Effective Transport Equations for non-Abelian Plasmas

37 Pages·1999·0.42 MB·English

CERN-TH-99-151
ECM-UB-PF-99-12
Effective Transport Equations for non-Abelian Plasmas
Daniel F. Litim (cid:3);a and Cristina Manuel y;b
aDepartament ECM & IFAE, Facultat de F(cid:19)(cid:16)sica
Univ. de Barcelona, Diagonal 647, E-08028 Barcelona, Spain.
bTheory Division, CERN, CH-1211 Geneva 23, Switzerland.
Abstract
Starting from classical transport theory, we derive a set of covariant equations de-
scribing the dynamics of mean (cid:12)elds and their statistical fluctuations in a non-
Abelian plasma in or out of equilibrium. A general procedure is detailed for
integrating-out the fluctuations as to obtain the e(cid:11)ective transport equations for
the mean (cid:12)elds. In this manner, collision integrals for Boltzmann equations are ob-
tained as correlators of fluctuations. The formalism is applied to a hot non-Abelian
plasmaclosetoequilibrium. Weintegrate-out explicitly thefluctuationswithtypical
momenta of the Debye mass, and obtain the collision integral in a leading logarith-
mic approximation. We also identify a source for stochastic noise. The resulting
dynamical equations are of the Boltzmann-Langevin type. While our approach is
based on classical physics, we also give the necessary generalizations to study the
quantum plasmas. Ultimately, the dynamical equations for soft and ultra-soft (cid:12)elds
change only in the value for the Debye mass.
(cid:3)
E-Mail: [email protected]
y
E-Mail: [email protected]
I. INTRODUCTION
This article presents in full detail an approach to the dynamics of non-Abelian plasmas
based on classical transport theory, the main results of which have been summarized in [1].
In the recent years, there has been an increasing interest in the dynamics of non-Abelian
plasmas at very high temperatures or high densities. Due to the asymptotic freedom of
quantum chromodynamics, one expects that quarks and gluons are no longer con(cid:12)ned under
such extreme conditions, but rather behave asfreeentities forming the so-called quark-gluon
plasma. Within the next few years, a lot of e(cid:11)orts will be given to detect experimentally this
new state of matter using heavy-ion colliders. Another domain of application concerns the
physics oftheearlyuniverse. Ifbaryogenesis can(cid:12)nallybeunderstoodwithinanelectroweak
scenario, an understanding of the physics of the electroweak model in the high temperature
regime where the spontaneous broken symmetry is restored, is essential for a computation
of the rate of baryon number violation.
It isthereforemandatoryto devise reliable theoretical toolsfor aquantitative description
ofnon-Abelianplasmasbothinoroutofequilibrium. Whilesomeprogresshasbeenachieved
in the recent years [2,3], we are still far away from having a satisfactory understanding of
the relevant relaxation and transport processes in non-Abelian plasmas, in particular when
it comes to out-of-equilibrium situations.
There are di(cid:11)erent approaches in the literature to study non-Abelian plasmas, ranging
from thermal (cid:12)eld theory to quantum transport equations or lattice studies. Even in the
close-to-equilibrium plasma, and for small gauge coupling, the situation is complicated due
to the non-perturbative character of long-wavelength excitations in the plasma. Most at-
tempts to tackle this problem are based on a quantum (cid:12)eld theoretical description of the
non-Abelian interactions [4,5]. It has been conjectured that the plasma close to equilibrium
allows for a description in terms of soft classical (cid:12)elds, as the occupation number for the soft
excitations are large. Surprisingly, a classical transport theory approach, as developed in
[6], has never been exploited in full detail for the non-Abelian case. The opposite holds true
for Coulomb plasmas, where all the essential transport phenomena have been studied longly
using techniques developed within (semi-)classical kinetic theory [7], while a quantum (cid:12)eld
theoretical approach has been undertaken only recently.
