A Generalized Boltzmann Kinetic Theory for Strongly Magnetized Plasmas with Application to Friction
AA Generalized Boltzmann Kinetic Theory for Strongly Magnetized Plasmaswith Application to Friction
Louis Jose and Scott D. Baalrud
Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242,USA (Dated: 17 August 2020)
Coulomb collisions in plasmas are typically modeled using the Boltzmann collision operator, or its variants,which apply to weakly magnetized plasmas in which the typical gyroradius of particles significantly exceedsthe Debye length. Conversely, O’Neil has developed a kinetic theory to treat plasmas that are so stronglymagnetized that the typical gyroradius of particles is much smaller than the distance of closest approachin a binary collision. Here, we develop a generalized collision operator that applies across the full range ofmagnetization strength. To demonstrate novel physics associated with strong magnetization, it is used tocompute the friction force on a massive test charge. In addition to the traditional stopping power component,this is found to exhibit a transverse component that is perpendicular to both the velocity and Lorentz forcevectors in the strongly magnetized regime, as was predicted recently using linear response theory. Goodagreement is found between the collision theory and linear response theory in the regime in which both apply,but the new collision theory also applies to stronger magnetization strength regimes than the linear responsetheory is expected to apply in.
I. INTRODUCTION
It is well known that magnetic fields influence thetransport of energy, momentum and particles in plas-mas. However, most plasmas are weakly magnetized inthe sense that the gyroradius of particles is much largerthan the Debye length. In this situation, the influenceof the magnetic field during Coulomb collisions is neg-ligible. In contrast, if the magnetic field is sufficientlystrong that the gyroradius of particles is smaller thanthe Debye length, the trajectories of colliding particlesare influenced by the magnetic field during a Coulombcollision. In this strongly magnetized regime, kinetic the-ory must account for gyromotion at the microscopic col-lision scale; i.e., within the collision operator. This isa formidable challenge in part because there is no ana-lytic solution of the two-body problem in the presence ofa magnetic field. Despite this difficulty, understandingthe influence of strong magnetization on transport pro-cesses remains interesting both from the point of view offundamental physics as well as its importance in manyexperiments, such as magnetic confinement fusion , non-neutral plasmas , ultracold neutral plasmas , magne-tized dusty plasmas , trapped antimatter and naturallyoccurring plasmas in planetary magnetospheres . Thiswork presents a generalized plasma kinetic theory thatapplies to conditions spanning from weak to strong mag-netization regimes.Regimes that characterize the influence of magne-tization on transport can be identified by comparingthe gyroradius ( r c = (cid:112) k B T /m/ω c ) with the Debyelength ( λ D = (cid:112) k B T / πe n ), Landau length times √ r L = √ e /k B T ) and Coulomb collision meanfree path ( λ col ) . Here, we apply the generalized ki-netic theory to compute the friction force on a mas-sive test charge moving in a magnetized one-component plasma (OCP). The magnetized OCP is described bytwo dimensionless parameters: the Coulomb couplingstrength, Γ = ( e /a ) / ( k B T ), where a = 3 / (4 πn ) isthe Wigner-Seitz radius and the magnetic field strength β = λ D /r c = ω c /ω p , where ω c = eB/mc is the gyrofre-quency and ω p = (cid:112) πe n/m is the plasma frequency.For a weakly coupled plasma (Γ < are summarized in table I.In traditional plasma kinetic theory, the transportproperties are obtained from the Boltzmann equation: ∂ t f + v · ∂ X f + em [ E + v /c × B ] · ∂ v f = C , in which theinteractions between particles occurring at microscopicscales are described by the collision operator ( C ). Thetraditional theory assumes that the gyromotion of theinteracting particles occurs at a length scale that is muchlarger than the Debye scale volume within which binarycollisions occur. As a result, the collision operator is in-dependent of the magnetic field, and the theory appliesto the unmagnetized and weakly magnetized regimes.O’Neil has developed a kinetic theory that treats the op-posite limit, in which particles are so strongly magnetizedthat the gyroradius is smaller than the Landau length.This theory considers single component plasmas andmakes use of a property of adiabatic invariance in a col-lisions that applies in the extremely magnetized regime(region 4). Other techniques to treat strongly magne-TABLE I: Four different transport regimes of a weaklycoupled one-component plasma r c > λ col λ D < r c < λ col r L < r c < λ D r c < r L < λ D a r X i v : . [ phy s i c s . p l a s m - ph ] A ug tized plasmas include Fokker-Planck equations , bi-nary collisions using perturbation theory , general-ized Lenard-Balescu theories and guiding-center ap-proximations .In this work, we present a generalized Boltzmann ki-netic theory that treats all four magnetization regimes.In this approach, a microscopic collision volume is definedin the coordinate space where the collisions occur. Col-liding particles enter the collision volume, interact andscatter out of it, having exchanged momentum and en-ergy inside the collision volume. The traditional Boltz-mann collision operator is obtained in the weak magneticfield limit and O’Neil’s collision operator is obtained asthe high magnetic field limit.The generalized collision operator predicts that strongmagnetization gives rise to new transport properties thatare not predicted by the traditional Boltzmann equation.To illustrate this, it is used to compute the friction forceon a massive test charge moving through a backgroundplasma in the presence of an external magnetic field ofvarying strength. The common understanding based onthe traditional kinetic theory is that the friction force actsantiparallel to the velocity vector of the test charge .The friction force is also commonly referred to as stop-ping power. The stopping power determines the energyloss of the projectile and has implications for the energydeposition of fusion products .The generalized kinetic theory predicts that a qual-itatively new effect arises when the plasma is stronglymagnetized. Instead of being aligned antiparallel to thevelocity vector, the friction force is predicted to shift,gaining a transverse component that is perpendicular tothe velocity vector in the plane formed by the velocityand magnetic field vectors. This transverse componentalters the trajectory and the stopping distance of theprojectile . It’s sign is predicted to change if the testcharge speed is either faster or slower than approximatelythe thermal speed of the plasma species with which itpredominately interacts. It acts to increase the gyrora-dius of fast particles and to decrease the gyroradius ofslow particles. Since the transverse force is perpendicu-lar to the velocity, it does not decrease the energy of theprojectile and is unnoticed in the conventional way of ob-taining the friction force from the energy loss of the pro-jectile . The existence of this transverse frictionforce was recently predicted using a different approachto kinetic theory based upon a linear dielectric responseformalism and was confirmed using first-principles MDsimulations . The assumptions inherent to the linearresponse approach limit its applicability to regimes 1-3in table 1. Good agreement between the two approachesis demonstrated throughout this range of conditions, butthe new collision based approach extends the range of ap-plicability because it also applies in the extremely mag-netized regime (region 4).The outline of the paper is as follows. In Sec. II, thegeneralized collision operator is derived and the tradi-tional Boltzmann collision operator and O’Neil’s collision operator are obtained in the appropriate asymptotic lim-its. Section III describes the theoretical formulation andnumerical evaluation of the friction force on a single testcharge. Section IV discusses the results of these com-putations in regimes of weak, strong and extreme mag-netization, along with a comparison of theories. SectionV gives a qualitative description of the transverse forcefrom the binary collision perspective. II. GENERALIZED COLLISION OPERATOR
