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

TL;DR

This paper analyzes a classical particle in a Wiener potential to test the accuracy of the replica variational approximation in predicting free energy and correlation functions, providing insights into its validity and ultrametricity assumptions.

Contribution

It offers an exact solution for the model's free energy and correlations, enabling a direct assessment of the replica variational approximation's accuracy and assumptions.

Findings

Replica scheme estimates free energy reasonably well.

Results support the validity of the ultrametricity assumption.

Exact calculations provide benchmarks for approximation methods.

Abstract

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.