Kac's Program and Relative Entropy Decay for Nonlinear Spin-Exchange Dynamics

TL;DR

This paper introduces a nonlinear spin-exchange model for Ising systems, proving convergence to equilibrium and exponential decay in relative entropy, using a kinetic theory approach and modified logarithmic Sobolev inequalities.

Contribution

It develops a new nonlinear exchange dynamics framework for Ising models and establishes convergence and decay rates using a novel kinetic theory approach.

Findings

Proves convergence of the nonlinear dynamics to the Ising equilibrium.

Shows exponential decay of relative entropy for weak interactions.

Establishes a uniform modified logarithmic Sobolev inequality for the associated particle system.

Abstract

We introduce and analyze a nonlinear exchange dynamics for Ising spin systems with arbitrary interactions. The evolution is governed by a quadratic Boltzmann-type equation that conserves the mean magnetization. Collisions are encoded through a spin-exchange kernel chosen so that the dynamics converge to the Ising model with the prescribed interaction and mean magnetization profile determined by the initial state. We prove a general convergence theorem, valid for any interaction and any transport kernel. Moreover, we show that, for sufficiently weak interactions, the system relaxes exponentially fast to equilibrium in relative entropy, with optimal decay rate independent of the initial condition. The proof relies on establishing a strong version of the Kac program from kinetic theory. In particular, we show that the associated Kac particle system satisfies a modified logarithmic Sobolev inequality with constants uniform in the number of particles. This is achieved by adapting the method of stochastic localization to the present conservative setting.