Derivations of the Boltzmann equation begin from ageneral description of the dynamics of N interacting par-ticles, but then apply a series of approximations to focuson average quantities of interest, and to make the prob-lem tractable by invoking properties of the dilute gas (orplasma) limit. Here, we follow a traditional derivationdue to Grad , insofar as it applies to arbitrary mag-netization. This includes defining reduced distributionfunctions, and making use of the dilute limit to justifybinary collisions, the molecular chaos approximation ,and local collisions that happen at microscopic space andtime scales. The departure from the traditional deriva-tion comes about by not making approximations of acertain geometry for the collision volume that are jus-tified only in the absence of strong magnetization, andby accounting for the Lorentz force when computing thebinary collision dynamics inside that volume. This leadsto a more general, but more computationally intensive,kinetic equation. It reduces to either the traditional re-sult or O’Neil’s result in the appropriate limits.Grad’s derivation begins from Liouville’s the-orem , which describes the phase space evo-lution of the N -particle distribution function( f ( N ) ( r , r · · · r N , v , v · · · v N , t )), but this descriptionis a complex computational problem because of thelarge number of degrees of freedom brought by thehuge number of particles in the system. As a way toreduce the computational complexity of the problemand to focus on physical processes of interest, reduceddistributions are defined by integrating a subset ofthe degrees of freedom. This results in the BBGKYhierarchy. These equations are not closed becausethe evolution equation for the n particle distribution( f ( n ) ( r , r · · · r n , v , v · · · v n , t )) contains the n + 1particle distribution.The Boltzmann equation can be derived from theBBGKY hierarchy by applying the closure f (3) = 0,which drops triplet correlations. Although this methodis accurate for particles interacting via short-range po-tentials, such as neutral gases, it leads to an infrared di-vergence in plasmas due to the long-range nature of theCoulomb interaction . This is usually corrected by intro-ducing an ad hoc cutoff to the impact parameter at theDebye length to model Debye screening. Recent work using a new closure of the BBGKY hierarchy has shownthat expanding about the deviations of correlations fromtheir equilibrium values,∆ f (3) = f (3) − f (3) o f (2) /f (2) o (1)rather than in terms of the strength of correlations, f (3) ,ensures that the exact equilibrium properties are main-tained at all orders of the hierarchy, including screening.Here, f o is the equilibrium distribution function. Thisexpansion shows that binary collisions occur via the po-tential of mean force , rather than the bare Coulombpotential. The potential of mean force asymptotes to theDebye-H¨uckel potential in the weakly coupled limit φ ( r ) = e e r e − r/λ D , (2)which is the only case we consider here. Here, r = | r | = | r − r | is the distance between the particles and e , e are the charges of the particles. We make use of thisnew closure, but otherwise follow Grad’s derivationof the Boltzmann equation.In this approach, the coordinate space is divided intoa microscopic volume where the collisions occur and anoutside region where no collisions occur. In this way,the particle distribution function is divided into two: ashort range component in which the microscopic colli-sions occur and a long range component (truncated dis-tribution function) which is uniform on the microscopiclength scale. In the small collision volume limit, the trun-cated distribution function is the observable particle dis-tribution function. The collision operator obtained afterfollowing these arguments is C = (cid:90) d v (cid:73) S ds u · ˆs f (2) ( r , r , v , v , t ) , (3)where u = v − v is relative velocity between the collid-ing particles and f (2) ( r , r , v , v , t ) is the two particledistribution function. The surface integral is defined bythe small region of space in which the two particles in-teract (collision volume). Here, ds is the infinitesimalarea on the surface and ˆs is the unit normal of the areaelement. The collision volume can be visualized in therelative frame with the coordinate system fixed to par-ticle 2 with particle 1 entering the collision volume asshown in Fig. 1.Depending on the sign of u · ˆs , the surface integral issplit into two terms, representing contributions from twosurfaces S + ( u · ˆs >
0) and S − ( u · ˆs < S + cor-respond to particles moving away from each other (post-collision state) and points on S − correspond to particlesmoving towards each other (precollision state). Assum-ing binary collisions occurring via the mean force withinthe interaction volume, the second order BBGKY equa-tion with the ∆ f (3) = 0 truncation provides a solutionwhereby f (2) ( r , r , v , v , t ) is a constant along the twoparticle trajectory. Thus, for points on the S + surface, f (2) can be replaced with the post collision coordinates: f (2) ( r , r , v , v , t ) → f (2) ( r (cid:48) , r (cid:48) , v (cid:48) , v (cid:48) , t ), where the FIG. 1: Illustration of a collision volume surroundingparticle 2 ( r ) during an interaction with particle 1 ( r ).variables with prime( (cid:48) ) represent postcollision states .On making these changes, the collision operator becomes C = (cid:90) d v (cid:90) S + ds | u · ˆs | f (2) ( r (cid:48) , r (cid:48) , v (cid:48) , v (cid:48) , t ) − (cid:90) d v (cid:90) S − ds | u · ˆs | f (2) ( r , r , v , v , t ) . (4)In order to get an explicit form of the collision op-erator, the integrals can be written over the same in-tegration range by the change of variable: ˆs → − ˆs inthe first integral, changing S + to S − . Further, we makethe assumption of molecular chaos ( Stosszahlansatz ): f (2) ( r , r , v , v , t ) = f (1) ( r , v , t ) f (1) ( r , v , t ), where f (1) ( r , v , t ) is the one particle distribution function . Wealso assume that the collision volume is small. This isjustified by the short timescale of the collision comparedto the larger timescale of evolution of f (1) ( r , v , t ). Inthis limit of a local collision r (cid:48) , r (cid:48) and r can be approx-imated by r . On making these approximations, ageneralized collision operator is obtained C = (cid:90) d v (cid:90) S − ds | u · ˆs | ( f (cid:48) f (cid:48) − f f ) , (5)in which the following abbreviated notations have beenapplied f (cid:48) ≡ f (1) ( r , v (cid:48) , t ) , f (cid:48) ≡ f (1) ( r , v (cid:48) , t ) ,f ≡ f (1) ( r , v , t ) , f ≡ f (1) ( r , v , t ) . Equation (5) is the expression for the generalized colli-sion operator. This result has been obtained in manyprior works during the path to derive the traditionalBoltzmann equation. The novelty here is to evaluateEq. (5) directly, rather than proceed to simplify it byinvoking arguments associated with either the weak orextreme magnetization limits. In order to evaluate thisexpression, the post collision velocities ( v (cid:48) and v (cid:48) ) needto be evaluated. This involves solving the two body dy-namics of the colliding particles inside the collision vol-ume for the initial velocities ( v and v ).Equation (5) is a 5-D integral: 3-D velocity space vol-ume and a 2-D surface in the coordinate space. Thesurface integral encloses a small region where collisionsoccur that is determined