Fourier's Law in a Generalized Piston Model

TL;DR

This paper derives Fourier's law in a generalized piston model using kinetic theory, analyzing particle-wall collisions, and confirms the law's validity through numerical simulations in specific limits.

Contribution

It introduces a simplified mechanical model to derive Fourier's law from kinetic theory, connecting microscopic collisions to macroscopic heat conduction.

Findings

Derivation of thermodynamic quantities for non-equilibrium states.

Validation of Fourier's law in the large n, mN/M>>1, m/M<<1 limits.

Good agreement between theoretical predictions and numerical simulations.

Abstract

A simplified, but non trivial, mechanical model -- gas of $N$ particles of mass $m$ in a box partitioned by $n$ mobile adiabatic walls of mass $M$ -- interacting with two thermal baths at different temperatures, is discussed in the framework of kinetic theory. Following an approach due to Smoluchowski, from an analysis of the collisions particles/walls, we derive the values of the main thermodynamic quantities for the stationary non-equilibrium states. The results are compared with extensive numerical simulations; in the limit of large $n$, $mN/M\gg 1$ and $m/M \ll 1$, we find a good approximation of Fourier's law.