Nonequilibrium dynamics of a stochastic model of anomalous heat transport

TL;DR

This paper investigates the nonequilibrium dynamics of a stochastic harmonic oscillator chain with conservative noise, revealing that the temperature evolution follows fractional diffusion and providing a hydrodynamic description of relaxation to the stationary state.

Contribution

The study extends previous work by deriving a hydrodynamic model for the covariance matrix evolution under generic boundary conditions, highlighting fractional diffusion in temperature dynamics.

Findings

Temperature profile evolves via fractional diffusion.

Hydrodynamic equations derived from covariance matrix dynamics.

Relaxation to stationary state characterized by fractional diffusion.

Abstract

We study the dynamics of covariances in a chain of harmonic oscillators with conservative noise in contact with two stochastic Langevin heat baths. The noise amounts to random collisions between nearest-neighbour oscillators that exchange their momenta. In a recent paper, [S Lepri et al. J. Phys. A: Math. Theor. 42 (2009) 025001], we have studied the stationary state of this system with fixed boundary conditions, finding analytical exact expressions for the temperature profile and the heat current in the thermodynamic (continuum) limit. In this paper we extend the analysis to the evolution of the covariance matrix and to generic boundary conditions. Our main purpose is to construct a hydrodynamic description of the relaxation to the stationary state, starting from the exact equations governing the evolution of the correlation matrix. We identify and adiabatically eliminate the fast variables, arriving at a continuity equation for the temperature profile T(y,t), complemented by an ordinary equation that accounts for the evolution in the bulk. Altogether, we find that the evolution of T(y,t) is the result of fractional diffusion.