An extension of the Kac ring model

TL;DR

This paper extends Kac's classical model to quantum spins, demonstrating exponential relaxation to equilibrium and analyzing entropy behavior in the large system limit.

Contribution

It introduces a quantum spin dynamics extension of Kac's model, providing rigorous proof of relaxation and entropy properties in the large system limit.

Findings

Magnetization follows an autonomous relaxation equation

Exponential relaxation to equilibrium magnetization

Monotonic increase of Gibbs-von Neumann entropy

Abstract

We introduce a unitary dynamics for quantum spins which is an extension of a model introduced by Mark Kac to clarify the phenomenon of relaxation to equilibrium. When the number of spins gets very large, the magnetization satisfies an autonomous equation as function of time with exponentially fast relaxation to the equilibrium magnetization as determined by the microcanonical ensemble. This is proven as a law of large numbers with respect to a class of initial data. The corresponding Gibbs-von Neumann entropy is also computed and its monotonicity in time discussed.