Coulomb Collision Effects on Linear Landau Damping

Coulomb collisions at rate ν produce slightly probabilistic rather than fully deterministic charged particle trajectories in weakly collisional plasmas. Their diffusive velocity scattering effects on the response to a wave yield an effective collision rate ν eff ν and a narrow dissipative boundary layer for particles with velocities near the wave phase velocity. These dissipative effects produce temporal irreversibility for times t > ∼ 1/ν eff during Landau damping of a small amplitude Langmuir wave.


I. INTRODUCTION
In a classic theoretical physics paper [1] Landau used a Laplace transform to introduce causality into the process of determining the time-asymptotic linear plasma response to a wave perturbation.This produced what is now known as Landau damping of the wave, which has been confirmed experimentally [2].It is often thought of as a "collisionless," entropy-producing process because: collisional effects are not involved in the derivation; the damping rate is independent of the collision rate; and the wave damping seems to imply irreversibility.
However, after a wave is turned on its phase information is transferred to a perturbed distribution of the plasmas' charged particles.That this initial state information still exists has been demonstrated by showing a second wave produces a plasma echo [3,4].But when sufficient Coulomb collisions are introduced, echoes are damped [5,6].Thus, irreversibility produced during Landau damping in a plasma is due to a collisional process.
As illustrated in Fig. 1, Coulomb collisions of a charged particle with other charged particles in a plasma mainly scatter the particle's velocity vector v.The characteristic Coulomb collision scattering rate ν is usually much smaller than the Landau damping rate γ L .In spherical velocity space coordinates speed v ≡ |v|, "pitch-angle" ϑ and phase angle ϕ, Coulomb collisions diffuse the pitchangle through a small angle δϑ ≡ ϑ 0 − ϑ ∼ √ ν ⊥ τ 1 and cause a typically smaller diffusive spread in the speed δv/v 0 ≡ (v 0 − v)/v 0 ∼ 2 ν τ 1 in a time τ 1/ν.These Coulomb collisional scattering effects are operative in and intrinsic to most weakly collisional plasmas.
Coulomb collision effects on the plasma response to a small amplitude wave can be illustrated by considering their effects on the phase Φ of a particle initially at x 0 , v 0 at time t 0 relative to a wave with Fourier representation φ(x 0 , t 0 ) = φ(k, ω)e i(k•x0−ωt0) there.Here, both ω and k are real.Along particle trajectories x = x 0 + v 0 τ , a short time τ ≡ t−t 0 after a wave is turned on the effective phase including the weak Coulomb collisional scattering * callen@engr.wisc.edu;http://www.cae.wisc.edu/∼calleneffects deduced from Eqs. ( 11)-( 13) below is The spherical velocity-space symmetry axis is taken to be along k so k In (1) ω − ku represents the Doppler-shifted wave frequency, kuτ [(ϑ 2 0 −ϑ 2 )/2+δv/u] is due to k • v 0 = ku and the two imaginary terms result from the dissipative pitchangle scattering and speed diffusion effects of Coulomb collisions.The combination of all the δϑ and δv terms yield an effective Coulomb collisional damping rate which causes decorrelation of particles from the wave and temporal irreversibility for τ > ∼ 1/ν eff .Collisional effects on the time-asymptotic Landau damping of a Langmuir (electron plasma) wave have been explored [6][7][8][9][10] using speed-diffusion-based collision operators that neglect the pitch-angle scattering effects of Coulomb collisions.Landau damping of the quasilinear wave energy has been interpreted physically [11].Diverse applications of the Landau theory are summarized in Ref. [12].Also, collisionless damping of a finite amplitude wave that traps particles in its sinusoidal potential has been studied [13][14][15][16][17][18].This paper introduces a novel Green function procedure for incorporating short time scale (τ 1/ν) velocity scattering effects of Coulomb collisions on electron trajectories after a small amplitude Langmuir wave is imposed.The procedure is analogous to approaches developed for a Brownian-motion-type isotropic collision operator [19] and for resonant broadening effects caused by strong plasma turbulence [20].The approach developed in this paper provides new perspectives on the temporal evolution, resonance broadening and irreversibility involved in Landau damping in weakly collisional plasmas.
This paper is organized as follows.The next section discusses the effects of weak Coulomb collisional scattering on the plasma kinetic equation and its solutions.The following section presents a concise derivation of the Landau damping of a Langmuir wave.Next, Section IV introduces a novel Green function solution of the linearized plasma kinetic equation that includes the weak Coulomb collisional scattering effects.The penultimate section discusses how the collisional effects in the Green function solution resolve the "collisionless" singularity where u = V ϕ ≡ ω/k.The final section discuses new perspectives on Landau damping that result from this analysis.

