Grand Canonical Evolution for the Kac Model

TL;DR

This paper analyzes a Kac model with particles interacting with an infinite reservoir, demonstrating exponential convergence to the grand canonical equilibrium, propagation of chaos, and deriving a Boltzmann-Kac type equation.

Contribution

It introduces a grand canonical Kac model with reservoir interactions, proving exponential convergence to equilibrium and establishing propagation of chaos.

Findings

Exponential convergence to the grand canonical equilibrium.

Spectral gap computed for the generator.

Propagation of chaos established.

Abstract

We study a model of random colliding particles interacting with an infinite reservoir at fixed temperature and chemical potential. Interaction between the particles is modeled via a Kac master equation \cite{kac}. Moreover, particles can leave the system toward the reservoir or enter the system from the reservoir. The system admits a unique steady state given by the Grand Canonical Ensemble at temperature $T=\beta^{-1}$ and chemical potential $\chi$. We show that any initial state converges exponentially fast to equilibrium by computing the spectral gap of the generator in a suitable $L^2$ space and by showing exponential decrease of the relative entropy with respect to the steady state. We also show propagation of chaos and thus the validity of a Boltzmann-Kac type equation for the particle density in the infinite system limit.