Our approach aims at (cid:12)lling this gap in the literature of classical non-Abelian plas-
mas. Here, we follow the philosophy of Klimontovitch [7], and our equations can be seen
as the generalization of classical kinetic theory for Abelian plasmas to non-Abelian ones.
Our essential contribution is considering the non-Abelian colour charges as dynamical vari-
ables and introducing the concept of ensemble average to the non-Abelian kinetic equations.
Equally important is the consistent treatment of the intrinsic non-linearities of non-Abelian
gauge interactions. The observation that Klimontovitch’s procedure leads to the Balescu-
Lenard collision integral for Coulomb plasmas has motivated earlier derivations of similar
(semi-)classical kinetic equations for non-Abelian plasmas [8{10]. However, these imple-
mentations are not fully consistent, and have never been worked out in all generality.
The starting point for a classical transport theory of non-Abelian plasmas is consider-
ing an ensemble of classical point particles carrying a non-Abelian charge. They interact
through self-consistent (cid:12)elds, that is, the (cid:12)elds generated by the particles themselves. The
1
microscopic dynamics is governed by the classical equations of motion given by the Wong
equations [11]. When the number of particles is large, one has to abandon a microscopic
description of the system in favour of a macroscopical one based on an ensemble average
of all the microscopic quantities. This leads naturally to a description in terms of aver-
aged quantities, and their statistical fluctuations. By averaging the microscopic dynamical
equations, we obtain e(cid:11)ective transport equations for mean quantities. These contain the
collision integrals of the macroscopic Boltzmann equation, which appear in this formula-
tion as statistical correlators of fluctuating quantities. By subtracting the exact microscopic
equations from the mean ones, we obtain the dynamical equations for the fluctuations them-
selves. Inprinciple, these two set of equations should be enough to consider all the transport
phenomena in the plasma.
This method is then applied in full detail to a thermal non-Abelian plasma close to equi-
librium, which allows to employ some approximations. For a small plasma parameter, the
two-particle correlators are small and can be neglected. In the case of small fluctuations,
the dynamical equations simplify considerably. These conditions are always met for a small
gauge coupling parameter, which, for simplicity, will be assumed throughout. Our approxi-
mate equations are the leading order ones in a consistent expansion in the gauge coupling.
However, we shall also see that the condition for a kinetic description to be valid could also
be met for large gauge couplings. After taking statistical averages, we are able to explicitly
integrate-out the fluctuations with momenta about the Debye mass. This gives the collision
integrals which appear in the transport equations for the mean (cid:12)elds. In addition to the
dissipative processes in the plasma described by the collision term, we are able to deduce
the stochastic source which prevents the system from abandoning equilibrium. This is an
important result, because it allows to prove explicitly that the fluctuation-dissipation theo-
rem holds, when switching from a microscopic to a macroscopic description of the system.
These (cid:12)ndings for classical plasmas can be generalized to the case of quantum ones. The
resulting dynamical equation match perfectly the e(cid:11)ective theory for the ultra-soft modes
as found by other approaches [12{14].
The lesson to be learned is thus two-fold: There exists a fully self-contained formalism to
study classical non-Abelian plasmas in the (cid:12)rst place, which opens in particular a door for
applications to out-of-equilibriumsituations. Second, this approachis -technically speaking-
much easier than approaches based on the full QFT. Some of the intrinsic complications of
a quantum (cid:12)eld theoretical description (like gauge-(cid:12)xing, ghost degrees of freedom) can be
avoided, and in the close to equilibrium plasma, the same e(cid:11)ective dynamical equations are
obtained.
The paper is organized as follows. We begin with a review of the microscopic picture,
based on the classical equations of motions for coloured point particles (sect. II). Changing
toamacroscopicdescriptionneedstheintroductionofastatisticalaverage, whichalsoallows
the computation of correlators of fluctuations (sect. III). This procedure is then applied to
the (cid:12)elds and the distribution function as to obtain dynamical equations for their mean
values and their fluctuations. The dynamical equations are given in its most general form.
Possible approximation schemes are detailed, and the interpretation of statistical correlators
in terms of collision integrals is given (sect. IV). The consistency of the procedure with the
requirements of gauge invariance is shown for the complete set of equations, and some
2
approximations to them (sect. V). In order to apply the formalism to a plasma close
to equilibrium, we discuss (cid:12)rst the relevant physical scales for both classical and quantum
plasmas(sect. VI). Thisisfollowedbyafullydetailedderivationofthemean(cid:12)elddynamical
equation for classical plasmas, which includes the integrating-out of the fluctuations with
momenta about the Debye mass, and the computation of the collision integral and the
related noise variable in a leading logarithmic approximation. Explicit expressions for the
non-local ultra-soft current and the colour conductivity are given as well (sect. VII). We
argue that these results can be translated to the case of quantum plasmas and detail the
necessary changes. Some comments on related work are added as well (sect. VIII). Finally,
we present our conclusions (sect. IX), deferring to the appendices some technical details
regarding the Darboux variables for SU(N) colour charges (appendix A), and the derivation
of a useful algebraic identity (appendix B).
II. MICROSCOPIC EQUATIONS FOR NON-ABELIAN CHARGED PARTICLES
Let us consider a system of particles carrying a colour charge Qa, where the colour index
runs from a = 1 to N2 − 1 for a SU(N) gauge group. Within a microscopic description,
the trajectories in phase space are known exactly. The trajectories x^((cid:28));p^((cid:28)) and Q^((cid:28)) for
every particle are solutions of their classical equations of motions, the Wong equations [11]
dx^(cid:22)
(cid:22)
m = p^ ; (2.1a)
d(cid:28)
dp^(cid:22)
m = gQ^aF(cid:22)(cid:23)p^ ; (2.1b)
a (cid:23)
d(cid:28)
dQ^a
m = −gfabcp^(cid:22)AbQ^c : (2.1c)
(cid:22)
d(cid:28)
Here, A denotes the microscopic gauge (cid:12)eld. The corresponding microscopic (cid:12)eld strength
(cid:22)
Fa and the energy momentum tensor of the gauge (cid:12)elds (cid:2)(cid:22)(cid:23) are given by
(cid:22)(cid:23)
Fa [A] = @ Aa −@ Aa +gfabcAbAc ; (2.2)
(cid:22)(cid:23) (cid:22) (cid:22) (cid:23) (cid:22) (cid:22) (cid:23)
(cid:2)(cid:22)(cid:23)[A] = 1g(cid:22)(cid:23)Fa F(cid:26)(cid:27) +F(cid:22)(cid:26)Fa(cid:23) (2.3)
4 (cid:26)(cid:27) a a (cid:26)
and fabc are the structure constants of SU(N). We set c = k = h(cid:22) = 1 and work in natural
B
units, unless otherwise indicated. Note that the non-Abelian charges are also subject to
dx^µ
dynamical evolution. Equation (2.1c) can be rewritten as D Q = 0, where D = D is
(cid:28) (cid:28) d(cid:28) (cid:22)
the covariant derivative along the world line, and Dac[A] = @ (cid:14)ac + gfabcAb the covariant
(cid:22) (cid:22) (cid:22)
derivative in the adjoint representation. With Q and Fa transforming in the adjoint rep-
a (cid:22)(cid:23)
resentation, the Wong equations can be shown to be invariant under gauge transformations.
