Effective dynamics of soft non-abelian gauge fields at finite temperature

We consider time dependent correlation functions of non-abelian gauge fields at finite temperature. An effective theory for the soft ($p\sim g^{2}T$) field modes is derived by integrating out the field modes with momenta of order $T$ and of order $gT$ in a leading logarithmic approximation. In this effective theory the time evolution of the soft fields is determined by a local Langevin-type equation. As an application, the rate for hot electroweak baryon number violation is estimated as $\mathrm{\Gamma }\sim g^2\log (1\mathrm{/}g)(g^{2}T{)}^4$. Furthermore, possible consequences for non-perturbative lattice computations of unequal time correlation functions are discussed.