Derivation of anomalous behavior from interacting oscillators in the high-temperature regime

TL;DR

This paper analyzes a microscopic model of interacting oscillators at high temperature, revealing anomalous fluctuation behaviors and deriving stochastic equations that describe their asymptotic dynamics.

Contribution

It extends previous harmonic chain results to a general nonlinear potential in the high-temperature regime, identifying new fluctuation phenomena.

Findings

Fluctuations of one field converge to Ornstein-Uhlenbeck process.

In stronger asymmetry, fluctuations follow stochastic Burgers equation.

Another field exhibits fractional diffusion with skewed Lévy process.

Abstract

A microscopic model of interacting oscillators, which admits two conserved quantities, volume, and energy, is investigated. We begin with a system driven by a general nonlinear potential under high-temperature regime by taking the inverse temperature of the system asymptotically small. As a consequence, one can extract a principal part (by a simple Taylor expansion argument), which is driven by the harmonic potential, and we show that previous results for the harmonic chain are covered with generality. We consider two fluctuation fields, which are defined as a linear combination of the fluctuation fields of the two conserved quantities, volume, and energy, and we show that the fluctuations of one field converge to a solution of an additive stochastic heat equation, which corresponds to the Ornstein-Uhlenbeck process, in a weak asymmetric regime, or to a solution of the stochastic Burgers equation, in a stronger asymmetric regime. On the other hand, the fluctuations of the other field cross from an additive stochastic heat equation to a fractional diffusion equation given by a skewed L\'evy process.