Quasi-linear transport approach to equilibration of quark-gluon plasmas
aa r X i v : . [ h e p - ph ] F e b Quasi-linear transport approach to equilibration of quark-gluon plasmas
Stanis law Mr´owczy´nski
Institute of Physics, Jan Kochanowski University,25-406 Kielce, ul. ´Swi¸etokrzyska 15, Poland andSo ltan Institute for Nuclear Studies, 00-681 Warsaw, ul. Ho˙za 69, Poland
Berndt M¨uller
Department of Physics & CTMS, Duke University, Durham, NC 27708, USA (Dated: 28-th January 2010)We derive the transport equations of quark-gluon plasma in the quasi-linear approximation. Theequations are either of the Balescu-Lenard or Fokker-Planck form. The plasma’s dynamics is as-sumed to be governed by longitudinal chromoelectric fields. The isotropic plasma, which is stable,and the two-stream system, which is unstable, are considered in detail. A process of equilibrationis briefly discussed in both cases. The peaks of the two-stream distribution are shown to rapidlydissolve in time.
I. INTRODUCTION
The quark-gluon plasma (QGP), which is produced inrelativistic heavy-ion collisions, is believed to be equili-brated within a time interval of order of 1 fm/ c or evenshorter [1]. Such a fast equilibration is naturally ex-plained assuming that the quark-gluon plasma is stronglycoupled [2–4]. Then, scattering processes are very fre-quent and relaxation times are short. However, the the-ory of high-energy density QCD [5] suggests that due tothe existence of a large momentum scale Q s , at whichthe gluon density saturates, the plasma is rather weaklycoupled at the early stage of the collision because ofasymptotic freedom. Experimental data on jet quench-ing indicate that the coupling constant α s ≤ . equilibrium quark-gluon plasma. Ourmotivation is rather different. We are interested in equi-libration of quark-gluon plasma, in particular in the equi-libration of the system which is initially unstable. Thus,we intend to study how fluctuating deviations from aquasi-stationary non-equilibrium state influence the sys-tem’s bulk or average momentum distribution. This ef-fect of back reaction is particularly important in the caseof unstable systems. The linear response theory describeshow unstable modes initially grow in the presence ofa non-equilibrium momentum distribution, but it saysnothing on how the modes modify the plasma momentumdistribution. Thus, the problem of equilibration cannotbe addressed in such a theory.Our objective here is to derive the transport equationswhere the bulk distribution function slowly evolves dueto the interaction with fluctuating chromodynamic fields.We actually consider only a simplified problem of QGP ina self-consistently generated longitudinal chromoelectricfield. This simplification is not much needed for isotropicplasma but it appears crucial to study anisotropic sys-tems. Taking into account only the longitudinal chromo-electric field, we obtain the transport equations of theFokker-Planck or Balescu-Lenard form which describethe effect of back reaction. A similar, but incompleteeffort was undertaken by Akkelin [39]. The derivationpresented here closely follows the procedure developedfor the electromagnetic plasma, where it is known as thequasi-linear theory or the theory of a weakly turbulentplasma [40–42]. The theory assumes that the distribu-tion function of plasma particles can be decomposed intoa large but slowly varying regular part and a small fluc-tuating or turbulent one which oscillates fast. The av-erage over the statistical ensemble of the turbulent partis assumed to vanish and thus the average of the distri-bution function equals its regular part. The turbulentcontribution to the distribution function obeys the colli-sionless transport equation while the transport equationof the regular part, which is of our main interest here,is determined by the fluctuation spectra. The fluctua-tions of chromodynamic fields, which are used to derivethe quasi-linear transport equations, were studied in [43]where stable and unstable plasma states were considered.The Fokker-Planck equation derived here is somewhatsimilar to the equation obtained in [44, 45]. It was usedthere to show that the chromomagnetized quark-gluonplasma exhibits an anomalous shear viscosity, as pres-ence of the domains of chromomagnetic field leads to themomentum transport in the plasma.Our paper is organized as follows. In Sec. II we presentthe QGP transport equations; the notation and conven-tions are introduced. The decomposition of the distri-bution functions into the regular and turbulent parts isdiscussed in Sec. III. The explicit expressions of thefluctuating distribution functions which obey collision-less transport equations are derived in Sec. IV. A generalform of the equations of the regular distribution functionsis found here as well. Further discussion splits into twoparallel parts: Sec. V is devoted to the stable isotropicplasma while in Sec. VI the unstable two-stream systemis discussed. Although we neglect transverse chromody-namic fields, the collision terms of transport equations,which are found here for an isotropic plasma, are verysimilar to those derived in [19, 21, 25, 25, 26, 29–33]. Asan application of the transport equations we derived, aprocess of equilibration of the isotropic plasma and of thetwo-stream system is discussed. The paper closes with asummary of our considerations and outlook. II. PRELIMINARIES
The transport theory of a quark-gluon plasma, whichforms the basis of our analysis, is formulated in terms ofparticles and classical fields. The particles - quarks, anti-quarks and gluons - should be understood as sufficientlyhard quasiparticle excitations of quantum fields of QCDwhile the classical fields are highly populated soft gluonicmodes. An excitation is called “hard”, when its momen-tum in the equilibrium rest frame is of order of the tem-perature T , and it is called “soft” when the momentumis of order gT with g being the coupling constant. Sincewe consider a weakly coupled quark-gluon plasma, thecoupling constant is assumed to be small g ≪
