Effective transport equations for non-Abelian plasmas

Daniel F. Litim

Institut f. Theoretische Physik,
Philosophenweg 16, D - 69120 Heidelberg.

I report about some recent results obtained in collaboration with Cristina Manuel (see also  her seminar  at this workshop). The main references are:

1. General approach
The aim is to systematically derive effective transport equations describing (non-Abelian) plasmas built out of quarks and gluons. A kinetic theory for non-Abelian plasmas is of particular interest for the physics of the early universe (like the baryon number violation rate), and possibly for the central region of heavy ion collisions. In either case a formalism is required able to cover both close-to-equlibrium and out-of-equilibrium situations.

Our approach is based on a classical  transport theory for coloured point particles obeying classical equations of motions, the Wong equations. These particles are coupled self-consistently to the Yang-Mills field. This leads immediately to a Boltzmann equation for the (microscopic) one-particle distribution function.

The new ingredience consists in taking the (Gibbs) ensemble average in phase space over the particles in order to obtain the effective (macroscopic) Boltzmann and Yang-Mills equations for the mean fields. At the same time, dynamical equations for the fluctuations about the mean values are obtained. This step has important conceptual implications, as the mean field equations, due to the intrinsic non-linearities, are not closed by themselves, but coupled to the statistical fluctuations about the mean fields. It appears that collision integrals for the mean fields are identified as correlators of flucutations, which can be seen as a derivation of collision integrals for Boltzmann equations. The coupled set of dynamical equations should be enough to study all transport phenomena occuring in the plasma. Most importantly, these equations do not rely on a close-to-equilibrium scenario.

2. The hot quark-gluon plasma close to equilibrium
Some simplifying approximations can be used when the approach is applied to a hot non-Abelian plasma close to equilibrium. In an expansion to leading order in the gauge coupling  at logarithmic accuracy, higher order correlators of fluctuations beyond quadratic order are suppressed and we recover the Boltzmann-Langevin equation first derived by Bodeker (see his seminar  at this workshop). This includes both a derivation of the (linearized) collision integral and of the source for stochastic noise.

3. Collision integral vs. noise and the fluctuation-dissipation theorem
In hep-ph/9906210, the source for stochastic noise has been derived directly form the effective dynamical equations, and ultimately from the Gibbs ensemble average. One might wonder whether this noise source is consistent with the fluctuation-dissipation theorem (FDT), which, in the close to equilibrium case, links the dissipative processes in the plasma to the underlying fluctuations. And indeed, under the assumption that the (linearized) dissipative term (i.e. the collision integral) is known, and assuming the FDT to hold, it is possible to derive the related noise term phenomenologically, hep-ph/9910348.  The resulting noise source coincides with our previous result,  which establishes that the above formalism ,and in particular Bodeker's effective theory, is consistent with the FDT.

Daniel F. Litim