  The conditions under which relaxation dynamics in the presence of quenched-in
disorder lead to rate-independent hysteresis are discussed. The calculation of
average hysteresis branches is reduced to the solution of the level-crossing
problem for the stochastic field describing quenched-in disorder. Closed
analytical solutions are derived for the case where the disorder is
characterized by Wiener-Levy statistics. This case is shown to be equivalent to
the Preisach model and the associated Preisach distribution is explicitly
derived, as a function of the parameters describing the original dynamic
problem.