1. In ourfurther considerations the quasiparticles are treated asclassical particles obeying Boltzmann statistics but the effect of quantum statistics can be easily taken into ac-count.The transport equations of quarks, antiquarks and glu-ons are assumed to be of the form D Q ( t, r , p ) − { F ( t, r ) , ∇ p Q ( t, r , p ) } = 0 , D ¯ Q ( t, r , p ) + 12 { F ( t, r ) , ∇ p ¯ Q ( t, r , p ) } = 0 , (1) D G ( t, r , p ) − { F ( t, r ) , ∇ p G ( t, r , p ) } = 0 . The (anti-)quark distribution functions Q ( t, r , p ) and¯ Q ( t, r , p ), which are N c × N c hermitean matrices, be-long to the fundamental representation of the SU( N c )group, while the gluon distribution function G ( t, r , p ),which is a ( N c − × ( N c −
1) matrix, belongs to the ad-joint representation. The distribution functions dependon the time ( t ), position ( r ) and momentum ( p ) vari-ables. There is no explicit dependence on the time-like( µ = 0) component of the four-vector p µ as the distri-bution functions are assumed to be non-zero only formomenta obeying the mass-shell constraint p µ p µ = 0.Because the partons are assumed to be massless, the ve-locity v equals p /E p with E p = | p | . D ≡ D + v · D isthe covariant substantive derivative given by the covari-ant derivative which in the four-vector notation reads D µ ≡ ∂ µ − ig [ A µ ( x ) , · · · ] with A µ ( x ) being the chromo-dynamic potential. The mean-filed terms of the transportequations (1) are expressed through the color Lorentzforce F ( t, r ) ≡ g (cid:0) E ( t, r ) + v × B ( t, r ) (cid:1) . The chromo-electric E ( t, r ) and chromomagnetic B ( t, r ) fields belongto either the fundamental or adjoint representation. Tosimplify the notation we use the same symbols D , D , D , E , and B for a given quantity in the fundamentalor adjoint representation. The symbol { . . . , . . . } denotesthe anticommutator.The collision terms are neglected in the transport equa-tions (1). The collisionless equations are applicable inthree physically different situations: when the distribu-tion function is of (local) equilibrium form; when thetimescale of processes of interest is much shorter thanthe average temporal separation of parton collisions; andwhen the system dynamics is dominated by the meanfield. In our study we refer to all three situations. Whenthe equilibration of isotropic plasma is discussed, it is cru-cial that the collision terms vanish in local equilibrium.In the case of unstable two-stream system, the effectsof collisions can be initially neglected, as the growth ofunstable modes is very fast. Later on, the strong fieldsbecome mostly responsible for the system’s evolutionThe transport equations are supplemented by theYang-Mills equations describing a self-consistent gener-ation of the chromoelectric and chromomagnetic fields.The equations read D · E ( t, r ) = ρ ( t, r ) D · B ( t, r ) = 0 , D × E ( t, r ) = − D B ( t, r ) , (2) D × B ( t, r ) = j ( t, r ) + D E ( t, r ) , where the color four-current j µ = ( ρ, j ) in the adjointrepresentation equals j µa ( t, r ) = − g Z d p (2 π ) p µ E p Tr h T a G ( t, r , p )+ τ a (cid:0) Q ( t, r , p ) − ¯ Q ( t, r , p ) (cid:1)i , (3)where τ a , T a with a = 1 , ... , N c − N c ) groupgenerators in the fundamental and adjoint representa-tions, normalized as Tr[ τ a τ b ] = δ ab and Tr[ T a T b ] = N c δ ab . The set of transport equations (1) and Yang-Mills equations (2) is covariant with respect to SU( N c )gauge transformations. III. REGULAR AND FLUCTUATINGQUANTITIES
We assume that the chromodynamic fields and distri-bution functions which enter the set of transport equa-tions can be decomposed into the regular and fluctuat-ing components. The quark distribution function is thuswritten down as Q ( t, r , p ) = h Q ( t, r , p ) i + δQ ( t, r , p ) , (4)where h· · ·i denotes ensemble average; h Q ( t, r , p ) i iscalled the regular part while δQ ( t, r , p ) is called the fluc-tuating or turbulent one. It directly follows from Eq. (4)that h δQ i = 0. The regular contribution is assumed tobe white, and it is expressed as h Q ( t, r , p ) i = n ( t, r , p ) I , (5)where I is the unit matrix in color space. Since the distri-bution function transforms under gauge transformationsas Q → U QU − , where U is the transformation matrix,the regular contribution of the form (5) is gauge inde-pendent. We also assume that |h Q i| ≫ | δQ | , |∇ p h Q i| ≫ |∇ p δQ | , (6)but at the same time (cid:12)(cid:12)(cid:12)(cid:12) ∂δQ∂t (cid:12)(cid:12)(cid:12)(cid:12) ≫ (cid:12)(cid:12)(cid:12)(cid:12) ∂ h Q i ∂t (cid:12)(cid:12)(cid:12)(cid:12) , |∇ δQ | ≫ |∇h Q i| . (7)Analogous conditions are assumed for the antiquark andgluon distribution functions. What concerns the chromo-dynamic fields, we assume in accordance with (5) thattheir regular parts vanish and thus h E ( t, r ) i = h B ( t, r ) i = 0 . (8)We substitute the distribution functions (4) into thetransport equations and the Yang-Mills equations andlinearize the equations in the fluctuating contributions. The linearized transport and Yang-Mills equations re-main rather complex. Therefore, we discuss here a sim-plified problem: we consider a QGP in the presence ofturbulent longitudinal chromoelectric fields, but neglectthe chromomagnetic and transverse chromoelectric fields.This simplification can be avoided for an isotropic plasmabut it is needed, as explained below, to make progress onan analytical treatment for anisotropic systems which areour main interest here. The simplified transport equa-tions then read D δQ ( t, r , p ) − g E ( t, r ) · ∇ p n ( t, r , p ) = 0 , D δ ¯ Q ( t, r , p ) + g E ( t, r ) · ∇ p ¯ n ( t, r , p ) = 0 , (9) D δG ( t, r , p ) − g E ( t, r ) · ∇ p n g ( t, r , p ) = 0 , where D ≡ ∂∂t + v · ∇ denotes from now on the material(not covariant) derivative.The equation describing the self-consistent generationof a longitudinal chromoelectric field is ∇ · E a ( t, r ) = ρ a ( t, r ) = − g Z d p (2 π ) δN a ( t, r , p ) , (10)where δN a ( t, r , p ) ≡ Tr (cid:2) τ a (cid:0) δQ ( t, r , p ) − δ ¯ Q ( t, r , p ) (cid:1) + T a δG ( t, r , p ) (cid:3) . (11)The linearized equations are formally Abelian but theyinclude a fundamentally non-Abelian effect, i. e. thegluon contribution to the color current. Therefore, thegluon-gluon coupling is partly taken into account. Thelinearized Yang-Mills equation corresponds to the multi-component electrodynamics of N c charges (in the so-called Heaviside-Lorentz system of units). The equations,however, are no longer manifestly covariant with respectto SU( N c ) gauge transformations. Nevertheless, our finalresults are gauge independent.We now substitute the distribution functions (4) intothe transport equations (1). Instead of linearizing theequations in the fluctuating contributions, we take theensemble average of the resulting equations and traceover the color indices. Thus we get D n − gN c Tr h E · ∇ p δQ i = 0 , D ¯ n + gN c Tr h E · ∇ p δ ¯ Q i = 0 , (12) D n g − gN c − h E · ∇ p δG i = 0 . Since the regular part of distribution function is assumedto be color neutral, see Eq. (5), the terms of the formTr[ h E · ∇ p n i ] vanish because the field E is traceless. Thetrace over color indices also cancels the terms originatingfrom covariant derivatives like Tr h [ A µ , δQ ] i . We finallynote that the regular distribution function n is gaugeindependent and so is Tr h E · ∇ p δQ i . IV. SOLUTION OF THE LINEARIZEDEQUATIONS
Due to the condition (7), the space-time dependenceof the regular distribution functions is neglected in thelinearized transport equations (9) and the equations be-come easily solvable. We solve Eq. (9) with the initialconditions δQ ( t = 0 , r , p ) = δQ ( r , p ) ,δ ¯ Q ( t = 0 , r , p ) = δ ¯ Q ( r , p ) , (13) δG ( t = 0 , r , p ) = δG ( r , p ) , using the one-sided Fourier transformation defined as f ( ω, k ) = Z ∞ dt Z d re i ( ωt − k · r ) f ( t, r ) . (14)The inverse transformation is f ( t, r ) = Z ∞ + iσ −∞ + iσ dω π Z d k (2 π ) e − i ( ωt − k · r ) f ( ω, k ) , (15)where the real parameter σ > ω is taken along a straight line in the complex ω − plane, parallel to the real axis, above allsingularities of f ( ω, k ). We note that − iωf ( ω, k ) = f ( t = 0 , k ) (16)+ Z ∞ dt Z d re i ( ωt − k · r ) ∂f ( t, r ) ∂t . The linearized transport equations (9), which aretransformed by means of the one-sided Fourier transfor-mation, are solved as δQ ( ω, k , p ) = i g E · ∇ p n ( p ) + δQ ( k , p ) ω − k · v ,δ ¯ Q ( ω, k , p ) = i g E · ∇ p ¯ n ( p ) − δ ¯ Q ( k , p ) ω − k · v , (17) δG ( ω, k , p ) = i g E · ∇ p n g ( p ) + δG ( k , p ) ω − k · v . We note that the color electric field E ( ω, k ) retains its fullfrequency and wave number dependence in these equa-tions. Inverting the one-sided Fourier transformation,one finds the solutions of linearized transport equationsas δQ ( t, r , p ) = g Z t dt ′ E (cid:0) t ′ , r − v ( t − t ′ ) (cid:1) · ∇ p n ( p ) + δQ ( r − v t, p ) ,δ ¯ Q ( t, r , p ) = − g Z t dt ′ E (cid:0) t ′ , r − v ( t − t ′ ) (cid:1) · ∇ p ¯ n ( p ) + δ ¯ Q ( r − v t, p ) , (18) δG ( t, r , p ) = g Z t dt ′ E (cid:0) t ′ , r − v ( t − t ′ ) (cid:1) · ∇ p n g ( p ) + δG ( r − v t, p ) , where we assumed that E ( ω, k ) is an analytic function of ω . With the help of solutions (18), the force terms in thetransport equations (12) become h E ( t, r ) · ∇ p δQ ( t, r , p ) i = g Z t dt ′ ∇ ip h E i ( t, r ) E j (cid:0) t ′ , r − v ( t − t ′ ) (cid:1) i∇ jp n ( p ) + ∇ ip h E i ( t, r ) δQ ( r − v t, p ) i , h E ( t, r ) · ∇ p δ ¯ Q ( t, r , p ) i = − g Z t dt ′ ∇ ip h E i ( t, r ) E j (cid:0) t ′ , r − v ( t − t ′ ) (cid:1) i∇ jp ¯ n ( p ) + ∇ ip h E i ( t, r ) δ ¯ Q ( r − v t, p ) i , (19) h E ( t, r ) · ∇ p δG ( t, r , p ) i = g Z t dt ′ ∇ ip h E i ( t, r ) E j (cid:0) t ′ , r − v ( t − t ′ ) (cid:1) i∇ jp n g ( p ) + ∇ ip h E i ( t, r ) δG ( r − v t, p ) i , We conclude that the transport equations (12) are determined by the correlation functions h E i ( t, r ) E j (cid:0) t ′ , r ′ ) i , h E i ( t, r ) δQ ( r ′ , p ) i , h E i ( t, r ) δ ¯ Q ( r ′ , p ) i , and h E i ( t, r ) δG ( r ′ , p ) i . To compute these functions, the state of the plasmamust be specified. Although we are mainly interested in an anisotropic plasma, we start with the isotropic case.Thereafter, we consider the two-stream system. V. ISOTROPIC PLASMA
For the case of isotropic plasma, the correlation functions of both longitudinal and transverse fields are well known[43]. Here we limit our considerations to longitudinal chromoelectric fields, whose correlation function is [43]: h E ia ( t, r ) E jb ( t ′ , r ′ ) i = g δ ab Z ∞ + iσ −∞ + iσ dω π Z d k (2 π ) Z ∞ + iσ ′ −∞ + iσ ′ dω ′ π Z d k ′ (2 π ) e − i ( ωt + ω ′ t ′ − k · r − k ′ · r ′ ) (20) × k i k ′ j k k ′ (2 π ) δ (3) ( k ′ + k ) ε L ( ω, k ) ε L ( ω ′ , k ′ ) Z d p (2 π ) f ( p )( ω − k · v )( ω ′ − k ′ · v ) , where f ( p ) ≡ n ( p ) + ¯ n ( p ) + 2 N c n g ( p ) and ε L ( ω, k ) isthe longitudinal chromodielectric function discussed inthe Appendix. Note that we do not assume that n ( p ),¯ n ( p ), and n g ( p ) are given by the thermal equilibriumdistributions, only that they are isotropic functions of p .Zeroes of ε L ( ω, k ) and ε L ( ω ′ , k ′ ) as well as those of thedenominators ( ω − k · v ) and ( ω ′ − k ′ · v ) contribute tothe integrals over ω and ω ′ . However, when the plasmasystem under consideration is stable with respect to lon- gitudinal modes, all zeroes of ε L lie in the lower half-planeof complex ω . Consequently, the contributions associatedwith these zeroes exponentially decay in time, and theyvanish in the long time limit of both t and t ′ .We are further interested in the long-time limit of h E i ( t, r ) E i ( t ′ , r ′ ) i . The only non-vanishing contributioncorresponds to the poles at ω = k · v and ω ′ = k ′ · v .This contribution reads h E ia ( t, r ) E jb ( t ′ , r ′ ) i = g δ ab Z d p (2 π ) Z d k (2 π ) e − i k · (cid:2) v ( t − t ′ ) − ( r − r ′ ) (cid:3) k i k j k f ( p ) | ε L ( k · v , k ) | , (21)The correlation functions like h E ia ( t, r ) δQ ( r ′ , p ′ ) i are not computed in ref. [43] but they can be readily inferredfrom the formulas given there. One finds h E ia ( t, r ) δQ ( r ′ , p ′ ) i = − g τ a Z ∞ + iσ −∞ + iσ dω π Z d k (2 π ) Z d k ′ (2 π ) e − i ( ωt − k · r − k ′ · r ′ ) k i k (2 π ) δ (3) ( k ′ + k ) ε L ( ω, k ) n ( p ′ ) ω − k · v ′ = ig τ a Z d k (2 π ) e − i k · ( v ′ t − r + r ′ )) k i k n ( p ′ ) ε L ( k · v ′ , k ) , (22)where the last equality holds in the long-time limit which is carried by the contribution corresponding to the pole ω = k · v ′ . Similarly, one finds h E ia ( t, r ) δ ¯ Q ( r ′ , p ′ ) i = g τ a Z ∞ + iσ −∞ + iσ dω π Z d k (2 π ) Z d k ′ (2 π ) e − i ( ωt − k · r − k ′ · r ′ ) k i k (2 π ) δ (3) ( k ′ + k ) ε L ( ω, k ) ¯ n ( p ′ ) ω − k · v ′ = − ig τ a Z d k (2 π ) e − i k · ( v ′ t − r + r ′ )) k i k ¯ n ( p ′ ) ε L ( k · v ′ , k ) , (23) h E ia ( t, r ) δG ( r ′ , p ′ ) i = − g T a Z ∞ + iσ −∞ + iσ dω π Z d k (2 π ) Z d k ′ (2 π ) e − i ( ωt − k · r − k ′ · r ′ ) k i k (2 π ) δ (3) ( k ′ + k ) ε L ( ω, k ) n g ( p ′ ) ω − k · v ′ = ig T a Z d k (2 π ) e − i k · ( v ′ t − r + r ′ )) k i k n g ( p ′ ) ε L ( k · v ′ , k ) . (24)Substituting the correlation functions (21, 22) into (19), one findsTr h E ( t, r ) · ∇ p δQ ( t, r , p ) i = g N c − Z t dt ′ ∇ ip Z d p ′ (2 π ) Z d k (2 π ) e i k · ( v − v ′ )( t − t ′ ) k i k j k f ( p ′ ) | ε L ( k · v ′ , k ) | ∇ jp n ( p )+ i g N c − ∇ ip Z d k (2 π ) k i k n ( p ) ε L ( k · v , k ) , (25)and analogous expressions for Tr h E ( t, r ) · ∇ p δ ¯ Q ( t, r , p ) i and Tr h E ( t, r ) · ∇ p δG ( t, r , p ) i . As shown in [43],Tr h E i ( t, r ) E j ( t ′ , r ′ ) i is gauge independent within the linear response approach. The same arguments used to showthis apply to Tr h E i ( t, r ) δQ ( r ′ , p ′ ) i . Thus, we conclude that the collision term of the transport equation (12),Tr h E ( t, r ) · ∇ p δQ ( t, r , p ) i , is gauge independent.Let us now discuss the first term on the r.h.s. of (25). Computing the integral over t ′ we getTr h E ( t, r ) · ∇ p δQ ( t, r , p ) i (1) = g N c − ∇ ip Z d p ′ (2 π ) Z d k (2 π ) k i k j k f ( p ′ ) | ε L ( k · v ′ , k ) | (26) × (cid:16) − i cos( k · ( v − v ′ ) t ) − k · ( v − v ′ ) + sin( k · ( v − v ′ ) t ) k · ( v − v ′ ) (cid:17) ∇ jp n ( p ) . The first term does not contribute to the integral because it is an odd function of k . Since in the limit t → ∞ wehave lim t →∞ sin( k · ( v − v ′ ) t ) k · ( v − v ′ ) = πδ (cid:0) k · ( v − v ′ ) (cid:1) , (27)one finally findsTr h E ( t, r ) · ∇ p δQ ( t, r , p ) i (1) = g π ( N c − ∇ ip Z d p ′ (2 π ) Z d k (2 π ) k i k j k f ( p ′ ) | ε L ( k · v ′ , k ) | δ (cid:0) k · ( v − v ′ ) (cid:1) ∇ jp n ( p ) . (28)Analogously, one computes Tr h E ( t, r ) · ∇ p δ ¯ Q ( t, r , p ) i (1) and Tr h E ( t, r ) · ∇ p δG ( t, r , p ) i (1) .The second term on the r.h.s. of (25) can be written asTr h E ( t, r ) · ∇ p δQ ( t, r , p ) i (2) = g N c − ∇ ip Z d k (2 π ) k i k ℑ ε L ( k · v , k ) | ε L ( k · v , k ) | n ( p ) , (29)because the term with ℜ ε L ( k · v , k ) is an odd function of k (see Appendix). Alternatively, one can argue thatthe r.h.s. of (29) has to be real as the l.h.s. is real. In the same way one finds: h E ( t, r ) · ∇ p δ ¯ Q ( t, r , p ) i (2) and h E ( t, r ) · ∇ p δG ( t, r , p ) i (2) .With the formulas derived above, the transport equations (12) can now be written either in the Balescu-Lenardform or the Fokker-Planck form. A. Balescu-Lenard equations
Using the formula (A.2) to express ℑ ε L through thedistribution function, the transport equations (12) getthe Balescu-Lenard form [42] D n ( t, r , p ) = ∇ p · S [ n, ¯ n, n g ] , D ¯ n ( t, r , p ) = ∇ p · ¯ S [ n, ¯ n, n g ] , (30) D n g ( t, r , p ) = ∇ p · S g [ n, ¯ n, n g ] , where, as previously, D is the material derivative, and S i [ n, ¯ n, n g ] ≡ Z d p ′ (2 π ) B ij ( v , v ′ ) × h ∇ jp n ( p ) f ( p ′ ) − n ( p ) ∇ jp ′ f ( p ′ ) i , ¯ S i [ n, ¯ n, n g ] ≡ Z d p ′ (2 π ) B ij ( v , v ′ ) (31) × h ∇ jp ¯ n ( p ) f ( p ′ ) − ¯ n ( p ) ∇ jp ′ f ( p ′ ) i ,S ig [ n, ¯ n, n g ] ≡ Z d p ′ (2 π ) B ijg ( v , v ′ ) × h ∇ jp n g ( p ) f ( p ′ ) − n g ( p ) ∇ jp ′ f ( p ′ ) i , with B ij ( v , v ′ ) ≡ g N c − N c Z d k (2 π ) k i k j k πδ (cid:0) k · ( v − v ′ ) (cid:1) | ε L ( k · v , k ) | , (32)and B ijg ( v , v ′ ) = 2 N c N c − B ij ( v , v ′ ) . (33)Since the interaction processes that are taken into ac-count conserve the numbers of particles of every species( q, ¯ q, g ), the transport equations in the Balescu-Lenardform (30) can be seen as continuity equations in momen-tum space with S , ¯ S , S g playing a role of currents. Oneobserves that for classical equilibrium functions f eq ( p ) , n eq ( p ) , ¯ n eq ( p ) , n eq g ( p ) ∼ e − E p /T , (34)the collision terms (31) vanish, as expected, because( v i − v ′ i ) B ij ( v , v ′ ) = 0 . (35)If ε L ( ω, k ) is replaced by unity, i.e. if one ignoresthe chromodielectric properties of the plasma, the ten-sor B ij ( v , v ′ ) is easily found to be B ij ( v , v ′ ) = g π N c − N c L | v − v ′ | (36) × (cid:18) δ ij − ( v i − v ′ i )( v j − v ′ j )( v − v ′ ) (cid:19) , with L ≡ Z dk/k = ln( k max /k min ) . (37)The parameter L is called the Coulomb logarithm andthe collision term with the tensor B ij ( v , v ′ ) of the form(36) is called the Landau collision term [42]. Estimating k max as the system temperature T and k min as the Debyemass m D ∼ gT , one finds L ∼ ln(1 /g ).It may appear strange that we start with the collision-less transport equations (1) to derive the collision terms.This procedure, which is commonly used in the plasmaliterature, is well justified, however, see e.g. Ref. [42].The collision terms, which are derived above, represent the effect of fluctuating soft fields on the hard quasipar-ticles. It is important to note that the collision terms aredominated, as it should be, by the soft wave vectors.Consequently, the collisions of quasiparticles involvingthe exchange of hard momenta, which are neglected inEqs. (1), do not need to be taken into account at lowestorder.