II. COLLISION OPERATOR AND ITS EFFECTS
For times t < ∼ 1/ν eff the lowest order, approximate (subscript a) Fokker-Planck (F-P) Coulomb collision operator acting on a perturbed distribution fe of electrons with charge q e = − e and mass m e is (see Appendix A) For Landau damping in the tail of the electron distribution [2] where v v T e ≡ 2T e /m e , ν ⊥ 2 (1 + Z i ) ν (electron collisions with electrons and ions of charge Z i ) and ν (v T e /v) 2 ν ν.The reference collision rate is Here, Λ is the ratio of the collective Debye shielding length λ D ≡ ( 0 T e /n e e 2 ) 1/2 (m) to the distance of closest approach during Coulomb collisions, and T e /e (eV) and n e (m −3 ) are the electron temperature and density.The ln Λ factor (typically > ∼ 10) represents the cumulative effect of all electron collisions within a Debye sphere.Also, ω p ≡ (n e e 2 /m e 0 ) 1/2 is the Langmuir electron plasma frequency.In typical plasmas n e λ 3 D 1 so the collision rate ν is much smaller than ω p and the Coulomb collision effects are weak.Note also that a very large number (∼ n e λ 3 D 1) of statistically independent Coulomb collisions contribute to ν and hence C a { fe } during the short time 1/ω p the electron traverses a Debye sphere.The complete F-P operator C{f e } [21] is the small momentum transfer limit [22] of the Boltzmann collision operator.

III. LANDAU DAMPING
Landau damping will be explored in an infinite, homogeneous, unmagnetized plasma that has only an electrostatic electric field force F e = − q e ∇φ.Assuming the equilibrium has no macroscopic electric field, the equilibrium PKE is 0 = C{f e0 }.The general solution of this equation is the isotropic Maxwellian distribution: . Introducing a small electric field perturbation Ẽ ≡ −∇ φ in (5), f e → f Me + fe , which yields a linearized perturbed PKE for the perturbed distribution fe (x, v, t): Neglecting collision effects and assuming e i(k•x−ωt) Fourier perturbations for φ and fe , this equation yields whose "resonant denominator" 1/(u−V ϕ ) indicates a singular response for particles that have speeds u ≡ k • v/k along k equal to the wave phase speed V ϕ ≡ ω/k.Since the solution in ( 7) is real, Vlasov [23] concluded there is no damping of wave-like perturbations in low collisionality plasmas.Landau [1] also neglected collision effects but used a Laplace transform for the time domain to employ causality to define the singularity in ( 7) by deforming the inverse Laplace transform contour to always be below this singularity.Using V ϕ → V ϕ + iγ/k, for small γ this formalism yields the Plemelj formula In this Landau prescription for resolving the singularity is the (real) principal value operator and δ[x] is the Dirac delta function.
The perturbed electron density induced by φ is obtained by integrating fe in (7) over all velocity space using (8).For V ϕ v T e this yields [22,24] ñe The kλ D 1 dispersion relation for Langmuir waves is obtained [24] by substituting this ñe into the perturbed Poisson equation −∇ 2 φ = ρq / 0 − ñe e/ 0 in which the small ion contribution has been neglected.Solving this dispersion relation for the complex frequency ω ≡ ω R + i γ yields the real frequency The imaginary delta function term in (8) produces γ L , which is usually smaller than ω p but much larger than ν.For example, for kλ D 0.3 and V ϕ /v T e 2.7 in the n e λ 3 D ∼ 10 4 plasmas in Ref. [2] where ν(v T e )/ω p ∼ 10 −4 , γ L /ω p 2 × 10 −2 and ν /ν ⊥ ∼ 1/30.

