Semi-classic Transport Theory For Soft And Ultra Soft Modes In Non-Abelian Plasma

We review a semi-classical transport theory for non-Abelian plasmas based on a classical picture of colored point particles. Within this formalism, kinetic equations for the mean particle distribution, the mean fields and their fluctuations are obtained using an ensemble-average in phase space. The framework permits the integrating-out of fluctuations in a systematic manner. This leads to the derivation transport equations of QCD. Consistency with the non-Abelian gauge symmetry is established, and systematic approximation schemes are worked out. In particular, the approach is applicable to both in- and out-of-equilibrium plasmas. The formalism is applied explicitly to a hot and weakly coupled QCD plasma slightly out of equilibrium. The physics related to color conductivity is deduced in very simple manner. Effective transport equations are computed to first and second order in moments of the fluctuations. To first order, they reproduce the seminal "hard-thermal-loop" and "hard-dense-loop" effective theory. To second order, the fluctuation induce collisions amongst the quasi-particles, leading to a Langevintype transport equation. A complementary Langevin approach is discussed as well. Finally, we show how the approach can be applied to hot and dense quark matter systems. Possible future applications are outlined.