B. Fokker-Planck equations
Sometimes it is more convenient to use the transportequations in the Fokker-Planck form. Following Ref. [42],one rewrites Eqs. (30) as (cid:0)
D − ∇ ip X ij ( v ) ∇ jp − ∇ ip Y i ( v ) (cid:1) n ( t, r , p ) = 0 , (cid:0) D − ∇ ip X ij ( v ) ∇ jp − ∇ ip Y i ( v ) (cid:1) ¯ n ( t, r , p ) = 0 , (38) (cid:0) D − ∇ ip X ijg ( v ) ∇ jp − ∇ ip Y ig ( v ) (cid:1) n g ( t, r , p ) = 0 , where X ij ( v ) ≡ g N c − Z d p ′ (2 π ) Z d k (2 π ) k i k j k f ( p ′ ) | ε L ( k · v ′ , k ) | πδ (cid:0) k · ( v − v ′ ) (cid:1) = Z d p ′ (2 π ) f ( p ′ ) B ij ( v , v ′ ) , (39) Y i ( v ) ≡ g N c − Z d k (2 π ) k i k ℑ ε L ( k · v , k ) | ε L ( k · v , k ) | (40)= − g N c − Z d p ′ (2 π ) Z d k (2 π ) k i k k · ∇ p ′ f ( p ′ ) | ε L ( k · v , k ) | πδ (cid:0) k · ( v − v ′ ) (cid:1) = − Z d p ′ (2 π ) ∇ jp ′ f ( p ′ ) B ij ( v , v ′ ) , and X ijg ( v ) = 2 N c N c − X ij ( v ) , (41) Y ig ( v ) = 2 N c N c − Y i ( v ) . (42)The equations (38) appear to be linear but actually theyare not: the coefficients X ij ( v ), Y i ( v ), X ijg ( v ) and Y ig ( v )depend on the distribution functions. When the dis-tribution functions are of the classical equilibrium form( f eq ( p ), n eq ( p ), ¯ n eq ( p ), n eq g ( p ) ∼ e − E p /T ), we have therelation Y i ( v ) = v i T X ij ( v ) . (43)Consequently, the Fokker-Planck collision terms vanishin equilibrium, as do the Balescu-Lenard collision terms.Since the system is assumed to be isotropic, X ij ( v )and Y i ( v ) can be expressed as follows: X ij ( v ) = a δ ij + b v i v j , (44) Y i ( v ) = c v i , (45)with a = 12 Z d p ′ (2 π ) f ( p ′ ) h δ ji − v j v i i B ij ( v , v ′ ) (46) b = 12 Z d p ′ (2 π ) f ( p ′ ) h v j v i − δ ji i B ij ( v , v ′ ) (47) c = − Z d p ′ (2 π ) v i ∇ jp ′ f ( p ′ ) B ij ( v , v ′ ) . (48)Because of the system’s isotropy, the coefficients a , b , c can depend only on v . In the ultrarelativistic limit,which is adopted here, v = 1, and consequently a , b , c are independent of v . We also note that in equilibriumthe coefficients are related as T c = a + b , (49)which follows from Eq. (43).When ε L ( ω, k ) is replaced, as previously, by unity onefinds that b = 0 and a ≡ g π ( N c − L Z ∞ dp p f ( p ) , (50) c ≡ − g π ( N c − L Z ∞ dp p df ( p ) dp . (51)Using the relations (A.5), the coefficient c can be ex-pressed in terms of the Debye mass as c = g π ( N c − L m D . (52)Furthermore, in equilibrium, a = c T .We note that in spite of our neglect of transverse chro-modynamic fields, the collision terms for the isotropicplasma derived here are very similar to those derived in[19, 21, 25, 26, 29–33]. C. Equilibration of an isotropic plasma
As an application of the Fokker-Planck equations (38)we discuss the problem of plasma equilibration. In thissection we limit our considerations to quarks, as the anal-ysis for antiquarks and gluons is very similar. We con-sider the system which is homogenous and mostly equi-librated but a small fraction ( λ ≪
1) of the particles,denoted by δn ( t, p ), is out of equilibrium. One asks onwhat time scale the system reaches the equilibrium. Thedistribution function is assumed to be of the form n ( t, p ) = (1 − λ ) n eq ( p ) + λ δn ( t, p ) . (53)In the course of equilibration n ( t, p ) tends to n eq ( p ).Since the particle number is conserved within the trans-port theory approach developed here, δn ( t, p ) is not re-duced to zero in the equilibration process but it tends to n eq ( p ).We define the rate of equilibration Γ through the rela-tion ∂n∂t = Γ δn . (54)We note that Γ is either positive, when δn grows goingto n eq , and it is negative, when δn decreases going to n eq . Using the Fokker-Planck equation (38), the defini-tion (54) givesΓ = 1 δn (cid:0) ∇ ip X ij ∇ jp + ∇ ip Y i (cid:1) δn . (55)Since the fraction of particles with non-equilibrium dis-tribution is assumed to be small, the coefficients a , b , c from the formulas (44, 45) are given by the equilibriumfunction n eq ∼ e − E p /T . Using the approximate expres-sion of a (50) with b = 0 and c = a/T , Eq. (55) is rewrit-ten as Γ = aδn (cid:16) ∇ p + 1 T v · ∇ p + 2 T E p (cid:17) δn . (56)The equilibration rate obviously depends of the formof δn . Here we consider the case where the small fractionof partons has an equilibrium distribution of temperature T which differs from the temperature T of the bulk of thepartons. Thus, δn ∼ e − E p /T . Then, the equilibrationrate (56) equalsΓ = a T − T T T E p ( E p − T ) . (57)For T = T , the whole system is in equilibrium and, asexpected, Γ = 0. When T > T , the distribution e − E p /T is steeper than e − E p /T . Equation (57) tells us that δn de-creases for E p < T and grows for E p > T during theequilibration process. When T < T , we have the oppo-site situation. In both cases, the slope of the distributionfunction δn tends to the slope of n eq . With the coefficient a given by (50), the formula (57) quantitatively predictshow fast the equilibrium is approached. VI. TWO-STREAM SYSTEM
