**
Mean Field Dynamics in a non-Abelian plasma from classical
transport theory **

This talk is divided in two parts, and it is based on work done in collaboration with D. Litim ( for more details, see the references below).

**1. Dissipation from fluctuations**

Our starting point is considering a system of classical point particles carrying a non-Abelian charge and which interact self-consistently. Their microscopic equations are the Wong equations. When the number of particles is large, one has to abandon a microscopic description of the system in favor of a macroscopic one. This is done by taking Gibbs ensemble averages of the microscopic equations. In this way, one can derive a set of covariant equations for both the mean fields and statistical fluctuations around them.

In the most general situation the mean field equations depend on correlators of fluctuations, while the dynamical evolution of a single fluctuation depends on higher order moments of fluctuations. A set of approximations must be prescribed in order to be able to integrate out the statistical fluctuations of the mean field equations.

We then consider a close to equilibrium plasma, and use the small gauge coupling constant limit together with the second moment approximation. These two set of approximations allow to linearize the fluctuation dynamics. After integrating out the fluctuations, one then ends up with the Boltzmann-Langevin equation of Bodeker's effective theory. This equation contains a linearized collision term and related white and Gaussian stochastic source.

**2. Fluctuations from dissipation**

We consider a Boltzmann-Langevin equation for a non-Abelian plasma close to equilibrium, assuming that we know very little about the microscopic processes in the plasma. The question raises as to what can be said on general grounds about the spectral functions of the noise. We then show that from the knowledge of the entropy and the (linearized) collision term one can deduce the spectral function of statistical fluctuations and the noise correlator, assuming the noise being white and Gaussian. In particular, we check that Bodeker's effective theory is consistent with the fluctuation-dissipation theorem.

This way of deriving the Boltzmann-Langevin equation is completely
analogous to the first derivations of the Langevin equation
to describe brownian motion. This can be done without
the knowledge of the microscopic processes that keep the
brownian particle moving. One only needs to know
the frictional forces acting on the particle and then use
the equipartition theorem. If the noise is white and Gaussian,
the noise correlator is then completely determined from the
above general considerations.

**References:**

1. D.F. Litim and C. Manuel, ``Mean Field Dynamics in non-Abelian Plasmas from Classical Transport Theory", Phys.Rev.Lett. 82 (1999) 4981; hep-ph/9902430.

2. D.F. Litim and C. Manuel, ``Effective Transport Equations for non-Abelian Plasmas", to be published in Nucl.Phys.B; hep-ph/9906210.

3. D.F. Litim and C. Manuel, ``Fluctuations from Dissipation in a Hot Non-Abelian Plasma", hep-ph/9910348.