IV. GREEN FUNCTION SOLUTION
In spherical velocity-space coordinates u ≡ v • k/k = v cos ϑ.Thus, if the collision operator in (3) operates on fe in (7), it yields which is even more singular than fe .This paper introduces a Green function procedure for solving (6) that collisionally resolves the singular region u V ϕ .
Since for short times δϑ ∼ √ ν ⊥ τ 1, the collision operator in (3) can be simplified using sin ϑ ϑ for ϑ 1.Then, the perturbed PKE in (6) becomes (10) This result is in the form L{ fe } = (e/T e ) (v •∇ φ)f Me in which L is a linear partial differential operator.Its solution can be obtained in terms of a Green function G(x, ϑ, v|x 0 , ϑ 0 , v 0 ; t − t 0 ) that solves the equation . This Green function represents the response at x, ϑ, v, t to a delta function source at x 0 , ϑ 0 , v 0 , t 0 .Assuming that fe = 0 initially, the Green function solution of (10) The characteristic curves of the first derivative (Vlasov) operators in (10) are obtained by solving dx/dt = v and dv/dt = 0. Using initial conditions x = x 0 and v = v 0 (i.e., v = v 0 and ϑ = ϑ 0 ) at the initial time t 0 , these first order ordinary differential equations yield the reversible, deterministic particle trajectory x = x 0 + v 0 τ in which τ ≡ t − t 0 ≥ 0. When collisional effects are weak the effects of the second derivative collision operator in (10) are separable from these first order characteristics.For ντ 1 the Coulomb-collision-induced characteristics are e −(ϑ−ϑ0) 2 /(ν ⊥ τ ) /(ν ⊥ τ /2) for pitch-angle scattering and e −(v−v0) 2 /(2v 2 0 ν τ ) /(v 0 2πν τ ) for speed diffusion.By construction, the complete Green function for the differential operator on the left hand side of ( 10) is thus in which H(τ ) = H(t − t 0 ) is the Heaviside step function.Since dH(τ )/dt = δ[τ ] and using the facts that lim operating on (12) with the linear operator L shows G satisfies the defining equation for the Green function for ϑ 0 1.Equation ( 12) is the "propagator" for charged particle trajectories.It includes both the reversible, deterministic motion x = x 0 + v 0 τ and probabilistic, irreversible diffusion of the velocity-space speed δv ∼ v 0 2 ν τ and pitch-angle δϑ ∼ √ ν ⊥ τ induced by Coulomb collisions.
The analogous temporal evolution of the integrand of I ν0 for ⊥ = 0 is illustrated in Fig. 3 for various ∆ u .This speed diffusion response is oscillatory for and always in phase with the Doppler-shifted wave for z = kuτ ωτ −1/3 ( 67 here).For ⊥ = 0 the effective collision frequency is Thus, as Fig. 3 shows, the speed diffusion response is damped for all ∆ u as [6] e − z 3 = e −ν 3 eff τ 3 for τ > 1/ν eff .The Padé approximate of these two temporally irreversible Coulomb collisional scattering effects yields the overall effective collision rate ν eff in (2).The ν 1/3 contribution to ν eff is often slightly larger even though on the tail of the electron distribution ν ν ⊥ .However, the physically different pitch-angle scattering at rate ν ⊥ and speed diffusion ν Coulomb collisional effects act simultaneously.Thus, in general both should be included in comprehensive analyses of the temporal evolution and irreversibility in linear Landau damping.