The two-stream configuration provides an interestingcase of an unstable plasma. The correlation function oflongitudinal chromoelectric fields, which is needed to de-rive the transport equations, was computed in [43]. Un-fortunately the correlation function for transverse fieldsis not known. This limits our considerations to longitu-dinal fields.The distribution function of the two-stream system ischosen as f ( p ) = (2 π ) n h δ (3) ( p − q ) + δ (3) ( p + q ) i , (58)where n is the effective parton density in a single stream.The distribution function (58) should be treated as anidealization of the two-peak distribution where the par-ticles have momenta close to q or − q .To compute ε L ( ω, k ) we first perform an integrationby parts in (A.1) and then substitute the distributionfunction (58) into the resulting formula. We obtain ε L ( ω, k ) = 1 − µ k − ( k · u ) k (cid:20) ω − k · u ) + 1( ω + k · u ) (cid:21) (59)= (cid:0) ω − ω + ( k ) (cid:1)(cid:0) ω + ω + ( k ) (cid:1)(cid:0) ω − ω − ( k ) (cid:1)(cid:0) ω + ω − ( k ) (cid:1)(cid:0) ω − ( k · u ) (cid:1) , where u ≡ q /E q is the stream velocity, µ ≡ g n/ E q and ± ω ± ( k ) are the four roots of the dispersion equation ε L ( ω, k ) = 0 which are explicitly given by ω ± ( k ) = 1 k (cid:20) k ( k · u ) + µ (cid:0) k − ( k · u ) (cid:1) ± µ r(cid:0) k − ( k · u ) (cid:1)(cid:16) k ( k · u ) + µ (cid:0) k − ( k · u ) (cid:1)(cid:17) (cid:21) . (60)One can show that 0 < ω + ( k ) ∈ R for any k , while ω − ( k ) is imaginary for k · u = 0 and k ( k · u ) < µ (cid:0) k − ( k · u ) (cid:1) . ω − represents the well-known two-stream electrostatic instability generated by a mechanism analogous to the Landaudamping. For k ( k · u ) ≥ µ (cid:0) k − ( k · u ) (cid:1) , the ω − mode is stable: 0 < ω − ( k ) ∈ R .The terms like h E ( t, r ) · ∇ p δQ ( t, r , p ) i , which enter the transport equations (12) are given by Eqs. (19). As for theisotropic plasma one needs to specify the the correlation functions h E ia ( ω, k ) E ib ( ω ′ , k ′ ) i , h E ( t, r ) δQ ( r ′ , p ′ ) i , etc. Thecorrelation function of the longitudinal fields h E ia ( ω, k ) E jb ( ω ′ , k ′ ) i was found in [43]: h E ia ( ω, k ) E jb ( ω ′ , k ′ ) i = − g δ ab n (2 π ) δ (3) ( k + k ′ ) k k i k j k h ωω ′ + ( k · u )( k ′ · u ) i (61) × ω − ( k · u ) (cid:0) ω − ω − ( k ) (cid:1)(cid:0) ω + ω − ( k ) (cid:1)(cid:0) ω − ω + ( k ) (cid:1)(cid:0) ω + ω + ( k ) (cid:1) × ω ′ − ( k ′ · u ) (cid:0) ω ′ − ω − ( k ′ ) (cid:1)(cid:0) ω ′ + ω − ( k ′ ) (cid:1)(cid:0) ω ′ − ω + ( k ′ ) (cid:1)(cid:0) ω ′ + ω + ( k ′ ) (cid:1) . We are particularly interested in the contributions of the unstable modes to the correlation function. For this reasonwe consider the domain of wave vectors obeying k · u = 0 and k ( k · u ) < µ (cid:0) k − ( k · u ) (cid:1) when ω − ( k ) is imaginaryand the mode is unstable. We write ω − ( k ) = iγ k with 0 < γ k ∈ R . The contribution coming from the modes ± ω − ( k )then equals [43] h E ia ( t, r ) E jb ( t ′ , r ′ ) i unstable = g δ ab n Z d k (2 π ) e i k ( r − r ′ ) k k i k j ( ω − ω − ) (cid:0) γ k + ( k · u ) (cid:1) γ k (62) × h(cid:0) γ k + ( k · u ) (cid:1) cosh (cid:0) γ k ( t + t ′ ) (cid:1) + (cid:0) γ k − ( k · u ) (cid:1) cosh (cid:0) γ k ( t − t ′ ) (cid:1)i . As Eq. (62) shows, the contribution of the unstablemodes to the field-field correlation function is spacetranslation invariant – it depends only on the difference( r − r ′ ). If the initial plasma is on average homogeneous,it remains so over the course of its evolution. The timedependence of the correlation function (62), however, isvery different from the spatial dependence. The elec-tric field grows exponentially and so does the correlationfunction, both in ( t + t ′ ) and ( t − t ′ ). The fluctuation spectrum also evolves in time as the growth rate of theunstable modes is wave-vector dependent. After a suffi-ciently long time the fluctuation spectrum will be domi-nated by the fastest growing modes.The correlation function h E ( t, r ) δQ ( r − v t, p ) i is, aspreviously, given by Eqs. (29). Since the dielectric func-tion (59) is real, the correlation functions h E ( t, r ) δQ ( r − v t, p ) i , h E ( t, r ) δ ¯ Q ( r − v t, p ) i and h E ( t, r ) δG ( r − v t, p ) i all vanish. Therefore,Tr h E ( t, r ) · ∇ p δQ ( t, r , p ) i = g Z t dt ′ ∇ ip h E i ( t, r ) E j (cid:0) t ′ , r − v ( t − t ′ ) (cid:1) i∇ jp n ( p )0= g N c − n Z t dt ′ ∇ ip Z d k (2 π ) e i kv ( t − t ′ ) k k i k j ( ω − ω − ) (cid:0) γ k + ( k · u ) (cid:1) γ k (63) × h(cid:0) γ k + ( k · u ) (cid:1) cosh (cid:0) γ k ( t + t ′ ) (cid:1) + (cid:0) γ k − ( k · u ) (cid:1) cosh (cid:0) γ k ( t − t ′ ) (cid:1)i ∇ jp n ( p ) . Performing the integration over t ′ and keeping only the real part, one findsTr h E ( t, r ) · ∇ p δQ ( t, r , p ) i = g N c − n ∇ ip Z d k (2 π ) k i k j k ( ω − ω − ) (cid:0) γ k + ( k · u ) (cid:1) γ k (cid:0) γ k + ( kv ) (cid:1) (64) × (cid:20) γ k sinh(2 γ k t ) + ( k · u ) (cid:16) γ k sinh(2 γ k t )+ ( k · v ) sin (cid:0) ( k · v ) t (cid:1) cosh( γ k t ) − γ k cos (cid:0) ( k · v ) t (cid:1) sinh( γ k t ) (cid:17)(cid:21) ∇ jp n ( p ) . Neglecting the oscillating terms, we finally getTr h E ( t, r ) · ∇ p δQ ( t, r , p ) i = g N c − n ∇ ip Z d k (2 π ) k i k j k ( ω − ω − ) (cid:0) γ k + ( k · u ) (cid:1) γ k (cid:0) γ k + ( kv ) (cid:1) sinh(2 γ k t ) ∇ jp n ( p ) . (65)In an analogous way one can obtain explicit expressions for h E ( t, r ) · ∇ p δ ¯ Q ( t, r , p ) i and h E ( t, r ) · ∇ p δG ( t, r , p ) i . Wedo not present these here, because they do not provide any new insight.Since we explicitly integrated over the distribution function (58) in deriving these results, we only give