The equation (2.1c) ensures the conservation under dynamical evolution of the set of N −1
Casimir of the SU(N) group.z
zFor SU(2), it is easy to verify explicitly the conservation of the quadratic Casimir QaQa. For
SU(3), both the quadratic and cubic Casimir dabcQaQbQc, where dabc are the symmetric structure
3
The colour current associated to each particle can be constructed once the solutions of
the Wong equations are known. For every single particle it reads
(cid:90)
dx^(cid:22)
j(cid:22)(x) = g d(cid:28) Q^ ((cid:28)) (cid:14)(4)(x−x^((cid:28))) : (2.4)
a a
d(cid:28)
Employing the Wong equations (2.1) we (cid:12)nd that j(cid:22) is covariantly conserved, D j(cid:22) = 0
(cid:22)
[11]. Similarly, the energy momentum tensor associated to a single particle is given by [11]
(cid:90)
dx^(cid:22)
t(cid:22)(cid:23)(x) = d(cid:28) p^(cid:23)((cid:28))(cid:14)(4)(x−x^((cid:28))) ; (2.5)
d(cid:28)
It is convenient to describe the ensemble of particles introducing a phase space density
which depends on the whole set of coordinates x(cid:22);p(cid:22) and Q . We de(cid:12)ne the function
a
(cid:90)
(cid:88)
n(x;p;Q) = d(cid:28) (cid:14)(4)(x−x^ ((cid:28))) (cid:14)(4)(p−p^((cid:28))) (cid:14)(N2−1)(Q−Q^ ((cid:28))) ; (2.6)
i i i
i
where the index i labels the particles. This distribution function is constructed in such a
way that the colour current
(cid:90)
p(cid:22)
(cid:22) 4 (N2−1)
J (x) = g d pd Q Q n(x;p;Q) (2.7)
a a
m
(cid:80)
coincides with the sum over all currents associated to the individual particles, J(cid:22) = j(cid:22),
a i a
and is covariantly conserved, D J(cid:22) = 0. It is convenient to make the following changes in
(cid:22)
the choice of the distribution function. We will de(cid:12)ne a new function f(x;p;Q) such that
the physical constraints like the on-mass shell condition, positive energy and conservation
of the group Casimirs are factored out into the phase space measure. We introduce the
momentum measurex
dP = d4p2(cid:18)(p )(cid:14)(p2 −m2) ; (2.8)
0
and the measure for the colour charges
dQ = d3Qc (cid:14)(Q Q −q ) ; (2.9)
R a a 2
in the case of SU(2). For SU(3) the measure is
dQ = d8Qc (cid:14)(Q Q −q )(cid:14)(d QaQbQc −q ) : (2.10)
R a a 2 abc 3
constants of the group, are conserved underthe dynamical evolution. Thelast conservation can be
checkedusing(2.1c)andaJacobi-likeidentitywhichinvolvesthesymmetricdabc andantisymmetric
fabc constants.
x
Note that in [1], we used a slightly di(cid:11)erent normalization of the measure. Here, the measure
has an additional factor of (2π)3.
4
For SU(N), N−1 (cid:14)-functions ensuring the conservation of the set of N−1 Casimirs have to
be introduced into the measure dQ. We have also introduced the representation-dependent
(cid:82)
normalization constant c into the measure dQ, which is (cid:12)xed by requiring dQ = 1 (see
R
appendix A). The constant C is de(cid:12)ned as
2
(cid:90)
dQQ Q = C (cid:14) ; (2.11)
a b 2 ab
and depends on the group representation of the particles.
With these conventions the colour current (2.7) reads now
(cid:90)
(cid:22) (cid:22)
J (x) = g dPdQp Q f(x;p;Q) ; (2.12)
a a
while the energy momentum tensor associated to the particles obtains as
(cid:90)
(cid:22)(cid:23) (cid:22) (cid:23)
t (x) = dPdQ p p f(x;p;Q) : (2.13)
We will now come to the dynamical equation of the microscopic distribution functions
n(x;p;Q) and f(x;p;Q), which will serve as the starting point for the subsequent formalism.
The dynamical equation for n(x;p;Q) is the same as for f(x;p;Q). This is so because the
physical constraints which we have factored out as to obtain f(x;p;Q) are not a(cid:11)ected by
the Wong equations. Employing (2.1), we (cid:12)nd
(cid:32) (cid:33)
@ @ @
p(cid:22) −gfabcAbQc −gQ Fa f(x;p;Q) = 0 ; (2.14a)
@x(cid:22) (cid:22) @Qa a (cid:22)(cid:23)@p
(cid:23)
which can be checked explicitly by direct inspection of (2.6) into (2.14a) (see appendix A of
[5]). In a self-consistent picture this equation is completed with the Yang-Mills equation,
(cid:22)(cid:23) (cid:23)
(D F ) (x) = J (x) ; (2.14b)
(cid:22) a a
and the current being given by (2.12). It is worth noticing that (2.14) is exact in the
sense that no approximations have been made so far. The e(cid:11)ects of collisions are included
inasmuch as the Wong equations do account for them, although (2.14a) looks formally like
a collisionless Boltzmann equation.
