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 (23)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• The statistics of critical points of Gaussian fields on large-dimensional spaces
• Exact calculation of multifractal exponents of the critical wave function of Dirac fermions in a random magnetic field
• Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation
• The spherical 2+p spin glass model: an analytically solvable model with a glass-to-glass transition
• Large times off-equilibrium dynamics of a particle in a random potential
• Variational theory of elastic manifolds with correlated disorder and localization of interacting quantum particles
• On Mean Field Glassy Dynamics out of Equilibrium
• The Complexity of Ising Spin Glasses
• Exactly solvable model of a quantum spin glass
• Renormalization Theory for Interacting Crumpled Manifolds
• Replica Fourier Transforms on Ultrametric Trees, and Block-Diagonalizing Multi-Replica Matrices
• Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kaehler manifolds
• Higher correlations, universal distributions and finite size scaling in the field theory of depinning
• Complexity of Random Energy Landscapes, Glass Transition and Absolute Value of Spectral Determinant of Random Matrices
• Spontaneous symmetry breaking of a hyperbolic sigma model in three dimensions
• Free energy landscapes, dynamics and the edge of chaos in mean-field models of spin glasses
• Directed polymers in a random medium: a variational approach
• On Supersymmetry Breaking in the Computation of the Complexity
• Localization of a polymer in random media: Relation to the localization of a quantum particle
• Stability of the Mezard-Parisi solution for random manifolds
• Large time dynamics and aging of a polymer chain in a random potential
• Computing the number of metastable states in infinite-range models

Cited by in corpus (30)

• Extreme value statistics of correlated random variables: a pedagogical review
• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• The statistics of critical points of Gaussian fields on large-dimensional spaces
• Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of $1/f$ Noises generated by Gaussian Free Fields
• Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution
• Topology trivialization and large deviations for the minimum in the simplest random optimization
• Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite dimensional Euclidean spaces
• Multifractality and Freezing Phenomena in Random Energy Landscapes: an Introduction
• High-Dimensional Random Fields and Random Matrix Theory
• Statics and dynamics of infinite-dimensional liquids and glasses: a parallel, compact derivation
• Hessian spectrum at the global minimum of high-dimensional random landscapes
• On an explicit construction of Parisi landscapes in finite dimensional Euclidean spaces
• Following the ground-states of full-RSB spherical spin glasses
• Liouville field theory and log-correlated Random Energy Models
• A spin glass model for reconstructing nonlinearly encrypted signals corrupted by noise
• Legendre Duality of Spherical and Gaussian Spin Glasses
• Manifolds in high dimensional random landscape: complexity of stationary points and depinning
• The density of stationary points in a high-dimensional random energy landscape and the onset of glassy behaviour
• Pre-freezing of multifractal exponents in Random Energy Models with logarithmically correlated potential
• One step replica symmetry breaking and extreme order statistics of logarithmic REMs
• Complexity of Gaussian random fields with isotropic increments: critical points with given indices
• Separability gap and large deviation entanglement criterion
• Operator Product Expansion in Liouville Field Theory and Seiberg type transitions in log-correlated Random Energy Models
• Manifolds pinned by a high-dimensional random landscape: Hessian at the global energy minimum
• A Solvable Mean Field Model of a Gaussian Spin Glass
• High-dimensional Gaussian fields with isotropic increments seen through spin glasses
• Critical point correlations in random gaussian fields
• Replica-symmetry breaking transitions in the large deviations of the ground-state of a spherical spin-glass
• Disordered statistical physics in low dimensions: extremes, glass transition, and localization
• Nonlinear analog of the complexity-stability transition in random dynamical systems: a replica calculation