Heat flow in a periodically forced, unpinned thermostatted chain

TL;DR

This paper proves the hydrodynamic limit for a one-dimensional thermostatted harmonic chain with periodic forcing, showing convergence of empirical profiles to a nonlinear diffusive PDE system under diffusive scaling.

Contribution

It establishes the hydrodynamic limit for a thermostatted harmonic chain with periodic forcing, including boundary conditions and nonlinear PDE description.

Findings

Empirical profiles converge to a nonlinear diffusive PDE system.

The system exhibits boundary conditions of Dirichlet and Neumann types.

Hydrodynamic limit holds under diffusive scaling.

Abstract

We prove the hydrodynamic limit for a one-dimensional harmonic chain of interacting atoms with a random flip of the momentum sign. The system is open: at the left boundary it is attached to a heat bath at temperature $T_-$, while at the right endpoint it is subject to an action of a force which reads as $\bar F + \frac 1{\sqrt n} \widetilde{\mathcal F} (n^2 t)$, where $\bar F \ge0$ and $\widetilde{\mathcal F}(t)$ is a periodic function. Here $n$ is the size of the microscopic system. Under a diffusive scaling of space-time, we prove that the empirical profiles of the two locally conserved quantities - the volume stretch and the energy - converge, as $n\to+\infty$, to the solution of a non-linear diffusive system of conservative partial differential equations with a Dirichlet type and Neumann boundary conditions on the left and the right endpoints, respectively.