Classical Particle in a Box with Random Potential: exploiting rotational symmetry of replicated Hamiltonian

TL;DR

This paper analyzes the thermodynamics of a classical particle in a spherical box with random potential, using symmetry methods to derive free energy expressions and explore phase diagrams, extending previous infinite-radius results.

Contribution

It introduces a symmetry-based approach to compute free energy directly for finite radius, avoiding variational approximations, and connects to spin glass models.

Findings

Phase diagram varies with box size and disorder type.

Method confirms previous results in the infinite-radius limit.

Reveals similarities with spherical spin glass models.

Abstract

We investigate thermodynamics of a single classical particle placed in a spherical box of a finite radius $R$ and subject to a superposition of a $N-$dimensional Gaussian random potential and the parabolic potential with the curvature $\mu>0$. Earlier solutions of $R\to \infty$ version of this model were based on combining the replica trick with the Gaussian Variational Ansatz (GVA) for free energy, and revealed a possibility of a glassy phase at low temperatures. For a general $R$, we show how to utilize instead the underlying rotational symmetry of the replicated partition function and to arrive to a compact expression for the free energy in the limit $N\to \infty$ directly, without any need for intermediate variational approximations. This method reveals striking similarity with the much-studied spherical model of spin glasses. Depending on the value of $R$ and the three types of disorder - short-ranged, long-ranged, and logarithmic - the phase diagram of the system in the $(\mu,T)$ plane undergoes considerable modifications. In the limit of infinite confinement radius our analysis confirms all previous results obtained by GVA.