by the range of the potential ofmean force; in this case, the Debye length. The integralcan be viewed as summing over all possible configura-tions in which particle 1 enters the collision volume andinteracts with particle 2, weighted by the postcollisionand the precollision velocity distributions. The surfaceintegral counts all the possible orientations in which theparticle enters the collision volume and the velocity inte-gral counts all the possible velocities of particle 2. Lim-iting the surface integral to the surface S − makes sureonly the precollision states are counted.Since we focus on weakly coupled plasmas for whichthe potential of mean force is the Debye-H¨uckel potential,the collision volume is characterized by the Debye lengthscale. The equations of motion for two charged particleswith masses m and m and charges e and e interactingin a uniform magnetic field B within the collision volumeare m d v dt = −∇ r φ ( r ) + e (cid:16) v c × B (cid:17) (6) m d v dt = −∇ r φ ( r ) + e (cid:16) v c × B (cid:17) . (7)Since the Debye-H¨uckel potential depends only on thedistance between the particles, it is useful to change thevariables to the center of mass, R = m r + m r m + m , (8) V = m v + m v m + m , (9)and the relative frame, r and u . Under this transforma-tion, the equations of motion for the center of mass andthe relative velocities are( m + m ) d V dt = m (cid:16) u c × B (cid:17)(cid:16) e m − e m (cid:17) +( e + e ) (cid:16) V c × B (cid:17) , (10) m d u dt = −∇ φ ( r ) + m (cid:16) u c × B (cid:17)(cid:16) e m + e m (cid:17) + m (cid:16) V c × B (cid:17)(cid:16) e m − e m (cid:17) . (11)These equations of motion describe how the precollisionstates transform to the postcollision states within the col-lision volume. The initial positions of the colliding par-ticles in the relative frame correspond to the surface ofthe collision volume. For initial conditions correspond-ing to a precollision state, u · ˆs <
0, the equations ofmotion are solved within the collision volume to obtainthe postcollision velocities associated with the location atwhich the particles leave the collision volume. However,as indicated in Fig. 1, the precise shape of the collisionvolume can be arbitrary. The only relevant characteris-tic is that it must be significantly larger than the rangeof the potential of mean force . As long as this condi-tion is met, both the precollision and postcollision states correspond to a condition in which there is negligible in-teraction between the particles. In practice, choosing acollision volume that is much larger than the range of thepotential of mean force increases the computational costassociated with evolving the particle trajectories, but itdoes not alter the momentum exchanged, which is theinput to the collision operator resulting from trajectorycalculation.Although the general case does not require a specifiedvolume, certain limits create symmetries that can be usedto simplify the problem through the specification of adefinite collision volume. We next consider two limitingcases: the traditional Boltzmann collision operator andO’Neil’s collision operator. A. Weakly magnetized limit: Boltzmann equation
In the unmagnetized and weakly magnetized regimes,the effect of the magnetic field in the collision volume isnegligible. The dynamics of the colliding particles aremodeled using the Debye-H¨uckel potential alone. Thespherical symmetry of the Debye-H¨uckel potential sug-gests that a sphere is an appropriate collision volume.FIG. 2: A spherical interaction volume around particle2 ( r ) in the presence of a repulsive interaction withparticle 1 ( r ). The disk surface is perpendicular to theprecollision relative velocity ( u ). Each point on thehemisphere ( S − ) can be projected to a point on the diskas shown making a one to one correspondence.The usual form of the collision operator can be ob-tained by introducing a plane perpendicular to the rela-tive velocity u of the colliding particles and intersectingthe sphere along a diameter with the coordinate systemcentered at particle 2 ( r ) as shown in Fig. 2. The S − surface is a hemisphere and the projection of it on thisplane is a disk surface. Points on this disk surface have aone-to-one correspondence with the points on the hemi-sphere. This enables a transformation of the integrationsurface from the hemisphere to the disk surface. Using b db dφ as the area element of the polar coordinates onthis disk, the collision operator can be recast as C = (cid:90) d v (cid:90) b db dφ u ( f (cid:48) f (cid:48) − f f ) , (12)where u = | u | and b can be identified as the impactparameter from Fig. 2. Substituting b db dφ = σd Ω,where d Ω = sin θdθdφ is the differential scattering crosssection and d Ω is the solid angle, we get C = (cid:90) d v (cid:90) σd Ω u ( f (cid:48) f (cid:48) − f f ) . (13)This is the traditional Boltzmann collision operator. Itis much simpler to evaluate than the general form ofEq. (5) because in the absence of a Lorentz force insidethe collision volume, classical mechanics provides a closedform expression for the differential scattering cross sec-tion from the scattering angle σ = b sin θ (cid:12)(cid:12)(cid:12)(cid:12) dbdθ (cid:12)(cid:12)(cid:12)(cid:12) (14)where θ = π −
2Θ andΘ = b (cid:90) ∞ r dr r (cid:20) − b r − φ ( r ) m u (cid:21) − / (15)is the scattering angle and r is the distance of closestapproach, obtained by finding the root of the denomina-tor of the integrand. In this case, the problem reduces tosolving the scattering angle integral (Eq. (15)). This ismuch simpler than the general case of solving the equa-tions of motion (Eqs. (10) and (11)) of the collidingparticles inside the collision volume, which are coupledordinary differential equations. B. Extremely magnetized limit: O’Neil equation
O’Neil developed a Boltzmann-like collision operator that accounts for the collisions between particles of a one-component plasma in the extremely magnetized regime(region 4). This was later used to calculate the tempera-ture anisotropy relaxation rate of a non-neutral plasma,and the predicted relaxation rate was validated experi-mentally . In this subsection, we show that O’Neil’sresults can be obtained from the generalized collision op-erator in the extremely magnetized limit by choosing thecollision volume to be a cylinder.The one-component plasma is a special case becausewhen the charge-to-mass ratio is the same, the centerof mass motion and relative motion are decoupled (Eqs.(10) and (11)). The resulting equations of motion in therelative frame are equivalent to that of a charged particlein a uniform magnetic field scattered by the potentialat the origin. The resulting trajectory is that of a helixbefore and after the collision, but where both the paralleland gyromotion can change due to the collision. The r r FIG. 3: Cylindrical interaction volume around particle2 ( r ) during an interaction with particle 1 ( r ) in thepresence of a strong magnetic field.natural geometry characterizing this motion is a cylinder,as depicted in Fig. 3.For this geometry of the collision volume, Eq. (5) takesthe form C = (cid:90) d v (cid:90) ρdρdθ | u · ˆz | ( f (cid:48) f (cid:48) − f f )+ (cid:90) d v (cid:90) ρdθdz | u · ˆ ρ | ( f (cid:48) f (cid:48) − f f ) . (16)The first term corresponds to collisions in which parti-cle 1 enters through the circular surface (blue), and thesecond term corresponds to collisions in which particle 1enters through the cylindrical surface (red) depicted inFig. 3. When the plasma is extremely magnetized, thegyroradius of particles is so small compared to the sizeof the collision volume ( λ D ) that motion is restricted toremain close to the guiding centers and scattering per-pendicular to the initial guiding centers is minimal. Inthis case, the contribution to the collision operator fromthe second term is negligible. In this limit C = (cid:90) d v (cid:90) ρdρdθ | u · ˆz | ( f (cid:48) f (cid:48) − f f ) . (17)This is O’Neil’s collision operator for extremely magne-tized plasmas . As there is no closed-form solution of theequations of motion of colliding particles in the presenceof a magnetic field, they are solved numerically to find thepostcollision states of the particles, except in the asymp-totic case of a very large magnetic field . Even thoughthe evaluation of the transport coefficients using O’Neil’stheory is more difficult than the traditional Boltzmanntheory, it is simpler than the general theory because theequations of motion are decoupled making the numer-ical calculation of the trajectories less computationallyexpensive. III. FRICTION FORCE
