  We determine the sphaleron transition rate using real time lattice
simulations of the classical system. An improved definition of the lattice
topological charge allows us to obtain a more reliable estimate of the
transition rate. For an SU(2) Yang-Mills-Higgs system in the broken phase we
find the transition rate to be strongly suppressed, and we have observed no
sphaleron transitions in the range of coupling constants used. For a pure SU(2)
Yang-Mills system in large volumes the rate behaves as $\kappa (\alpha_w T)^4$,
with $\kappa$ slightly decreasing as the lattice spacing is reduced. If the
lattice size is reduced to about twice the magnetic screening length, the rate
is suppressed by finite-size effects, and $\kappa$ is approximately
proportional to the lattice spacing. Our rate measurements are supplemented by
analysis of gauge field correlation functions in the Coulomb gauge.
