  We measure the diffusion rate of Chern-Simons number in the (1+1)-dimensional
Abelian Higgs model interacting with a realistic heat bath for temperatures
between 1/13 and 1/3 times the sphaleron energy. It is found that the measured
rate is close to that predicted by one-loop calculation at the lower end of the
temperature range considered but falls at least an orderof magnitude short of
one-loop estimate at the upper end of that range. We show numerically that the
sphaleron approximation breaks down as soon as the gauge-invariant two-point
function yields correlation length close to the sphaleron size.