The generalized collision operator can be used to com-pute the macroscopic transport properties of the plasmain all magnetization strength regimes. To illustrate this,we compute the friction force acting on a massive projec-tile, taken to be a single test charge, moving through amagnetized one-component plasma. Understanding howfriction is modified in the presence of a strong magneticfield is fundamentally important and it also has directimplications in many magnetized plasma experimentssuch as non-neutral plasmas , ultracold neutral plas-mas , magnetic confinement fusion and naturally occur-ring plasmas in planetary magnetospheres . It is also thefundamental process controlling macroscopic transport ofmomentum.A projectile moving through the plasma is acted uponby friction in addition to the Lorentz force from the exter-nal magnetic field. The friction force is due to Coulombcollisions with the background plasma. In the unmag-netized and weakly magnetized regimes, the Boltzmannequation predicts that the friction force is antiparallelto the velocity of the projectile and is commonly knownas the stopping power . It was recently predicted thata qualitatively new effect occurs in strongly magnetizedplasmas: the friction force obtains a transverse compo-nent that is perpendicular to the velocity vector of theprojectile, in the plane formed by v and B . This predic-tion was made using linear response theory, and was latertested using molecular dynamics simulations . Here, wetest the generalized collision operator by comparing withthe predictions of linear response theory in the stronglymagnetized regime. We also compute the friction forcein the extremely magnetized regime, showing that thetransverse component of the friction force exists, and islarge, in this regime. This extends the regime of mag-netization over which this phenomenon has been studiedbecause the linear response theory from is not expectedto apply in the extremely magnetized regime due to aclose-collision cutoff in that theory. A. Theory
Consider a massive projectile slowing down on a mag-netized one-component plasma. Since the projectile isvery massive compared to the background plasma, thegyromotion of the projectile happens at a larger spa-tial scale than the size of the collision volume. Whilethe Lorentz force significantly influences the backgroundplasma, it has a negligible influence on the massive testcharge during the collision and can therefore be accu-rately excluded from the equations of motion for the testcharge. The equations of motion from Eqs. (10) and (11)then reduce to( m + m ) d V dt = e (cid:16) V c × B (cid:17) − e m m (cid:16) u c × B (cid:17) (18) m d u dt = −∇ φ ( r ) + e m m (cid:16) u c × B (cid:17) − e m m (cid:16) V c × B (cid:17) . (19) Here the charge on the massive projectile is taken to bethe same as the charge of the background plasma parti-cles ( e = e = e ). The friction force is F = R /n ,where n is the density of the projectile and R is thefriction force density obtained by taking the momentummoment of the collision operator, R = (cid:90) d v (cid:90) d v (cid:90) S − ds | u · ˆs | m v ( f (cid:48) f (cid:48) − f f ) . (20)Equation (20) can be simplified using the principleof detailed balance ( (cid:82) d v d v ds | u · ˆs | m v f (cid:48) f (cid:48) = (cid:82) d v d v ds | u · ˆs | m v (cid:48) f f ). The derivation of the prin-ciple of detailed balance typically relies upon the invari-ance of the collision dynamics under time-reversal andspace inversion to find an inverse collision . But for asystem with an externally generated magnetic field, thetime-reversal symmetry is no longer valid (the system hasthe time-reversal invariance only if the reversal of currentdirection that produces the magnetic field is accountedfor) making it difficult to find an inverse collision. Never-theless, the system of charged particles with an externallygenerated magnetic field is expected to follow detailedbalance because many collisions can be lumped togetherto produce a result with the same consequence as an in-verse collision. This argument is similar to that madein polyatomic gases, which are another system in whicha inverse binary collision does not exist . Although aproof has not been developed for the magnetized plasmacase, as it has for the polyatomic gas , we adopt thedetailed balance relation as a postulate, as others havedone .After applying the detailed balance, the friction forcedensity can be recast as R = (cid:90) d v (cid:90) d v (cid:90) S − ds | u · ˆs | m ( v (cid:48) − v ) f f , (21)where v (cid:48) is the postcollision velocity of the projectile,obtained by solving the equations of motion. Since theprojectile is a single particle, its distribution is a Diracdelta function. The background plasma distribution istaken as a uniform Maxwellian distribution. Thus, f = n δ ( v − v ) ,f = n π / v T exp (cid:18) − v v T (cid:19) , where n is the density of the background plasma and v T = (cid:112) k B T /m is the thermal velocity of the back-ground plasma. On making these substitutions we get, R = n n m π / v T (cid:90) d v (cid:90) S − ds | u · ˆs | ( v (cid:48) − v ) exp (cid:18) − v v T (cid:19) . (22)Three components of the friction force are obtainedfrom the friction force density using the following defini-tions F v = R · ˆv n , (23a) F × = R · ( ˆv × ˆn ) n , (23b) F n = R · ˆn n , (23c)where ˆn is the unit vector perpendicular to v and B defined as ˆn = ˆv × ˆb / sin θ , ˆ b = B / | B | is the unit vectorin the direction of the magnetic field and θ is the anglebetween v and B . Here, − F v is the stopping power, F × is the transverse force and F n is the friction forcecomponent along the direction of the Lorentz force. B. Numerical evaluation
The integrals for computing different components ofthe friction force are five-dimensional: three in the ve-locity space and two in the coordinate space. They weresolved numerically using Monte Carlo integration. Thecomputational difficulty is that the coupled differentialequations (Eq. (18) and Eq. (19)) describing the two-body interaction in a magnetic field must be solved nu-merically to compute the change in velocity of the pro-jectile ( v (cid:48) − v ) for each Monte Carlo integration point.Because the parameter-space is five-dimensional, a verylarge number of integration points is required for conver-gence. In our computations, the number ranged from 10 to 10 . In order to solve the integrals numerically, theequations were first made dimensionless by normalizingthe time with the plasma frequency, distance with theDebye length and velocity with the Debye length timesthe plasma frequency. Using the scaled variables, R = M k B T n √ π / Γ / λ D (cid:90) ˜ v sin θ v dθ v dφ v d ˜ v (cid:90) ˜ S − ˜ R s sin θ R s dθ R s dφ R s | ˜u · ˆs | ( ˜v (cid:48) − ˜v ) exp (cid:18) − ˜ v (cid:19) . (24)Here, the collision volume is taken as a sphere and the in-tegrals are written in spherical polar coordinates for bothvelocity and space (recall that the shape of the collisionvolume is unimportant in the general theory, so long as itis large compared to the range of the interparticle force).Since the potential falls off exponentially on the Debyelength scale, the radius of the sphere ( R s ) is taken as 2.5Debye lengths for computations in which the Coulombcoupling strength is Γ = 0 . .