For the energy momentum tensor of the gauge (cid:12)elds we (cid:12)nd
@ (cid:2)(cid:22)(cid:23) = −F(cid:23)(cid:22)Ja : (2.15)
(cid:22) a (cid:22)
On the other hand, using (2.14a) and the de(cid:12)nition (2.13) we (cid:12)nd that
(cid:22)(cid:23) (cid:23)(cid:22) a
@ t = F J : (2.16)
(cid:22) a (cid:22)
which establishes that the total energy momentum tensor is conserved, @ ((cid:2)(cid:22)(cid:23) +t(cid:22)(cid:23)) = 0.
(cid:22)
To (cid:12)nish the review of the microscopic description of the system, let us recall the gauge
symmetry properties of the set of equations (2.14) (see [5] for more details). From the
5
de(cid:12)nition of f(x;p;Q) it follows that it transforms as a scalar under a ((cid:12)nite) gauge trans-
formation, f0(x;p;Q0) = f(x;p;Q). This implies the gauge covariance of (2.14b) because
the current (2.7) transforms like the vector Q in the adjoint. The non-trivial dependence
a
of f on the non-Abelian colour charges implies that the partial derivative @ f does not yet
(cid:22)
transform as a scalar, but rather its covariant derivative D f, which is given by
(cid:22)
D [A]f(x;p;Q) (cid:17) [@ −gfabcQ A @Q]f(x;p;Q) : (2.17)
(cid:22) (cid:22) c (cid:22);b a
Notice that (2.17) combines the (cid:12)rst two terms of (2.14a). (Here, and in the sequel we use
the shorthand notation @ (cid:17) @=@x(cid:22), @p (cid:17) @=@p(cid:22) and @Q (cid:17) @=@Qa.) The invariance of
(cid:22) (cid:22) a
the third term in (2.14a) follows from the trivial observation that Q Fa is invariant under
a (cid:22)(cid:23)
gauge transformations. This establishes the gauge invariance of (2.14a).
III. STATISTICAL AVERAGES
If the system under study contains a large number of particles it is impossible to follow
their individual trajectories. One has then to switch to a statistical description of the
system.
In this section, we describe in detail the statistical average to be used in the sequel.
As we are studying classical point particles in phase space, the appropriate statistical av-
erage corresponds to the Gibbs ensemble average for classical systems. We will review the
main features of this procedure, de(cid:12)ned in phase space. Let us remark that this deriva-
tion is completely general, valid for any classical system, and does not require equilibrium
situations.
We will introduce two basic functions. The (cid:12)rst one is the phase space density function
N which gives, after integration over a phase space volume element, the number of particles
contained in that volume. In a microscopic description it is just a deterministic quantity,
and a function of the time t, the vectors x and p, and the set of canonical (Darboux)
variables φ and π associated to the colour charges Q (see appendix A). For SU(N), there
are N(N − 1)=2 pairs of canonical variables, which we denote as φ = ((cid:30) ;:::;(cid:30) )
1 N(N−1)=2
and π = ((cid:25) ;:::;(cid:25) ). The microscopic phase space density is given by
1 N(N−1)=2
(cid:88)
N(x;p;φ;π) = (cid:14)(3)(x−x^ (t)) (cid:14)(3)(p−p^ (t)) (cid:14)(φ−φ^ (t)) (cid:14)(π −π^ (t)) ; (3.1)
i i i i
i
wherethesumrunsoverallparticlesofthesystem, and(x^ ;p^ ;φ^ ;π^ )referstothetrajectory
i i i i
of the i-th particle in phase space. Then Ndxdpdφdπ gives the number of particles at time
t in an in(cid:12)nitesimal volume element of phase space around the point z = (x;p;φ;π)
Let usnowde(cid:12)ne thedistribution functionF ofthemicrostates ofasystem ofLidentical
classical particles. Due to Liouville’s theorem, dF=dt = 0. Thus, it can be normalized as
(cid:90)
dz dz :::dz F(z ;z ;:::;z ;t) = 1 : (3.2)
1 2 L 1 2 L
Here z denotes all the phase space variables associated to the particle i. For simplicity
i
we have considered that there is only one species of particles in the system, although the
generalization to several species of particles is rather straightforward.
