A stochastic model of anomalous heat transport: analytical solution of the steady state

TL;DR

This paper presents an exact analytical solution for the temperature profile and energy current in a one-dimensional harmonic crystal with stochastic noise, revealing anomalous heat transport behavior.

Contribution

It provides the first exact solution of the steady state covariance matrix for this model, showing independence of the temperature profile from noise rate and deriving the energy current scaling.

Findings

Temperature profile is independent of noise rate fectively constant across the system.

Energy current scales as 1/rac{sqrt{\,gamma N}}.

Analytical results match numerical solutions for finite system sizes.

Abstract

We consider a one-dimensional harmonic crystal with conservative noise, in contact with two stochastic Langevin heat baths at different temperatures. The noise term consists of collisions between neighbouring oscillators that exchange their momenta, with a rate $\gamma$. The stationary equations for the covariance matrix are exactly solved in the thermodynamic limit ($N\to\infty$). In particular, we derive an analytical expression for the temperature profile, which turns out to be independent of $\gamma$. Moreover, we obtain an exact expression for the leading term of the energy current, which scales as $1/\sqrt{\gamma N}$. Our theoretical results are finally found to be consistent with the numerical solutions of the covariance matrix for finite $N$.