# An open microscopic model of heat conduction: evolution and non-equilibrium stationary states

## TL;DR

This paper models heat conduction in a one-dimensional oscillator chain with stochastic momentum exchanges, demonstrating diffusive behavior, uphill diffusion, and deriving macroscopic equations for energy and stretch profiles.

## Contribution

It introduces a microscopic stochastic model that captures non-equilibrium stationary states and derives macroscopic diffusive equations, including uphill diffusion phenomena.

## Key findings

- Energy and stretch profiles converge to diffusive solutions
- Stationary temperature profile can have a maximum inside the chain
- Derived non-stationary coupled diffusive equations

## Abstract

We consider a one-dimensional chain of coupled oscillators in contact at both ends with heat baths at different temperatures, and subject to an external force at one end. The Hamiltonian dynamics in the bulk is perturbed by random exchanges of the neighbouring momenta such that the energy is locally conserved. We prove that in the stationary state the energy and the volume stretch profiles, in large scale limit, converge to the solutions of a diffusive system with Dirichlet boundary conditions. As a consequence the macroscopic temperature stationary profile presents a maximum inside the chain higher than the thermostats temperatures, as well as the possibility of uphill diffusion (energy current against the temperature gradient). Finally, we are also able to derive the non-stationary macroscopic coupled diffusive equations followed by the energy and volume stretch profiles.