Long time, large scale limit of the Wigner transform for a system of linear oscillators in one dimension

TL;DR

This paper analyzes the asymptotic behavior of the Wigner transform for a 1D system of linear oscillators with weak noise, showing convergence to fractional or classical heat equations depending on pinning conditions.

Contribution

It proves the convergence of the Wigner transform to fractional or classical heat equations in the long-time, large-scale limit for a 1D oscillator system with weak noise.

Findings

Unpinned case leads to a fractional heat equation with exponent 3/4.

Pinned case results in a standard heat equation.

Identifies the scaling limits for the Wigner transform in different physical regimes.

Abstract

We consider the long time, large scale behavior of the Wigner transform $W_\eps(t,x,k)$ of the wave function corresponding to a discrete wave equation on a 1-d integer lattice, with a weak multiplicative noise. This model has been introduced in Basile, Bernardin, and Olla to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum. The kinetic limit for the Wigner transform has been shown in Basile, Olla, and Spohn. In the present paper we prove that in the unpinned case there exists $\gamma_0>0$ such that for any $\gamma\in(0,\gamma_0]$ the weak limit of $W_\eps(t/\eps^{3/2\gamma},x/\eps^{\gamma},k)$, as $\eps\ll1$, satisfies a one dimensional fractional heat equation $\partial_t W(t,x)=-\hat c(-\partial_x^2)^{3/4}W(t,x)$ with $\hat c>0$. In the pinned case an analogous result can be claimed for $W_\eps(t/\eps^{2\gamma},x/\eps^{\gamma},k)$ but the limit satisfies then the usual heat equation.