# Diffusion limit for a kinetic equation with a thermostatted interface

## TL;DR

This paper proves that under diffusive scaling, the energy density in a harmonic chain with an interface converges to a heat equation with a fixed boundary temperature, linking microscopic stochastic dynamics to macroscopic heat flow.

## Contribution

It establishes the diffusion limit for a kinetic phonon equation with an interface, connecting microscopic stochastic models to macroscopic heat equations with boundary conditions.

## Key findings

- Solutions converge to a heat equation with boundary temperature T
- Validates the diffusive limit for the phonon Boltzmann equation
- Provides a rigorous link between microscopic and macroscopic models

## Abstract

We consider a linear phonon Boltzmann equation with a reflecting/transmitting/absorbing interface. This equation appears as the Boltzmann-Grad limit for the energy density function of a harmonic chain of oscillators with inter-particle stochastic scattering in the presence of a heat bath at temperature $T$ in contact with one oscillator at the origin. We prove that under the diffusive scaling the solutions of the phonon equation tend to the solution $\rho(t,y)$ of a heat equation with the boundary condition $\rho(t,0)\equiv T$.