6
The statistical average of any function G de(cid:12)ned in phase space is given by
(cid:90)
hGi = dz dz ;:::dz G(z ;z ;:::;z )F(z ;z ;:::;z ;t) : (3.3)
1 2 L 1 2 L 1 2 L
The one-particle distribution function is obtained from F as
(cid:90)
f (z ;t) = V dz :::dz F(z ;z ;:::;z ;t) ; (3.4)
1 1 2 L 1 2 L
where V denotes the phase space volume. Correspondingly, the two-particle distribution
function is
(cid:90)
f (z ;:::;z ;t) = V2 dz :::dz F(z ;z ;:::;z ;t) ; (3.5)
2 1 k 3 L 1 2 L
and similarly for the k-particle correlators. A complete knowledge of F would allow us to
obtain all the set of (f ;f ;:::;f ) functions; this is, however, not necessary for our present
1 2 L
purposes.
Notice that we have allowed for an explicit dependence on the time t of the function
F, as this would be typically the case in out of equilibrium situations. We will drop this t
dependence from now on to simplify the formulas.
Using the above de(cid:12)nition one can obtain the mean value of the microscopic phase space
(cid:80)
density. Microscopically N(z) = L (cid:14)(z −z^), where z^ describes the trajectory in phase
i=1 i i
space of the particle i. The statistical average of this function is
(cid:90)
(cid:88)L L
hN(z)i = dz dz ;:::;dz F(z ;z ;:::;z ) (cid:14)(z −z^) = f (z) : (3.6)
1 2 L 1 2 L i 1
V
i=1
The second moment hN(z)N(z0)i can equally be computed, and it is not di(cid:14)cult to see
that gives
L L(L−1)
hN(z)N(z0)i = (cid:14)(z −z0)f (z)+ f (z;z0) : (3.7)
V 1 V2 2
Let us now de(cid:12)ne a deviation of the phase space density from its mean value
(cid:14)N(z) (cid:17) N(z)−hN(z)i : (3.8)
By de(cid:12)nition h(cid:14)N(z)i = 0, although the second moment of this statistical fluctuation does
not vanish in general, since
h(cid:14)N(z)(cid:14)N(z0)i = hN(z)N(z0)i−hN(z)ihN(z0)i : (3.9)
If the number of particles is large L (cid:29) 1 then one then (cid:12)nds
(cid:18) (cid:19) (cid:18) (cid:19)
2
L L
h(cid:14)N(z)(cid:14)N(z0)i = (cid:14)(z −z0)f (z)+ g (z;z0) ; (3.10)
1 2
V V
where the function g (z;z0) = f (z;z0)−f (z)f (z0) measures the two-particle correlations
2 2 1 1
in the system. For a completely uncorrelated system g = 0.
2
7
Notice that the above statistical averages are well de(cid:12)ned in the thermodynamic limit,
L;V ! 1 but L=V remaining constant. Higher order moments and higher order correlators
can as well be de(cid:12)ned.
We (cid:12)nally point out that the the function N(x;p;φ;π) agrees with the microscopic
function f(x;p;Q) introduced earlier, except for a representation-dependent normalization
constant c as introduced in sect. II. We will also swallow the density factors L=V into the
R
(cid:22)
mean functions f. Those small changes in the normalizations allow to simplify slightly the
notations of the equations.
