Derivation of Anomalous Behavior from Interacting Oscillators in the High-Temperature Regime

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 Lévy process.