Available Energy from Diffusive and Reversible Phase Space Rearrangements
AAvailable Energy from Diffusive and Reversible Phase Space Rearrangements
E. J. Kolmes, a) P. Helander, and N. J. Fisch Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey,USA Max-Planck Institut f¨ur Plasmaphysik, 17491 Greifswald, Germany (Dated: 6 April 2020)
Rearranging the six-dimensional phase space of particles in plasma can release energy. The rearrangementmay happen through the application of electric and magnetic fields, subject to various constraints. Themaximum energy that can be released through a rearrangement of a distribution of particles can be calledits available or free energy. Rearrangement subject to phase space volume conservation leads to the classicGardner free energy. Less free energy is available when constraints are applied, such as respecting conservedquantities. Also, less energy is available if particles can only be diffused in phase-space from regions of highphase-space density to regions of lower phase-space density. The least amount of free energy is available ifparticles can only be diffused in phase space, while conserved quantities still need to be respected.
I. INTRODUCTION
Waves injected into plasma can be amplified, extract-ing energy from the plasma. Similarly, internal modeswithin the plasma can grow by extracting energy fromthe plasma. There are multiple and distinct ways inwhich this energy can be accessed, and the open ques-tion is how much energy can possibly be accessed. Theground state of the system can be defined as the state ofleast energy accessible respecting any constraints. Thefree energy or the accessible energy can then be definedas the difference between the initial state energy and theground state energy.In his classic work, Gardner calculated the free en-ergy when plasma could be rearranged while preservingthe six-dimensional phase space densities or volumes. Ifthe distribution is divided into separate phase space vol-umes, the volume preservation constraint means that it isnot possible to rearrange the system to any lower-energystate than the one in which the lowest-energy regions ofphase space are occupied by the most-populated avail-able phase space elements. Each volume carries with itits initial number of particles, so putting that volumein a six-dimensional location of lower energy preservesthat volume phase space density as well. This is knownas Gardner restacking, representing a particular phasespace rearrangement. The fully restacked phase space isthe ground state subject to phase space conservation; thereleasable energy is known as the Gardner free energy.This problem was approached using variational tech-niques by Dodin and Fisch. Dodin and Fisch also calcu-lated the free energy, including the additional constraintthat the total current be preserved. The free energy re-specting current conservation, of course, would be lessthan the Gardner free energy. Recently, Helander cal-culated the free energy, conserving like Gardner the six-dimensional phase space densities, but conserving as wellother quantities, particularly those quantities that affect a) Electronic mail: [email protected] the motion of individual particles in phase space.
Forexample, in a magnetized system, quantities like the firstor second adiabatic invariants µ and J might be con-served. The free energy constrained by conservation ofthe adiabatic invariants of the motion, of course, wouldsimilarly be less than the Gardner free energy.However, even under Hamiltonian dynamics, rear-rangements of plasma can appear not to conserve phasespace volume, when the distribution functions are viewedwith finite granularity. Viewed at finite granularity,waves can diffuse particles. Importantly, these diffusivewave-particle interactions can release particle energy tothe waves. The energy accessible via diffusive operationsin phase space is less than that accessible via restacking.The property of free energy constrained by phase spacediffusion was first posed by Fisch and Rax. The operation of diffusing particles between volumesin phase space, rather than interchanging these volumes,has actually had applicability to a rather wide rangeof problems. Mathematical treatments of rearrange-ment by diffusion describe theories of income inequality, altruism, and physical chemistry. The maximal ex-tractable energy under diffusive phase space rearrange-ments in plasma was addressed by Hay, Schiff, andFisch.