IV. AVERAGING THE MICROSCOPIC EQUATIONS
A. The mean (cid:12)elds and the fluctuations
In this section we perform the step from a microscopic to a macroscopic formulation of
the problem. Using the prescription explained in sect. III, we take statistical averages of the
microscopic equations (2.14). This implies that the distribution function f(x;p;Q), which
in the microscopic picture is a deterministic quantity, has now a probabilistic nature and
can be considered as a random function, given by its mean value and statistical (random)
fluctuation about it. Let us de(cid:12)ne the quantities
(cid:22)
f(x;p;Q) = f(x;p;Q)+(cid:14)f(x;p;Q) ; (4.1a)
J(cid:22)(x) = J(cid:22)(cid:22)(x)+(cid:14)J(cid:22)(x) ; (4.1b)
a a a
Aa(x) = A(cid:22)a(x)+aa(x) ; (4.1c)
(cid:22) (cid:22) (cid:22)
where the quantities carrying a bar denote the mean values, e.g. f(cid:22) = hfi, J(cid:22) = hJi and
A(cid:22) = hAi, while the mean value of the statistical fluctuations vanish by de(cid:12)nition, h(cid:14)fi = 0,
h(cid:14)Ji = 0 and hai = 0. This separation into mean (cid:12)elds and statistical, random fluctua-
tions corresponds e(cid:11)ectively to a split into low frequency (long wavelength) modes asso-
ciated to the mean (cid:12)elds, and high frequency (short wavelength) modes associated to the
fluctuations.(cid:3)(cid:3) We also split the (cid:12)eld strength tensor as
Fa = F(cid:22)a +fa ; (4.2a)
(cid:22)(cid:23) (cid:22)(cid:23) (cid:22)(cid:23)
F(cid:22)a = Fa [A(cid:22)] ; (4.2b)
(cid:22)(cid:23) (cid:22)(cid:23)
fa = (D(cid:22) a −D(cid:22) a )a +gfabcabac ; (4.2c)
(cid:22)(cid:23) (cid:22) (cid:23) (cid:23) (cid:22) (cid:22) (cid:23)
usingalsoD(cid:22) (cid:17) D [A(cid:22)]. Thetermfa containstermslinearandquadraticinthefluctuations.
(cid:22) (cid:22) (cid:22)(cid:23)
Note that the statistical average of the (cid:12)eld strength hFa i is given by hFa i = F(cid:22)a +
(cid:22)(cid:23) (cid:22)(cid:23) (cid:22)(cid:23)
gfabchabaci, due to quadratic terms contained in fa .
(cid:22) (cid:23) (cid:22)(cid:23)
In the same light, we split the energy momentum tensor of the gauge (cid:12)elds into the part
from the mean (cid:12)elds and the fluctuations, according to
(cid:3)(cid:3)
In the close-to-equilibrium plasma (sect. VII), we identify the relevant momentum scales
explicitly.
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(cid:2)(cid:22)(cid:23) = (cid:2)(cid:22)(cid:22)(cid:23) +(cid:18)(cid:22)(cid:23) (4.3a)
(cid:2)(cid:22)(cid:22)(cid:23) = 1g(cid:22)(cid:23)F(cid:22)a F(cid:22)(cid:26)(cid:27) +F(cid:22)(cid:22)(cid:26)F(cid:22)a(cid:23) ; (4.3b)
4 (cid:26)(cid:27) a a (cid:26)
(cid:18)(cid:22)(cid:23) = 1g(cid:22)(cid:23)F(cid:22)a f(cid:26)(cid:27) +F(cid:22)(cid:22)(cid:26)fa +F(cid:22)(cid:23)(cid:26)fa + 1g(cid:22)(cid:23)f f(cid:26)(cid:27);a +f(cid:22)(cid:26)fa(cid:23) ; (4.3c)
2 (cid:26)(cid:27) a a (cid:26)(cid:23) a (cid:26)(cid:22) 4 (cid:26)(cid:27);a a (cid:26)
The term (cid:18)(cid:22)(cid:23) contains the fluctuations up to quartic order. Due to the non-linear character
of the theory, we (cid:12)nd that the ensemble average of the energy momentum tensor reads
h(cid:2)(cid:22)(cid:23)i = (cid:2)(cid:22)(cid:22)(cid:23) +h(cid:18)(cid:22)(cid:23)i.