the transportequations (12) for the two-stream system in the Fokker-Planck form: (cid:0) D − ∇ ip X ij ( t, v ) ∇ jp (cid:1) (cid:26) n ( t, r , p )¯ n ( t, r , p ) (cid:27) = 0 , (cid:0) D − ∇ ip X ijg ( t, v ) ∇ jp (cid:1) n g ( t, r , p ) = 0 , (66)where X ij ( t, v ) ≡ g N c − N c n Z d k (2 π ) k i k j k ( ω + γ k ) (cid:0) γ k + ( k · u ) (cid:1) γ k (cid:0) γ k + ( kv ) (cid:1) sinh(2 γ k t ) , (67)and X ijg ( t, v ) ≡ N c X ij ( t, p ) / ( N c − u , with the latter beingchosen along the axis x . The only non-vanishing compo-nent of X ij ( t, v ) is then X xx ( t, v ). Neglecting the depen-dence of X xx ( t, v ) on p and assuming that the system ishomogenous, the Fokker-Planck equation (66) for quarksbecomes a one-dimensional diffusion equation ∂n ( t, p ) ∂t = D ( t ) ∂ n ( t, p ) ∂p x , (68)with the diffusion coefficient D ( t ) ≡ X xx ( t ) dependingon time approximately as D ( t ) = d e γt , (69)where d and γ are constants.If the distribution function is initially of the form n ( t = 0 , p ) = 2 π ˜ n δ ( p x − q ) , (70) where ˜ n is independent of p x , the solution of the diffusionequation (68) is found as n ( t, p ) = ˜ n s πγd ( e γt −
1) exp (cid:20) − γ ( p x − q ) d ( e γt − (cid:21) . (71)The distribution function (71) is normalized in such away that Z dp x π n ( t, p ) = ˜ n . According to the solution (71), the electric field growingdue to the electrostatic instability rapidly washes out thepeak-like structures of the two-stream distribution func-tion (58). It should be understood, however, that thesolution (71) is valid only for time intervals which aresufficiently short that the distribution function used tocompute the coefficient X ij ( t, v ) is not much differentfrom the function (58). Nevertheless, the solution (71)shows how the equilibration process commences.1 VII. SUMMARY AND OUTLOOK
We have developed here the quasi-linear transport the-ory of a weakly coupled quark-gluon plasma. Our mainmotivation was to study the equilibration of plasmas thatare initially unstable. The field fluctuation spectrum,which is found within the linear response approach, de-termines the evolution of the regular distribution func-tions. More specifically, the fluctuations of chromody-namic fields provide collision terms to the transport equa-tions of the regular distribution functions. We havelimited our considerations to longitudinal chromoelectricfields, as then the field correlation functions are knownfor both the isotropic and two-stream systems. The col-lision terms were found in either the Balescu-Lenard orFokker-Planck form. In the case of an isotropic plasma weshowed how the system equilibrates when a small fractionof particles has a different temperature than the bulk.The case of the two-stream system is more interest-ing. The Fokker-Planck equation could be approximatelywritten as an equation of diffusion in momentum space.The diffusion coefficient, which is given by the chromo-electric fields for the two-stream instability, exponentiallygrows in time. We found the exact solution of the diffu-sion equation, which showed that the peak-like structuresin the parton momentum distribution dissolve rapidly.In nonrelativistic plasmas it is often a well justified ap-proximation to keep only longitudinal electric fields andto neglect magnetic and transverse electric fields [40, 41].In the case of ultrarelativistic plasmas, this is no longertrue. If initially the fields are purely longitudinal, thetransverse fields are automatically generated, and theyare dynamically important. Therefore, the ultrarelativis-tic plasma considered here, where the transverse fieldsare neglected, should be rather treated as a toy modelwhich we have studied mostly for the sake of analyti-cal tractability. With this simplified example we havebeen able to elucidate some general features of the prob-lem. Physically better motivated situations will requiresubstantial numerical work, which is less conducive togeneral insights.The considerations presented here clearly demonstratethe usefulness of the quasi-linear transport theory forthe study of equilibration processes of quark-gluon plas-mas. As mentioned in the Introduction, numerical stud-ies indicate that the unstable chromomagnetic plasmamodes play an important role at the early stage of thequark-gluon plasma produced in relativistic heavy-ioncollisions. Therefore, it would be of considerable inter-est to compute the correlation functions of transversefields in arbitrary anisotropic plasmas in order to de-rive the relevant transport equations. As explained in[43], there is no conceptual difficulty in such a compu-tation, but one has to invert the matrix Σ ij ( ω, k ) ≡− k δ ij + k i k j + ω ε ij ( ω, k ). This is easily done forisotropic plasmas but for anisotropic plasmas one obtains a rather complex expression which is very cumbersomefor further analytic calculations [46]. Except for somespecial cases, numerical methods seem to be unavoid-able. Such computational studies are beyond the scopeof the present work but progress in this direction will behopefully reported soon. Acknowledgments
St. M. is grateful to the Physics Department of DukeUniversity, where this project was initiated, for warmhospitality during his visit. This work was supportedin part by the U. S. Department of Energy under grantDE-F02-05ER41367.