Helander’s recent calculation of the plasma free en-ergy obeying phase space conservation, with the motionof individual particles constrained by adiabatic invari-ants, now points to a natural generalization: the free en-ergy under diffusion in phase space, but with the motionof individual particles similarly constrained by adiabaticinvariants. This paper will discuss that generalization.Lest this be thought to be an academic exercise, we notethat it is realized, for instance, in the quasilinear theoryof plasma waves and instabilities with frequencies belowthe cyclotron frequency. The available energies subjectto both diffusion and adiabatic motion constraints shouldbe more limited than the available energy subject onlyto either constraint.The paper is organized as follows: Section II describesthese different available energies. Section III introducesa simple, discrete model which illustrates the different a r X i v : . [ phy s i c s . p l a s m - ph ] A p r ways in which these energies can be extracted. Section IVshows how that simple model can be modified to describea more concrete plasma system. Finally, Section V dis-cusses the context and implications of these ideas. II. THE FOUR CLASSES OF AVAILABLE ENERGY
Let ∆ W G denote the accessible energy in Gardner’srestacking problem, in which phase space volume con-servation is the only restriction. Let ∆ W G | µ denote He-lander’s available energy for the version of this problemin which one or more conservation law constraints are in-cluded. Let ∆ W D denote the maximum extractable en-ergy in the variant of Gardner’s problem in which phasespace elements must be diffusively averaged rather thanbeing exchanged (“restacked”). Finally, let ∆ W D | µ de-note the maximum extractable energy for the diffusiveproblem with conservation laws.Here, a “diffusive exchange” refers to an operation inwhich the populations f a and f b of two equal-volume re-gions of phase space are mixed so that both populationsare ( f a + f b ) / This leaves us with four distinct available energies.The condition for the plasma to be in a ground state is thesame in the restacking and diffusion problems: for anypair of equal-volume phase space elements with differentpopulations, the higher-population element must occupya region of phase space with no more energy than thelower-population element. If there is a conserved quan-tity µ (or, in general, a vector of conserved quantities µ ), then the condition is identical except that it is onlynecessary to consider pairs of phase space elements withthe same values of µ (or µ ).Although Gardner restacking, even with constraints,leads to a unique ground state, that is not the case withdiffusive exchange. It is possible through diffusive ex-change to reach more than one possible ground state fromthe same initial configuration. These ground states willoften have different energies. Note that ∆ W D and ∆ W D | µ are defined as the maximum possible extractable energy,which is the difference between the initial energy and theenergy of the lowest-energy accessible ground state.Given the same initial phase space configuration, dif-fusive exchange and restacking will typically not lead tothe same ground state, the extractable energy will dif-fer too. It is clear that the available energies withoutadditional conservation-law restrictions will always be atleast as large as their restricted counterparts. That is,∆ W G | µ ≤ ∆ W G (1)∆ W D | µ ≤ ∆ W D . (2)Moreover, phase space restacking can always access atleast as much energy as diffusive exchanges can (strictlymore, if they are nonzero). To see this, note that both µ (cid:15)f f f f f f f f f FIG. 1. Schematic of a simple discrete system, with varyingenergy and some additional coordinate µ . kinds of exchange move the system toward the sameground state condition, where the most highly popu-lated phase space volumes are at the lowest energies,but that diffusive exchanges reduce the difference be-tween the more- and less-highly populated phase spacevolumes. This reasoning is equally applicable with andwithout conservation laws, so it follows that∆ W D ≤ ∆ W G (3)∆ W D | µ ≤ ∆ W G | µ , (4)with equality holding only when both sides vanish. III. SIMPLE DISCRETE MODEL