B. Dynamical equations for the mean (cid:12)elds and the fluctuations
We perform now the step from the microscopic to the macroscopic Boltzmann equation,
taking the statistical average of (2.14). This yields the dynamical equation for the mean
values,
(cid:16) (cid:17)
p(cid:22) D(cid:22) −gQ F(cid:22)a @(cid:23) f(cid:22)= h(cid:17)i+h(cid:24)i : (4.4a)
(cid:22) a (cid:22)(cid:23) p
We have made use of the covariant derivative of f as introduced in (2.17). The macroscopic
Yang-Mills equations are obtained as
D(cid:22) F(cid:22)(cid:22)(cid:23) +hJ(cid:23) i = J(cid:22)(cid:23) : (4.4b)
(cid:22)
fluc
In (4.4), we collected all terms quadratic or cubic in the fluctuations into the functions
(cid:17)(x;p;Q);(cid:24)(x;p;Q) and J (x). They read explicitly
fluc
(cid:17)(x;p;Q) (cid:17) gQ p(cid:22)@(cid:23)fa (cid:14)f(x;p;Q) ; (4.5a)
a p (cid:22)(cid:23)
(cid:16) (cid:17)
(cid:24)(x;p;Q) (cid:17) gp(cid:22)fabcQc @Qab (cid:14)f(x;p;Q) +gaaab@(cid:23)f(cid:22)(x;p;Q) ; (4.5b)
a (cid:22) (cid:22) (cid:23) p
(cid:110) (cid:16) (cid:17)(cid:111)
Ja;(cid:23)(x) (cid:17) g fdbcD(cid:22)(cid:22) a a(cid:23) +fabca (D(cid:22)(cid:22)a(cid:23) −D(cid:22)(cid:23)a(cid:22)) +gfcdea(cid:22)a(cid:23) (x): (4.5c)
fluc ad b;(cid:22) c b;(cid:22) c d e
The corresponding equations for the fluctuations are obtained by subtracting (4.4) from
(2.14). The result is
(cid:16) (cid:17)
p(cid:22) D(cid:22) −gQ F(cid:22)a @(cid:23) (cid:14)f = gQ (D(cid:22) a −D(cid:22) a )ap(cid:22)@pf(cid:22);
(cid:22) a (cid:22)(cid:23) p a (cid:22) (cid:23) (cid:23) (cid:22) (cid:23)
+gp(cid:22)a fabcQ @Qf(cid:22)+(cid:17) +(cid:24) −h(cid:17) +(cid:24)i (4.6a)
b;(cid:22) c a
(cid:16) (cid:17)
a
D(cid:22)2a(cid:22) −D(cid:22)(cid:22)(D(cid:22) a(cid:23)) + 2gfabcF(cid:22)(cid:22)(cid:23)a +Ja;(cid:22) −hJa;(cid:22)i = (cid:14)Ja;(cid:22) : (4.6b)
(cid:23) b c;(cid:23) fluc fluc
The above set of dynamical equations is enough to describe all transport phenomena in the
plasma.
While the dynamics of the mean (cid:12)elds (4.4) depends on correlators quadratic and cubic
in the fluctuations, the dynamical equations for the fluctuations (4.6) do in addition depend
on higher order terms (up to cubic order) in the fluctuations themselves. The dynamical
equations for the higher order correlation functions are contained in (4.6). To see this,
consider for example the dynamical equation for the correlators h(cid:14)f (cid:14)fi. After multiplying
(4.6a) with (cid:14)f and taking the statistical average, we obtain
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