01. Here, M is the ratio of the mass of the projec-tile to that of the background plasma particle ( M = m m )and the variables with tilde (˜) on top represent scaledvariables.A variety of Monte Carlo integration techniques areavailable to reduce the large number of sample pointsrequired in the integration routine. One common tech-nique is the transformation method , but this requiresan approximate analytic expression for the change in mo-mentum of the projectile during a collision. As there isno known analytic expression for this problem, a differ-ent technique is desirable. Instead, we use an adaptiveMonte Carlo integration technique - VEGAS . Inthis method, the integration variables are recast in anattempt to make the integrand a constant and a MonteCarlo integration is performed. These two steps are iter-ated several times. The algorithm uses the informationabout the integrand from one iteration to optimize thechange of variable for the next iteration.Friction force curves were obtained by evaluating theintegrals (Eqs. (23a), (23b) and (23c)) based upon tra-jectory calculations with initial conditions selected from the adaptive Monte Carlo algorithm. Each point in this5-D integral corresponds to an initial velocity of the back-ground particle and an initial position in the relative co-ordinates ˜r i = ˜ R s sin θ R s cos φ R s ˜ R s sin θ R s sin φ R s ˜ R s cos θ R s (25) ˜v i = ˜ v sin θ v cos φ v ˜ v sin θ v sin φ v ˜ v cos θ v (26)and the initial velocity of the projectile ˜v i is taken as ˜v .The unit normal vector is ˆs = ˜r i / ˜ R s . Using Eq. (8), ˜v and ˜v i were transformed to the relative and the center ofmass coordinates. For initial states that satisfy ˜u · ˆs < . The trajectory calculations werestopped when the particle crossed the collision volume,i.e, | ˜r | > ˜ R s and the change of projectile velocity ( ˜v (cid:48) − ˜v )was calculated. Twenty iterations of the VEGAS gridadaptation and the integral estimate were made. Thefinal result and error was obtained by taking the weightedaverage of the last 10 iterations with the weight chosen tobe the inverse of the variance of each of those iterations.Results from evaluation of the friction force in differenttransport regimes are discussed in the following section. IV. RESULTS
In this section, we discuss the results from calcula-tion of the friction on the projectile and compare it withthe traditional Boltzmann theory and the linear responsetheory. The traditional Boltzmann theory is valid for theunmagnetized and weakly magnetized regimes and pre-dicts that the stopping power is the only non-zero compo-nent of the friction force. The expression for the stoppingpower when the interaction is modeled using the Debye-H¨uckel potential in the traditional Boltzmann collisiontheory is F v = − n m v T √ π | v | (cid:90) duu σ (1) ( u ) (cid:104) e − ( u −| v | ) /v T (cid:16) u | v | v T − (cid:17) + e − ( u + | v | ) /v T (cid:16) u | v | v T + 1 (cid:17)(cid:105) , (27)where σ (1) = 4 π (cid:90) ∞ bdb cos Θ( b, u ) (28)is the momentum-transfer scattering cross section andΘ( b, u ) is the scattering angle (Eq. (15)).Linear response theory is valid in all the transportregimes except the extremely magnetized regime. It com-putes the friction force from the induced electric field as-sociated with the wake generated by the movement ofprojectile in the plasma. The result is F = − e π (cid:90) d k k k Im (cid:26) (cid:15) ( k , k · v ) (cid:27) , (29)where k is wave vector andˆ (cid:15) ( k , k · v ) = 1 + 1 k λ D (cid:20) k · v | k || | v T exp (cid:18) − k ⊥ v T ω c (cid:19)(cid:21)(cid:34) ∞ (cid:88) n = −∞ I n (cid:18) k ⊥ v T ω c (cid:19) Z (cid:18) k · v − nω c | k || | v T (cid:19)(cid:35) (30)is the linear dielectric response function of the plasma.Here, Z is the plasma dispersion function , I n is the n th order modified Bessel function of the first kind, and k || and k ⊥ are parallel and perpendicular componentsof the wave vector with respect to the direction of themagnetic field. Since the linear response theory does notaccount for the short range collisions, it would lead toa logarithmic divergence . This is typically avoided bychoosing a high wave number cut off for the k integral, k max = m ( v T + v ) /e , which is approximately the in-verse of distance of closest approach . A. Unmagnetized and weakly magnetized plasma
In order to test the generalized collision operator, andour numerical implementation, we first compute the fric-tion force in the unmagnetized and weakly magnetized (a) = 0.1 BCO LR GCO (b) = 0.01
Projectile speed, v ( v T ) S t o pp i n g p o w e r , F v ( k B T / a × ) FIG. 4: Stopping power ( − F v ) of a massive projectile( M = 1000) in a light background plasma with thecoupling strengths (a) Γ = 0 . .
01 usingthe generalized collision operator (GCO). Also shownare the predictions using the linear response theory(LR) and the traditional Boltzmann collision operator(BCO)plasma regimes and compare the results with the ac-cepted results from the Boltzmann equation. Our com-putations are consistent with the expectation from theBoltzmann equation that, in this regime, only the stop-ping power component F v is non-zero . Figure 4 com-pares the stopping power curve obtained using the gen-eralized collision operator to the result of the traditionalBoltzmann collision operator with the Debye-H¨uckel po-tential as well as the results of linear response theory for Γ = 0 . .
01. Since the influence of the mag-netic field during collisions is negligible, the magneticfield was taken as zero in the equations of motion (Eq.(18) and Eq. (19)).Results from the generalized and traditional Boltz-mann collision operators agree to within numerical tol-erances. Of course, this is expected since the traditionalBoltzmann collision operator is a limiting case of the gen-eralized collision operator. Nevertheless, this comparisonalso helps to verify the generalized collision operator andour numerical implementation.The linear response predicts a slightly larger stoppingpower than the predictions by the binary collision mod-els. Sources of discrepancy between these approachesinclude the absence of the velocity-dependent screening(dynamic screening) in the Debye-H¨uckel potential usedfor modeling the binary collisions, as well as uncertaintyin the short-range cut off length (Landau length) usedin the linear response theory to avoid the logarithmicdivergence caused by neglecting strong nonlinear scat-tering associated with close collisions . The strengthsand weaknesses of these models were previously studiedusing molecular dynamics simulations , and the re-sults shown in Fig. 4 are consistent with these previousstudies. B. Strongly magnetized plasma