In order to get an intuitive sense for the four availableenergies – that is, the restacking energy with and with-out a conservation law and the diffusive-exchange energywith and without a conservation law – it is helpful toconstruct a simple model that shows concretely how thedifferent energies can play out.To that end, consider a collection of discrete states,indexed by their energies and by some coordinate µ , asshown schematically in Figure 1. Associate the statesin the n th column with energy (cid:15) n , with (cid:15) i ≤ (cid:15) j ∀ i < j .One could construct a grid of this kind with any numberof columns and rows. Physically, previous work presentsthe discrete versions of these problems in two ways. First, it can model an intrinsically discrete physical sys-tem, like transitions between atomic energy levels stimu-lated by lasers. Second, it can model a system with con-tinuous phase space (e.g., a plasma) being mixed withsome finite granularity.In this scenario, it is straightforward to understandthe four available energies described in Section II. If µ isconserved, different rows are not allowed to interact, andthe total available energy (either ∆ W G | µ or ∆ W D | µ ) isthe sum of the available energies of the individual rows,considered independently. If µ is not conserved, theneither the diffusive or the Gardner energy minimizationproblem must consider the system as a whole.When constructing examples of this kind, it quicklybecomes apparent that conservation laws affect the out-come of diffusive and Gardner relaxation in qualitativelydifferent ways. Consider the following phase-space con-figuration, with two energy levels (with the left-hand col-umn corresponding to a lower energy) and two differentvalues of µ : 0 10 0 . Suppose the separation between the two energy levels is ε . Then ∆ W G | µ = ∆ W G = ε and ∆ W D | µ = ε/
2. If µ is not conserved,, the lowest diffusively accessible energycan be reached by the sequence0 10 0 → / /
20 0 → / / / W G = (3 / ε . At this point, we note that thereis a qualitative difference between discrete and continu-ous systems. In the example just given, the ground statecontradicts a theorem by Gardner that the distributionfunction can only depend on energy alone. This contra-diction is a consequence of the fact that energetically neu-tral exchanges are possible between the two low-energyboxes in the diagram. It disappears if the distributionfunction is required to be a smooth function of a contin-uous energy variable.This example worked because, in the absence of any µ constraint, diffusive processes can take advantage of theadditional phase space to more efficiently transfer ma-terial (or quanta, etc., depending on how this model isinterpreted physically) more efficiently from high-energyto low-energy states. One might imagine that this a spe-cial property of phase space that is initially unoccupied.However, similar behavior appears when the initial pop-ulation configurations for each µ are the same. Consider,for example, the following 2 × . Suppose, as before, that the rows correspond to differentvalues of µ and that the columns correspond to energystates that are separated by some energy ε . Three of thefour measures of available energy are immediately clear:∆ W G | µ = ∆ W G = 2 ε and ∆ W D | µ = ε .Hay, Schiff, and Fisch showed that there are onlya finite number of candidates for the optimal sequence of diffusive steps to relax a system to equilibrium. Afterenumerating all possible candidates, it is possible to showthat the lowest possible energy state accessible throughdiffusive steps can be reached by the sequence0 10 1 → / / → / / / → / / / / → / / / / , so ∆ W D = (5 / ε .It seems that there is some qualitative way in whichincreasing the number of accessible states improves thediffusive available energy without necessarily improvingthe Gardner available energy. However, it is nontrivialto write down a strong condition relating ∆ W G | µ , ∆ W G ,∆ W D | µ , and ∆ W D that captures this intuition.One might expect, after looking at a number of exam-ples involving small systems, that ∆ W D | µ / ∆ W D wouldalways be less than or equal to ∆ W G | µ / ∆ W G (in otherwords, that restricting the size of accessible phase spacewould always have a fractionally more severe impact onthe energy accessible via diffusive exchange). In fact, thisinequality holds for any system with two energy levels (a2 × N grid of states).To see this, first note that in a 2 × N system, ∆ W D | µ =(1 / W G | µ . If only two cells can be exchanged, thenit is either favorable to exchange them (in which casethe Gardner exchange moves the difference between thecells’ populations from one to the other and the diffusiveexchange accomplishes half that) or it is not (in whichcase neither does anything).Now consider the same system without any µ con-straints. It is possible to divide the initial populationvalues { f ij } into the N largest and N smallest values –or, more precisely, into equally large sets A and B suchthat for any a ∈ A and b ∈ B , f a ≥ f b . These setscan then be subdivided into the elements starting in thelower-energy state ( A and B ) and those starting in thehigher-energy state ( A and B ). A and B will havethe same number of elements, so every element of A canbe paired with one unique element of B . If each of thesepairs are exchanged, then all members of A will be in thelower-energy state, and the