Thermal Equilibrium with the Wiener Potential: Testing the Replica Variational Approximation

We consider the statistical mechanics of a classical particle in a one-dimensional box subjected to a random potential which constitutes a Wiener process on the coordinate axis. The distribution of the free energy and all correlation functions of the Gibbs states may be calculated exactly as a function of the box length and temperature. This allows for a detailed test of results obtained by the replica-variational-approximation scheme. We show that this scheme provides a reasonable estimate of the averaged free energy. Furthermore our results shed more light on the validity of the concept of approximate ultrametricity which is a central assumption of the replica variational method.