ( X , X ) = ˜ a ( T π ( X ) , T π ( X )) , (5.146) HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION ˜ a is independent of x k , we see that the components of ˜ α must become large as x k becomeslarge. Because D L does not depend on x , let alone x k , by adjusting x k , ˜ D can always be mademuch larger than D L . This rules out the possibility that ˜ D could be a good approximation to D L . We have succeeded in showing that there is a Hamiltonian Fokker-Planck equation that governs thedynamics of the Lorentz plasma. This suggests that pitch angle scattering dynamics is Hamiltonianin a stochastic sense. On the other hand, we have also shown that our Hamiltonian Fokker-Planckequation does not have an exact energy conservation law. Instead the mean kinetic energy growson a time scale proportional to τ /τ ac , where τ is the Fokker-Planck time step and τ ac = ω − p .When inquiring as to why C HL produces energy, it is useful to remember the following factabout the derivation of Fokker-Planck equations. In these derivations, one tacitly (or explicitly,as we have done) divides the time axis into τ -second intervals and studies the dynamics of anelectron on a typical such interval. Because the positions of the ions are assumed to be statisticallyindependent, this problem is reduced to studying the interaction of a single electron with a singleDebye screened ion on a τ -second time interval. Given the location of the ion, most electrons willboth begin and end their τ -second journeys well outside of the support of the ion’s potential. Eachof these electrons will not suffer any change in its kinetic energy. However, there are some electronsthat will either start or end within the support of the ion’s potential. Each of these electrons willsuffer a change in their kinetic energy as a result of either climbing out of or falling into the ion’spotential well. Thus, an ensemble of electrons will not precisely conserve its kinetic energy over a τ -second time interval.While this reasoning sheds some light on the energy-production problem, it is still not com-pletely satisfactory. Yes, an ensemble of electrons will only approximately conserve its kineticenergy over τ -second time intervals. However, it is not physically true that the ensemble’s mean HAPTER
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AMILTONIAN MECHANICS OF STOCHASTIC ACCELERATION C HL (see Eq. (5.117)). This would correspond to electron heating, which is impossible because there isno energy impinging on the Lorentz plasma.The key to a complete understanding of the energetics of CHL is a careful scrutinization of theMarkov assumption. This assumption artificially eliminates any memory an electron might have ofits past after each τ -second time interval. As a result, within the Markov model, the following non-physical dynamical process is possible. After moving for τ seconds, an electron finds itself withinthe support of an ion’s potential. Whereas this electron should begin the next τ -second interval byclimbing out of this potential, instead it forgets the locations of all ions during the previous step,and, with high probability, fails to shed any of the kinetic energy it gained. This type of unphysicalbehavior allowed within the context of the Markov model is ultimately the source of the artificialheating predicted by Eq. (5.117). Moreover, it can be shown that this heating is not present in theclassical Fokker-Planck equation for the Lorentz plasma because the probability that one of theunphysical processes just discussed occurs tends to zero as τ ac /τ tends to zero.We believe this explanation of why C HL causes slow artificial heating suggests a way to rec-oncile the stochastic Hamiltonian approach with energy conservation. The idea is to slightly relaxthe Markov assumption. In the rare event that electron ends a τ -second time interval within thesupport of an ion’s potential, then the electrostatic field it sees in the next τ -second interval shouldconsist of the the old ion’s potential plus the potential of a new ion. Each time a new ion is drawnat random, it should never been chosen so that the support of its potential intersects the electron’slocation. By allowing for this small memory effect, the unphysical electron trajectories that cause C HL to produce energy would be eliminated. Moreover, the evolution of a given electron wouldstill be given by iterated symplectic mappings on phase space, and this ought to keep things withinthe realm of stochastic Hamiltonian mechanics. A challenging, yet enticing aspect of this possibleroute to overcoming the shortcomings of this Chapter would be developing the theory of stochasticHamiltonian processes with memory. HAPTER
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