In the strongly magnetized regime, the magnetic fieldinfluences the collision dynamics, causing the numericalevaluation of the trajectories of colliding particles to be-come much more computationally expensive. The com-putational expense was reduced by optimizing the num-ber of integration points per iteration of VEGAS ( neval )and setting the tolerance for the trajectory calculations( tol ) in order to achieve a chosen numerical accuracy ofthe computed friction force coefficients. The toleranceof the trajectory calculation is set by both the relativetolerance and the absolute tolerance, which were takento be the same. Figure 5 shows an example convergencetest for the transverse force on a projectile in a back-ground plasma of Γ = 0 . β = 50 and having speed0 . v T and 2 v T . In (a), the tolerance was chosen to be10 − and convergence with respect to the number of in-tegration points was established. In (b), the number ofintegration points for each iteration was chosen to be 10 and convergence with respect to the tolerance was es-tablished. As expected, convergence is obtained as thenumber of integration points increases, as well as whenthe tolerance of the trajectory calculation decreases. Thenumber required for convergence was observed to dependon the projectile speed, as well as the Γ and β param-eters. Nevertheless, a tolerance of 10 − and number ofintegration points for each iteration of 10 was sufficientto obtain convergence to less than 1% throughout thestrongly magnetized and extremely magnetized regimes.Figure 6 shows the friction force curves obtained usingthe generalized collision operator and the linear responsetheory for the coupling strength Γ = 0 . β = 10 for different orientations ofthe initial projectile velocity with respect to the mag-netic field. Only a qualitative agreement can be reachedbetween the curves from these two theories, because ofthe shortcomings of the two models that were discussedin the previous subsection.The main result of this work is shown in panels (b)and (e) of Fig. 6. This shows that a significant trans-verse component of the friction force is predicted by theGCO computations. The existence of this componentwas recently predicted by the linear response approach.Our results demonstrate that this effect is captured in themore complete description from the collision operator ofa kinetic theory. It also demonstrates that the effect iscaptured by the binary collision approach. The two ap-proaches predict qualitatively similar behavior, but havequantitative differences at a similar level to what wasobserved in the unmagnetized case. This is expectedat Γ = 0 . Number of integration points ( neval ) (a) v = 0.2 v T v = 2 v T Tolerance ( tol ) (b) T r a n s v e r s e f o r c e , F × ( k B T / a × ) = 0.1, = 50, = 22.5° FIG. 5: Transverse force ( F × ) on a projectile in a lightbackground plasma of Γ = 0 . β = 50 as a functionof (a) number of integration points per iteration ( neval )and (b) tolerance in trajectory calculation ( tol ). Thevelocity of the projectile makes an angle of 22 . ◦ withthe direction of the magnetic field and initial speed of v = 0 . v T (green circle) and v = 2 v T (red diamond).two approaches would be expected to merge as the cou-pling strength decreases.In linear response theory, the friction force on the pro-jectile is due to the induced electric field associated withthe wake generated by the projectile in the backgroundplasma. But in the binary collision theory, the frictionforce is the net force acting on the projectile from sub-sequent binary interactions with the background plasma.The linear response theory attributes the origin of thetransverse force to the way in which the Lorentz forceon the background plasma influences the instantaneouslygenerated wake. In contrast, the generalized collision op-erator captures the transverse force by accounting forthe gyromotion of the background particles while inter-acting with the projectile. Even though these two arecompletely different approaches, they both are equallycapable of capturing the physics of transverse force inthis regime.On comparing the stopping power curves ( − F v ) fromFig. 6 with those for the weakly magnetized regime fromFig. 4, qualitative changes are observed. The position ofthe peak shifts to a lower speed, and the magnitude of theforce decreases at the Bragg peak, but decays less rapidlywith speed. The stopping power is also observed to de-pend on the orientation of the projectile velocity with0 (a) F v LRGCO 0 2 4 62101234 (b) F × (c) = 22.5° F n (d) (e) (f) = 45° Projectile speed v ( v T ) F r i c t i o n f o r c e ( k B T / a × ) FIG. 6: Friction force on a massive projectile ( M = 1000) slowing down on a light background plasma with couplingstrength Γ = 0 . β = 10 for different initial projectile velocity angles with respect to themagnetic field θ = 22 . ◦ [a, b, and c] and θ = 45 ◦ [d, e, and f]. The generalized collision operator results (GCO) andlinear response theory curve (LR).respect to the magnetic field. The friction force alongthe direction of the Lorentz force ( F n ) is much smallerthan either the stopping power or transverse force. Pointscomputed at most velocities are consistent with zero towithin the estimated accuracy of the data, but there area few points at which the computed force appears to benon-zero. This is a qualitative distinction with the pre-dictions of linear response theory and will be studied ingreater detail in future work. C. Extremely magnetized plasma
Figure 7 shows the friction force curves obtained usingthe generalized collision operator and the linear responsetheory for the coupling strength Γ = 0 . β = 50 for different orientations ofthe initial projectile velocity with respect to the mag-netic field. On comparing with the friction force curvesin the strongly magnetized regime, the magnitude of thefriction force is predicted to increase and the peak ofthe stopping power curve shift slightly to lower projec-tile speeds. Similar to the case of strongly magnetizedregime, the friction force in the direction of the Lorentzforce, F n is much smaller than either the stopping power or transverse force.Linear response theory assumes that the interactionsare weak and are small angle collisions. In order to avoidthe divergence in the theory caused by the strong large-angle collisions an ad hoc short-range cut off is intro-duced at the Landau length. These assumptions breakdown in the extremely magnetized regime. In this regime,the gyroradius is the smallest length scale and the parti-cles are bound to the magnetic field lines. The collisionsbetween the particles are strong and are large-angle col-lisions. However, the physics of strong interactions arecaptured by the binary collision theory. This makes thegeneralized collision operator a strong candidate to un-derstand the physics of the extremely magnetized plas-mas. Although the k max cutoff used in the linear responsetheory is not expected to apply in the extremely magne-tized regime, Fig. 7 shows a similar level of agreement be-tween linear response theory and the GCO as is observedin the strongly magnetized regime shown in Fig. 6. It isunknown if this a fortuitous agreement particular to thiscombination of Γ and β , or if it will also extend to yetstronger magnetization.1 (a) F v LRGCO 0 2 4 64202468 (b) F × (c) = 22.5° F n (d) (e) (f) = 45° Projectile speed v ( v T ) F r i c t i o n f o r c e ( k B T / a × ) FIG. 7: Friction force on a massive projectile ( M = 1000) slowing down on a light background plasma with couplingstrength Γ = 0 . β = 50 for different initial projectile velocity angles with respect to themagnetic field θ = 22 . ◦ [a, b, and c] and θ = 45 ◦ [d, e, and f]. The generalized collision operator results (GCO) andlinear response theory curve (LR). V. DISCUSSION
