  We derive, from first principle, the Fokker-Planck equation describing the
non-perturbative real-time evolution of a gauge field at high temperatures on
the spatial scale $(g^2T)^{-1}$ and the time scale $(g^4T)^{-1}$, where $g$ is
the gauge coupling and $T$ is the temperature. The knowledge of the effective
dynamics on such spatial and time scales is crucial for the understanding and
quantitative calculation of the baryon number violation rate at high
temperatures in the electroweak theory. The Fokker-Planck equation, which
describes the diffusive motion of soft components of the gauge field in the
space of field configurations, is shown to be gauge-invariant and consistent
with the known thermodynamics of soft gauge modes. We discuss the ways the
Fokker-Planck equation can be made useful for numerical simulations of
long-distance dynamics of hot gauge theories.