V. COLLISIONAL RESOLUTION OF SINGULARITY AT u = Vϕ
Coulomb collision effects can be neglected for short times by taking z 1 and ⊥ z 1 limits inside the I ν in (15).Using lim z 1 e −δv 2 /(4 zv 2 0 ) ) yields a temporally reversible result for times t 1/ν eff : Figure 4 shows the behavior of this collisionless result.Averaging (21) in time t over the effective sinusoidal particle period 2π/ku yields a type of "phase mixed" I ν .As indicated in Fig. 4, its real part yields 1/∆ u except near ∆ u = 0 where it vanishes.Its imaginary part is largest for |∆ u | < ∼ 1/kut.Since sin(kut ∆ u )/∆ u is a correlation function, its kut 1 limit yields a delta function: lim kut→∞ sin(kut ∆ u )/∆ u .= π δ[∆ u ].Thus, the time-asymptotic limit of this phase mixed is Landau's Laplace-transform-based prescription in ( 8) that is used in collisionless (Vlasov) plasma models.However, in the presence of weak Coulomb collisional scattering effects the short time criterion t 1/ν eff for obtaining (21) and the kut → ∞ limit are incompatible.As will be shown next, the phase mixing approximation and t → ∞ requirements are not needed to obtain (8) for most plasmas which by definition include weak, nonnegligible, intrinsic Coulomb collisional scattering effects.
When Coulomb collision effects are included in I ν , in general the integrals indicated in (15) have to be evaluated numerically.The behavior of the = 0, ϑ = 0 time history integral I ν0 in the dissipative resonance region is shown in Fig. 5. Its key properties for t 1/ν eff = (ν ⊥ ω/2) −1/2 = 1/(ω 1/2 ⊥ ), which show that for The last result is obtained by numerical integration and accurate to < 0.01% for ⊥ < 0.1.The speed u where Im{I ν0 } peaks is larger than ϕ Coulombcollision-induced pitch-angle scattering causes the effective u v (1 − ϑ 2 /2) to be smaller than v (see Fig. 1).
The analogous behavior of the ⊥ = 0, ϑ = 0 time history integral I ν0 in the dissipative resonance region |(u − V ϕ )/u| ∼ 1/3 is shown in Fig. 6.Its key prop- show that for 1 its real part is P{1/∆ u } and its

Im
Re The last result is obtained by numerical integration and accurate to < 0.01% for < 0.1.The speed u where Im{I ν0 } peaks is at u = 0 because at ϑ = 0 the collisioninduced speed diffusion is in the u direction (see Fig. 1).With collisional speed reduction the Im{I ν0 } peak moves very slightly to ∆ u / 1/3 ∼ − 1/3 (see Appendix A).
Thus, for times t > ∼ 1/ν eff Coulomb collisions produce two important effects on the collisionless I ν shown in Fig. 4. First, far away from the resonance (i.e., for |∆ u | π/kut) they smooth its real part to the phase mixed result Re{I ν } 1/∆ u and cause its imaginary part to become exponentially small.Second, they limit progression of the singular layer in Fig. 4 to an ever narrower region as kut increases beyond ω/ν eff ; they instead produce a resonance-broadened dissipative region where |∆ u | < ∼ ν eff /ω.As shown in ( 22)-( 27), Im{I ν } has delta-function-type properties for small ⊥ and : max{I ν } ∼ ω/ν eff 1 and ∞ −∞ d∆ u I ν = iπ.Thus, for weakly collisional plasmas where ⊥ 1 and 1 lim This collision-based result is Landau's prescription in (8).