This section provides a qualitative description of thephysical origin of the transverse friction force due tostrong magnetization from the binary collision perspec-tive. Binary collision theory calculates the friction forcevia the change in momentum of the projectile after asequence of elementary binary collisions with the back-ground plasma particles that includes all possible scatter-ing events. When the gyroradius of the colliding particlesare larger than the characteristic scattering length ( λ D ),the influence of the magnetic field during the collisionsare negligible. In this case, background plasma particlescollide with the projectile from all the directions withequal probability. The net change in momentum from thecollisions in all the directions except parallel or antipar-allel to the projectile velocity will be zero. For instance,take the projectile velocity to be in the +ˆ x direction asshown in panel (a) of Fig. 8. Here, the projectile is la-beled (P) and the background particles are numbered r to r . The change in momentum of the projectile froma collision with a background particle having velocity v ˆ y ( r ) is canceled by the change in momentum from a col-lision with a background particle having velocity − v ˆ y ( r ). This leads to the conclusion that there is no trans- verse component of the friction. In contrast, the changein momentum of the projectile from collisions with thebackground particle having velocity − v ˆ x ( r ) is greaterthan from a collision with velocity + v ˆ x ( r ) because ofthe larger magnitude of the relative velocity. Thus thefriction force is antiparallel to the projectile velocity.When the plasma is strongly magnetized or extremelymagnetized, background particles are bound to the mag-netic field lines, effectively making their motion 1-D. Thisrestricts the approach of background particles to the pro-jectile along the ± ˆ b direction, breaking the symmetry ofparticles approaching uniformly from all directions, as inthe unmagnetized and weakly magnetized cases. Panel(b) of Fig. 8 shows the collision of a projectile with fourdifferent background particles. The projectile (P) veloc-ity vector makes an angle θ with respect to the magneticfield and the background particles are numbered r to r .Consider the case that the velocity vector of the pro-jectile makes an acute angle to the magnetic field (0 ◦ ≤ θ ≤ ◦ ). The net change in momentum of the pro-jectile along the magnetic field direction from collisionswith the background plasma particles approaching fromthe +ˆ b direction will be higher than that of the colli-sion with background plasma particle approaching fromthe − ˆ b direction, resulting in a force anti parallel to the2 Y X
FIG. 8: Illustration of collisions of the massive projectilewith the background particles - (a) unmagnetized andweakly magnetized (b) strongly magnetized andextremely magnetized transport regimes.direction of magnetic field. For example consider the col-lision between particle 1 ( r ) and particle 2 ( r ). Boththe particles have equal speed, but the relative velocity ishigher for the collision with particle 2, resulting in higherexchange of momentum.The projectile also experiences a net force in the − ˆ x di-rection. This can be understood by comparing collisionsbetween the projectile and particle 3 ( r ) or particle 4( r ). The projectile experiences more change in momen-tum from the collision with particle 3 than collision withparticle 4 because the projectile is moving towards theparticle 3. The conclusion of these arguments is thatthe presence of the magnetic field breaks the symmetryabout the velocity vector, causing there to be both a stop-ping power component antiparrallel to the velocity anda transverse component perpendicular to the velocity inthe plane of v and B .The force on the projectile in the ˆ y direction is ex-pected to be zero because the projectile has no compo-nent of the velocity in this direction, other than its gyro-motion, and the symmetry of momentum exchange withparticles entering from either ± ˆ y directions is expectedto balance. For instance, the force on the projectile inthe ˆ y direction from a background particle moving alongthe magnetic field line at | y | ˆ y is canceled by the forcefrom the background particle moving along the field lineat −| y | ˆ y . The above discussion considered oblique angles be-tween v and B , but when the projectile moves eitherperpendicular or parallel to the magnetic field, symme-try about the projectile velocity vector is expected to re-turn, and the transverse component of the friction forceto vanish. Consider a projectile moving perpendicular tothe magnetic field in the ˆ x direction. In this case, theprojectile is not expected to experience any net force inthe ˆ z direction as the momentum exchanged in collisionswith background particles approaching from the +ˆ b and − ˆ b directions are antisymmetric. In this case, the projec-tile experiences a force in the − ˆ x direction only. Thus,a projectile moving perpendicular to the magnetic fieldonly has a stopping power component and no transverseforce. Similar arguments of symmetry can be made tounderstand why there is also no transverse componentwhen the projectile velocity aligns along the magneticfield. Although the solutions in the previous section fo-cused only on oblique angles, these symmetry propertieswere confirmed, and they have also been shown to holdin both the previous linear response calculations andmolecular dynamics simulations . VI. CONCLUSION
This work has developed a generalized collision opera-tor applicable to plasmas in the presence of a magneticfield of arbitrary strength. It consists of a 3-D velocityspace integral and a 2-D spatial integral on the surface ofa collision volume, inside of which the particles interactvia the potential of mean force. The size of the collisionvolume is determined by the range of the potential ofmean force. The generalized collision operator incorpo-rates the magnetic field into the collisions by includingthe Lorentz force acting on the colliding particles in theequations of motion describing the binary collision. Thetraditional Boltzmann collision operator for the unmag-netized and the weakly magnetized plasma and O’Neil’sBoltzmann-like collision operator for the extremely mag-netized plasma were obtained from the generalized colli-sion operator by simplifying the collision geometry andequations of motion for the interacting particles in thelimits of no magnetic field and high magnetic field, re-spectively.The generalized collision operator was applied to com-pute the friction force acting on a massive projectile mov-ing through the magnetized OCP. The numerical imple-mentation of the generalized collision operator was ver-ified by comparing results from the unmagnetized andweakly magnetized plasma cases with the accepted re-sults of the traditional Boltzmann kinetic theory. Thefriction force curve for the strongly magnetized plasmawas compared with previous results from the linear re-sponse theory. The work also extended the computationof the friction force to the extremely magnetized trans-port regime which was not attainable using the linearresponse theory. This shows the versatility of the gener-3alized collision operator to obtain the transport proper-ties of the plasmas in all the transport regimes.By extending the kinetic theory to the transportregimes with the gyroradius smaller than the charac-teristic scattering length ( λ D ), we now have a theoryto understand the basic properties of the plasmas inthe strongly magnetized and the extremely magnetizedtransport regimes. Capturing the influence of the mag-netic field on binary collisions has described a new phys-ical effect - the transverse friction force. The transverseforce mixes the parallel and perpendicular velocity com-ponents of the projectile. How the mixing of these com-ponents changes other macroscopic transport propertiesis unknown. The generalized collision operator is a strongcandidate to explore this question.Many experiments in which strongly magnetized plas-mas are found, such as ultracold neutral plasmas , nonneutral plasmas , and magnetized dusty plasma experi-ments exhibit strong Coulomb coupling (Γ > . This will be explored in future work. VII. DATA AVAILABILITY STATEMENT
The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest.
