  In hot non-Abelian gauge theories, processes characterized by the momentum
scale $g^2 T$ (such as electroweak baryon number violation in the very early
universe) are non-perturbative. An effective theory for the soft
($|\vec{p}|\sim g^2 T$) field modes is obtained by integrating out momenta
larger than $g^2 T$. Starting from the hard thermal loop effective theory,
which is the result of integrating out the scale $T$, it is shown how to
integrate out the scale $gT$ in an expansion in the gauge coupling $g$. At
leading order in $g$, one obtains Vlasov-Boltzmann equations for the soft field
modes, which contain a Gaussian noise and a collision term. The 2-point
function of the noise and the collision term are explicitly calculated in a
leading logarithmic approximation. In this approximation the Boltzmann equation
is solved. The resulting effective theory for the soft field modes is described
by a Langevin equation. It determines the parametric form of the hot baryon
number violation rate as $\Gamma = \kappa g^{10} \log(1/g) T^4$, and it allows
for a calculation of $\kappa$ on the lattice.
