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

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.

46 pages, 4 figures; This version corrects a few more typos discovered in the published version

References in corpus (4)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• Free energy landscapes, dynamics and the edge of chaos in mean-field models of spin glasses
• Computing the number of metastable states in infinite-range models
• Large time dynamics and aging of a polymer chain in a random potential

Cited by in corpus (4)

• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite dimensional Euclidean spaces
• On an explicit construction of Parisi landscapes in finite dimensional Euclidean spaces
• The density of stationary points in a high-dimensional random energy landscape and the onset of glassy behaviour