ACKNOWLEDGMENTS
The authors thank Dr. J´erˆome Daligault for helpfulconversations during the development of this work, Dr.Trevor Lafleur for providing the codes used to computethe linear response theory curves and Prof. Peter Lepagefor suggestions with implementation of VEGAS.This material is based upon work supported by theAir Force Office of Scientific Research under Award No.FA9550-16-1-0221, by the U.S. Department of Energy,Office of Fusion Energy Sciences, under Award No. DE-SC0016159, and the National Science Foundation underGrant No. PHY-1453736. R. Aymar, P. Barabaschi, and Y. Shimomura, “The ITER de-sign,” Plasma Physics and Controlled Fusion , 519–565 (2002). B. R. Beck, J. Fajans, and J. H. Malmberg, “Measurement of col-lisional anisotropic temperature relaxation in a strongly magne-tized pure electron plasma,” Phys. Rev. Lett. , 317–320 (1992). X. L. Zhang, R. S. Fletcher, S. L. Rolston, P. N. Guzdar, andM. Swisdak, “Ultracold plasma expansion in a magnetic field,”Phys. Rev. Lett. , 235002 (2008). E. Thomas, R. L. Merlino, and M. Rosenberg, “Magnetized dustyplasmas: the next frontier for complex plasma research,” PlasmaPhysics and Controlled Fusion , 124034 (2012). J. Fajans and C. Surko, “Plasma and trap-based techniques forscience with antimatter,” Physics of Plasmas , 030601 (2020). K. K. Khurana, M. G. Kivelson, V. M. Vasyliunas, N. Krupp,J. Woch, A. Lagg, B. H. Mauk, and W. S. Kurth, “The con-figuration of Jupiter’s magnetosphere,” in
Jupiter: The Planet,Satellites and Magnetosphere , Vol. 1, edited by F. Bagenal, T. E.Dowling, and W. B. McKinnon (2004) pp. 593–616. S. D. Baalrud and J. Daligault, “Transport regimes spanningmagnetization-coupling phase space,” Phys. Rev. E , 043202(2017). J. H. Ferziger, H. G. Kaper, and H. G. Kaper,
Mathematicaltheory of transport processes in gases (North-Holland, 1972). T. ONeil, “Collision operator for a strongly magnetized pure elec-tron plasma,” Physics of Fluids , 2128–2135 (1983). S. Cohen, E. Sarid, and M. Gedalin, “Collisional relaxation ofa strongly magnetized ion-electron plasma,” Physics of Plasmas , 082105 (2019). S. Cohen, E. Sarid, and M. Gedalin, “Fokker-planck coefficientsfor a magnetized ion-electron plasma,” Physics of Plasmas ,012311 (2018). D. Montgomery, L. Turner, and G. Joyce, “Fokker-planck equa-tion for a plasma in a magnetic field,” Physics of Fluids ,954–960 (1974). D. Montgomery, G. Joyce, and L. Turner, “Magnetic field depen-dence of plasma relaxation times,” Physics of Fluids , 2201–2204 (1974). D. H. Dubin, “Parallel velocity diffusion and slowing-down ratefrom long-range collisions in a magnetized plasma,” Physics ofPlasmas , 052108 (2014). C. Dong, H. Ren, H. Cai, and D. Li, “Effects of magnetic fieldon anisotropic temperature relaxation,” Physics of Plasmas ,032512 (2013). C. Dong, H. Ren, H. Cai, and D. Li, “Temperature relaxation ina magnetized plasma,” Physics of Plasmas , 102518 (2013). N. Rostoker, “Kinetic equation with a constant magnetic field,”Physics of Fluids , 922–927 (1960). A. A. Ware, “Electron fokker-planck equation for collisions withions in a magnetized plasma,” Phys. Rev. Lett. , 51–54 (1989). L. Pitaevskii and E. Lifshitz,
Physical Kinetics (Butterworth-Heinemann, 2012). D. H. E. Dubin, “Test particle diffusion and the failure of integra-tion along unperturbed orbits,” Phys. Rev. Lett. , 2678–2681(1997). H. Nersisyan, C. Toepffer, and G. Zwicknagel,
Interactions be-tween charged particles in a magnetic field (Springer-VerlagBerlin Heidelberg, 2007). D. Sigmar and G. Joyce, “Plasma heating by energetic particles,”Nuclear Fusion , 447–456 (1971). T. Lafleur and S. D. Baalrud, “Transverse force induced by amagnetized wake,” Plasma Physics and Controlled Fusion ,125004 (2019). T. Lafleur and S. D. Baalrud, “Friction in a strongly magne-tized neutral plasma,” Plasma Physics and Controlled Fusion , 095003 (2020). H. B. Nersisyan and G. Zwicknagel, “Binary collisions of chargedparticles in a magnetic field,” Phys. Rev. E , 066405 (2009). H. B. Nersisyan, G. Zwicknagel, and C. Toepffer, “Energy loss ofions in a magnetized plasma: Conformity between linear responseand binary collision treatments,” Phys. Rev. E , 026411 (2003). H. B. Nersisyan, M. Walter, and G. Zwicknagel, “Stopping powerof ions in a magnetized two-temperature plasma,” Phys. Rev. E , 7022–7033 (2000). C. Cereceda, M. de Peretti, and C. Deutsch, “Stopping powerfor arbitrary angle between test particle velocity and magneticfield,” Physics of plasmas , 022102 (2005). D. J. Bernstein, T. Lafleur, J. Daligault, and S. D. Baalrud,“Friction force in strongly magnetized plasmas,” submitted toPhys. Rev. Lett. (2020). S. Harris,
An introduction to the theory of the Boltzmann equa-tion (Courier Corporation, 2004). H. Grad, “Principles of the kinetic theory of gases,” in
Thermo-dynamik der Gase/Thermodynamics of Gases (Springer, 1958) pp. 205–294. C. Cercignani, R. Illner, and M. Pulvirenti,
The mathematicaltheory of dilute gases , Vol. 106 (Springer-Verlag New York, 1994). S. D. Baalrud and J. Daligault, “Mean force kinetic theory: Aconvergent kinetic theory for weakly and strongly coupled plas-mas,” Physics of Plasmas , 082106 (2019). S. D. Baalrud and J. Daligault, “Effective potential theory fortransport coefficients across coupling regimes,” Phys. Rev. Lett. , 235001 (2013). S. D. Baalrud and J. Daligault, “Extending plasma transporttheory to strong coupling through the concept of an effectiveinteraction potential,” Physics of Plasmas , 055707 (2014). Considering collisions in strongly magnetized one-componentplasmas, Dubin has argued that strong magnetization causes par-ticles to recollide multiple times . Such multi-body interactionsviolate the molecular chaos approximation. Further research willbe required to understand the role of recollisions in the frictionforce problem considered in this work, or to adapt the generalizedcollision operator to incorporate recollision dynamics. T. ONeil and P. Hjorth, “Collisional dynamics of a strongly mag-netized pure electron plasma,” Physics of fluids , 3241–3252(1985). M. E. Glinsky, T. M. ONeil, M. N. Rosenbluth, K. Tsuruta, andS. Ichimaru, “Collisional equipartition rate for a magnetized pureelectron plasma,” Physics of Fluids B: Plasma Physics , 1156–1166 (1992). S. D. Baalrud and J. Daligault, “Effective potential kinetic the-ory for strongly coupled plasmas,” AIP Conference Proceedings , 130001 (2016). F. Reif,
Fundamentals of statistical and thermal physics (Wave-land Press, 2009). C. Cercignani and M. Lampis, “On the h-theorem for polyatomicgases,” Journal of Statistical Physics , 795–801 (1981). W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flan-nery,
Numerical recipes 3rd edition: The art of scientific com-puting (Cambridge University Press, 2007). G. Peter Lepage, “A new algorithm for adaptive multidimen-sional integration,” Journal of Computational Physics , 192 –203 (1978). P. Lepage, “gplepage/vegas: vegas version 3.4.2,” (2020). P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber-land, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson,W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wil-son, K. Jarrod Millman, N. Mayorov, A. R. J. Nelson, E. Jones,R. Kern, E. Larson, C. Carey, ˙I. Polat, Y. Feng, E. W. Moore,J. Vand erPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Hen-riksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H.Ribeiro, F. Pedregosa, P. van Mulbregt, and SciPy 1. 0 Contrib-utors, “SciPy 1.0: Fundamental Algorithms for Scientific Com-puting in Python,” Nature Methods , 261–272 (2020). B. D. Fried and S. D. Conte,
The plasma dispersion function:the Hilbert transform of the Gaussian (Academic Press, 1961). D. R. Nicholson,
Introduction to plasma theory (Wiley, 1983). S. Ichimaru,
Statistical Plasma Physics, Volume I: Basic Prin-ciples (CRC Press, 2004). D. J. Bernstein, S. D. Baalrud, and J. Daligault, “Effects ofcoulomb coupling on stopping power and a link to macroscopictransport,” Physics of Plasmas , 082705 (2019). P. E. Grabowski, M. P. Surh, D. F. Richards, F. R. Graziani,and M. S. Murillo, “Molecular dynamics simulations of classicalstopping power,” Physical review letters111