Appendix
We discuss here the longitudinal chromodielectric per-meability ε L ( ω, k ) which is known to be ε L ( ω, k ) = 1 + g k Z d p (2 π ) k · ∇ p f ( p ) ω − k · v + i + . (A.1)Applying the identity1 x ± i + = P x ∓ iπδ ( x )to Eq. (A.1), one immediately finds ℑ ε L ( ω, k ) ℑ ε L ( ω, k ) = − g k Z d p (2 π ) πδ ( ω − k · v ) k · ∇ p f ( p ) . (A.2)If the plasma is isotropic ∇ p f ( p ) can be expressed as ∇ p f ( p ) = df ( p ) dE p v . (A.3)And if the partons are additionally masslees, the integralin (A.1) factorizes into the angular integral and the inte-gral over p ≡ | p | . Then, one finds the real and imaginaryparts of the longitudinal chromodielectric permeability ε L ( ω, k ) as ℜ ε L ( ω, k ) = 1 + m D k (cid:20) − ω | k | ln (cid:12)(cid:12)(cid:12)(cid:12) ω + | k | ω − | k | (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , ℑ ε L ( ω, k ) = π k − ω ) m D ω | k | , (A.4)where the Debye mass m D is m D ≡ − g π Z ∞ dp p df ( p ) dp . (A.5)2 [1] U. W. Heinz, AIP Conf. Proc. (2005) 163.[2] V. M. Bannur, Eur. Phys. J. C , 169 (1999).[3] M. Gyulassy and L. McLerran, Nucl. Phys. A , 30(2005).[4] E. Shuryak, Prog. Part. Nucl. Phys. , 48 (2009).[5] L. D. McLerran and R. Venugopalan, Phys. Rev. D ,2233 (1994).[6] G. Y. Qin, J. Ruppert, C. Gale, S. Jeon, G. D. Mooreand M. G. Mustafa, Phys. Rev. Lett. , 072301 (2008).[7] J. L. Nagle, arXiv:0805.0299 [nucl-ex].[8] St. Mr´owczy´nski, Acta Phys. Polon. B , 427 (2006).[9] A. Dumitru and Y. Nara, Phys. Lett. B , 89 (2005).[10] A. Dumitru, Y. Nara and M. Strickland, Phys. Rev. D , 025016 (2007).[11] P. Arnold and G. D. Moore, Phys. Rev. D , 025006(2006).[12] H. T. Elze and U. W. Heinz, Phys. Rept. , 81 (1989).[13] St. Mr´owczy´nski, Phys. Rev. D , 1940 (1989).[14] J. P. Blaizot and E. Iancu, Phys. Rept. , 355 (2002).[15] St. Mr´owczy´nski and M. H. Thoma, Phys. Rev. D ,036011 (2000).[16] St. Mr´owczy´nski, A. Rebhan and M. Strickland, Phys.Rev. D , 025004 (2004).[17] M. H. Thoma, in Quark-Gluon Plasma 2, edited byR.C. Hwa (World Scientific, Singapore, 1995).[18] U. Kraemmer and A. Rebhan, Rept. Prog. Phys. , 351(2004).[19] A. V. Selikhov, Phys. Lett. B , 263 (1991) [Erratum-ibid. B , 398 (1992)].[20] A. V. Selikhov and M. Gyulassy, Phys. Lett. B , 373(1993).[21] A. V. Selikhov and M. Gyulassy, Phys. Rev. C , 1726(1994).[22] X. Zhang and J. Li, Phys. Rev. C , 964 (1995).[23] X. Zheng and J. Li, Phys. Lett. B , 45 (1997).[24] D. B¨odeker, Phys. Lett. B , 351 (1998).[25] D. B¨odeker, Nucl. Phys. B , 502 (1999).[26] P. Arnold, D. T. Son and L. G. Yaffe, Phys. Rev. D ,105020 (1999).[27] P. Arnold, D. T. Son and L. G. Yaffe, Phys. Rev. D , 025007 (1999).[28] P. Arnold and L. G. Yaffe, Phys. Rev. D , 125014(2000).[29] D. F. Litim and C. Manuel, Phys. Rev. Lett. , 4981(1999).[30] D. F. Litim and C. Manuel, Nucl. Phys. B , 237(1999).[31] M. A. Valle Basagoiti, arXiv:hep-ph/9903462.[32] J. P. Blaizot and E. Iancu, Nucl. Phys. B , 183 (1999).[33] Yu. A. Markov and M. A. Markova, J. Phys. G , 1869(2001).[34] Yu. A. Markov and M. A. Markova, Annals Phys. ,172 (2002).[35] Y. A. Markov, M. A. Markova and A. N. Vall, AnnalsPhys. , 93 (2004).[36] Yu. A. Markov, M. A. Markova and A. N. Vall, AnnalsPhys. , 282 (2005).[37] Yu. A. Markov and M. A. Markova, Nucl. Phys. A ,162 (2006).[38] Yu. A. Markov and M. A. Markova, Nucl. Phys. A ,443 (2007).[39] S. V. Akkelin, Phys. Rev. C , 014906 (2008).[40] A. A. Vedenov, E. P. Velikhov and R. Z. Sagdeev, Usp.Fiz. Nauk, , 701 (1961) [in Russian]; Sov. Phys. Usp. , 332 (1961).[41] A. A. Vedenov, Atomnaya Energiya , 5 (1962) [in Rus-sian]; J. Nucl. Energy C , 169 (1963).[42] E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Pergamon Press, Oxford, 1981).[43] St. Mr´owczy´nski, Phys. Rev. D , 105022 (2008).[44] M. Asakawa, S. A. Bass and B. M¨uller, Phys. Rev. Lett. , 252301 (2006).[45] M. Asakawa, S. A. Bass and B. M¨uller, Prog. Theor.Phys. , 725 (2006).[46] The problem greatly simplifies for longitudinal fieldswhen, instead of the matrix Σ ij ( ω, k ), one deals withthe scalar function k i k j Σ ij ( ω, k ) = ω k i k j ε ij ( ω, k ) = ω k ε L ( ω, kk