system will be in a groundstate, having released energy ∆ W G = || A || ε .If those same pairs of cells are diffusively averagedrather than being exchanged, the released energy will be(1 / W G . This will often not be the optimal diffusivestrategy, but it demonstrates that ∆ W D ≥ (1 / W G ,which is enough to show (for the 2 × N case) that∆ W D | µ / ∆ W D ≤ ∆ W G | µ / ∆ W G .However, this inequality does not always hold for sys-tems with more than two allowed energy levels. Provingfor a particular case that ∆ W D | µ / ∆ W D > ∆ W G | µ / ∆ W G is often somewhat involved; given the other three avail-able energies, it requires establishing an upper bound for∆ W D . Consider the following configuration, with threeenergy levels and two allowed values for µ : X + 1 X
00 2 1 . Suppose the three columns correspond to energies 0, ε ,and 2 ε , and consider the limit where X → ∞ .When µ conservation is enforced, the upper row is in itsground state, does not contribute to ∆ W G | µ or ∆ W D | µ .The contribution from the lower row gives ∆ W G | µ = 3 ε .Applying the criteria for extremal sequences from Hay,Schiff, and Fisch for a three-level system, and checkingall possible candidates, it is possible to show that theoptimal sequence respecting µ conservation is X + 1 X
00 2 1 → X + 1 X / / → X + 1 X / / / , so that ∆ W D | µ = (7 / ε .In the absence of µ conservation, the Gardner availableenergy is ∆ W G = ( X + 1) ε . In the limit where X is verylarge, the only thing that will determine ∆ W D will bethe fraction of X that can be moved from the second col-umn to the first. There is no strategy that can move morethan half of the content of one cell to another using diffu-sive exchanges; in other words, lim X →∞ ∆ W D = ( X/ ε .Then, for this example, ∆ W D | µ / ∆ W D > ∆ W G | µ / ∆ W G . IV. EXAMPLE: INHOMOGENEOUS MAGNETIC FIELD
In many scenarios, the kind of simple phase-space gridconsidered in the previous section may need to be mod-ified. However, much of the intuition remains the same.One case that was discussed by Helander for the contin-uous restacking problem was a plasma in an inhomoge-neous magnetic field, with conservation of the first adi-abatic invariant µ = mv ⊥ / B . To illustrate the fourfree energies, consider for simplicity the case where thephysical volume is divided into two halves: one in which B = B = const; and another in which B = B = const,with B > B . Suppose the field is straight and its direc-tion does not vary, and suppose the energy in the direc-tion parallel to the field can be ignored. For simplicity,we consider a discrete version of this system. The dis-crete version can be constructed by averaging the plasmadistribution function over finite regions of phase space,and then restricting the Gardner and diffusive exchangeoperations to act on these macroscopic regions.There are two major ways in which this scenario differsfrom those discussed in Section III. First, for any given µ ,the volume of accessible phase space is now proportionalto B . This can be shown by calculating the appropriateJacobian determinant. Intuitively, it can be understoodin terms of the geometry of phase space when µ is con-served. For a given B and a given µ (or a given smallrange in µ ), the allowed region in phase space traces out acircle (or thin ring) in the v ⊥ = v x × v y plane. If µ is held (1 / µ (3 / µ (5 / µ (7 / µ (cid:15)µ FIG. 2. Schematic of the discrete restacking problem for aspatial region divided between an area with constant field B and an area with constant field B = 2 B . Dashed linesseparate different values of µ . For each value of µ , the singlebox occupies the phase space on the low-field side and thepair of boxes occupy the phase space on the high-field side. fixed and B is changed from B to B , then the regiontransforms to become a ring with a larger radius, with acorrespondingly larger phase-space volume. The discretesystem is composed of a series of boxes with equal phase-space volumes. Therefore, for any given µ , there will bea larger number of phase-space boxes on the higher-field,higher-energy side of the system than on the lower-field,lower-energy side.In addition, there is now a coupling between the in-variant µ and the energy ε . For any given value of B , ε ∝ µ . Moreover, if a particle has energy ε on thelower-field side, and if it is then moved to the higher-field side without changing µ , it must then have energy ε = µ ( B /B ) ε . When µ is conserved this is a two-energy-level system, but for different values of µ the dif-ference between the two energy levels will be different. Asa result, for this scenario, analogous to Figure 1, Figure 2represents the magnetic field constraint.This picture can be used to illustrate the behavior dis-cussed by Helander. For example, if µ is conserved, thenany distribution that is