VI. DISCUSSION AND CONCLUSIONS
The analysis in this paper shows that when Coulomb collisional scattering effects are included, the deterministic, reversible particle motion propagator δ[x − x 0 − v 0 τ ] is replaced by the probabilistic, irreversible Green function in (12).This causes the singular factor u/(u − V ϕ ) in (7) to be replaced by the integral I ν defined in (15), as indicated in (14).Since the t 1/ν eff limit of I ν yields the Landau prescription in (8), the Landau damping rate γ L in ( 9) is also obtained with this Coulomb-collisionbased Green function approach.Thus, a Laplace transform procedure is neither needed nor appropriate when the intrinsic, weak Coulomb collisional scattering effects that are intrinsic to most plasmas are included.
These results do not change the time-asymptotic Landau singularity resolving prescription in (8) or Langmuir wave damping rate.However, the Green function analysis presented here facilitates exploration of the plasma response to a wave on various time scales.The Coulombcollision-influenced time-history integral I ν includes: 1) the short-time-scale (t 1/ν eff ) temporally reversible collisionless result in (21), 2) evolution into an irreversible, dissipative, resonance-broadened response for t > ∼ 1/ν eff (e.g., as in damping of echoes [5,6]) and 3) the long but finite time-scale (t 1/ν eff ) Coulomb collisional justification in (28) of the Landau prescription in (8) for resolving the u = V ϕ singularity.
The collisionless limit of the time-history integral given in (21) has properties that are superficially similar to the Landau prescription in (8), especially after this collisionless I ν is phase mixed in time.Namely, as shown in Fig. 4, its real and imaginary parts are strongly peaked in the ∼ π/kut resonance region and its real part decays approximately as 1/∆ u .However, there is a very important difference: the collisionless I ν in ( 21) is temporally reversible whereas the imaginary part of (8) indicates irreversibility.A temporally irreversible response is obtained from (21) only in the time-asymptotic limit kut → ∞.This is consistent with Landau's use of the Laplace transform, which introduces causality.
In the presence of weak Coulomb collisional scattering effects the short time criterion t 1/ν eff for obtaining (21) and the time-asymptotic limit kut → ∞ are incompatible.This paper has demonstrated that the probabilistic Coulomb collisional scattering effects determine the minimum width of the wave-particle resonance for t > ∼ 1/ν eff and that the plasma response evolves into a temporally irreversible, dissipative state for t 1/ν eff .These results are the linear theory predictions for the long time scale plasma response to a small amplitude wave that should occur in most physically relevant plasmas which are intrinsically weakly collisional.Does Landau damping increase entropy?Addressing this question requires a quasilinear analysis that is beyond the scope of this paper.However, Figs. 2 and 3 show the instantaneous fe response is in phase with the applied potential φ for all ∆ u up until the Coulomb-collisioninduced effective collision time 1/ν eff , after which it is damped.This suggests entropy changes little for t 1/ν eff but then increases for t > ∼ 1/ν eff .The temporal irreversibility is caused by the Coulomb collisional scattering effects that resolve the wave-particle singularity in the resonance region |(u − V ϕ )/u| < ∼ ν eff /ω, as indicated by the localized imaginary part of I ν in Figs. 5 and 6.
A wave of finite amplitude traps nearly resonant electrons in its sinusoidal potential.The oscillation frequency for such trapped electrons is [13][14][15][16][17] ) (e φ/T e ) 1/2 .Trapping dominates when resonant electrons complete a bounce period.Thus, Coulomb collision scattering effects are dominant for times t up to the minimum of 1/ν eff or 2π/ω trap .They are also critical for producing temporal irreversibility for finite times t 1/ν eff in weakly collisional plasmas when particle trapping becomes important or even dominant.
The Landau analysis [1] of wave damping is a linear theory.However, it has been demonstrated recently [16][17][18] that when the collisionless Vlasov equation is used, obtaining temporal irreversibility in Landau damping on long time scales requires consideration of nonlinear effects.In particular, third order (echo-type responses) and higher order terms produce heteroclinic (temporally irreversible) solutions in which dynamical chaos ensues when a KAM-type condition for localized solutions is violated on very long but finite time scales.When the probabilistic Coulomb collisional scattering effects discussed in this paper are large enough they damp echoes [5,6].It remains to be determined how these collisional effects interact with and potentially modify the nonlinear effects discussed in Refs.[16][17][18] for most physically relevant plasmas which are intrinsically weakly collisional.
The novel Green function procedure developed in this paper for exploring Coulomb collisional scattering effects on the temporal evolution of the plasma response to a wave and the wave-particle resonance in weakly collisional plasmas should be useful for other plasma applications.It could be particularly helpful in determining the effective collision frequency that is relevant for resonance broadening and temporal irreversibility in plasmas.

FIG. 1 .
FIG. 1. Thick arrows indicate small Coulomb collisional scattering of the pitch-angle ϑ0 about ϑ and speed v0 about v.The wave phase speed in the k direction is Vϕ ≡ ω/k.Particles with u ≡ v • k/k = Vϕ are resonant with the wave.

FIG. 2 .
FIG. 2. Real part of = 0 integrand i e −i ∆uz /(1 − i ⊥ z 2 ) in Iν0 integral which represents the in-phase distribution function response to a wave as a function of dimensionless time z ≡ kuτ wave turn on for various Here ⊥ = 10 −4 .

3 FIG. 3 .
FIG. 3. Real part of the⊥ = 0 integrand i e −i ∆uz− z 3in the Iν0 integral which represents the in-phase distribution function response to a wave as a function of dimensionless time z ≡ kuτ after wave turn on for various ∆u.Here = 10 −4 /30.