a function of µ alone is a groundstate, and such states will have spatial densities that areproportional to B . This is in some sense counterintuitive,since for any given µ a higher field means a higher energy.However, the intuition can be restored with reference toFigure 2. If f is a function of µ alone, then each of theboxes in a given row must have the same population. Itis then clear that distributions of this kind will give upno energy not only under both µ -conserving exchanges,but also under µ -conserving diffusive exchanges.Moreover, it is clear why the total spatial density varia-tions in such a state must be proportional to B : there areproportionally more boxes on the higher-field side thanon the lower-field side for each choice of µ , and thereforethere must be a proportionally higher total population(in other words, the sum of the population over all µ fora given side of the system will be proportional to B ).Helander also showed that f = f ( µ ) ground states willhave temperatures proportional to B ( x ). This can beunderstood in terms of Figure 2 in a similar way. Sup-pose f = f ( µ ). Then if the distribution function hassome structure at an energy ε on the low-field side, itmust have the same structure at energy ( B /B ) ε on thehigh-field side, since for a given µ the low-field and high-field regions of phase space have energies proportional totheir local values of B . Given that the density of vol-umes is higher in the high field region proportionately to( B /B ), it follows that the pressure in high field regionsis then ( B /B ) the pressure of the low field region.This is, of course, only one particular ground state,which occurs when all accessible states have equal occu-pation for each µ . It is also possible to release free energywhen the initial states are populated differently in sucha way that there is population inversion, ( ∂f /∂(cid:15) ) µ ≥ , , ε free en-ergy, whereas the diffusive solution allows only (3 / ε freeenergy (where ε is the energy gap between the states).More generally, the difference between the diffusive andrestacking solutions is similar to what was discussed inSection III. However, the details do turn out somewhatdifferently. For instance, recall the first example fromSection III: 0 10 0 . It is possible to construct a six-cell analog of this, withone populated cell on the high-field side and two possiblechoices of µ , with a structure along the lines of the firsttwo rows of Figure 2. It might look something like this:0 10 0 00or this: 0 00 0 10 . But now the solutions will be different. For instance,it now makes a difference to all four available energieswhether the populated cell was for the larger or smallervalue of µ , since these now have different energies anddifferent gaps between their energies and the low-fieldstates at the same µ . V. SUMMARY AND DISCUSSION
Depending on what processes act on a plasma, anddepending on how they are constrained, there are dif-ferent ways of quantifying the energy that can be ex-tracted from the plasma. This paper has discussed andcompared four such available energies: the unconstrainedrestacking energy (∆ W G ), the restacking energy withconservation laws (∆ W G | µ ), the diffusive-exchange en-ergy (∆ W D ), and the diffusive-exchange energy with con-servation laws (∆ W D | µ ). The first three of these had beendiscussed previously in the literature. The last is first ex-plicitly described here but is implicit in any quasilineartheory of low-frequency waves and instabilities, includinggyrokinetics.Some facts about the relative sizes of these energies arestraightforward; for instance, imposing a conservation-law constraint always reduces the available energy orleaves it unchanged, and diffusive exchange can never ac-cess more energy than Gardner restacking. However, thedetails of how these constraints play out can be complex,as we show. A specific example of an inhomogeneousmagnetic field illustrates how these energies differ.These energies are best thought of as upper boundsrather than as predictions for how much energy a givenprocess actually will extract from a distribution. Whichof these four upper bounds is most appropriate in a spe-cific setting will depend on the details of the physicalprocess being considered. The inclusion or omission ofconserved quantities like the adiabatic invariants are rel-atively straightforward to determine for a given physicalscenario. The difference between diffusion and restackingis more subtle, but should depend on the different char-acteristic scales in phase space. For example, if diffusionwere accomplished by injecting certain waves, then thecharacteristic scales in phase space might be set by thecharacteristic scales associated with the applied fields. ACKNOWLEDGMENTS
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