Effective Hamiltonian approach to kinetic Ising models: Application to an infinitely long-range Husimi-Temperley model

TL;DR

This paper develops an effective Hamiltonian approach for kinetic Ising models, applying it to the Husimi-Temperley model, deriving a nonlinear PDE for the effective Hamiltonian, and confirming the evolution of magnetization aligns with mean field theory.

Contribution

It introduces a nonlinear master equation for the effective Hamiltonian density, enabling analysis of large systems without computational infeasibility, and applies it to the Husimi-Temperley model.

Findings

Reproduces known numerical results for finite N

Derives an exact nonlinear PDE in the thermodynamic limit

Shows magnetization evolution matches mean field predictions

Abstract

The probability distribution (PD) of spin configurations in kinetic Ising models has been cast in the form of the canonical Boltzmann PD with a time-dependent effective Hamiltonian (EH). It has been argued that in systems with extensive energy EH depends linearly on the number of spins $N$ leading to the exponential dependence of PD on the system size. In macroscopic systems the argument of the exponential function may reach values of the order of the Avogadro number which is impossible to deal with computationally, thus making unusable the linear master equation (ME) governing the PD evolution. To overcome the difficulty, it has been suggested to use instead the nonlinear ME (NLME) for the EH density per spin. It has been shown that in spatially homogeneous systems NLME contains only terms of order unity even in the thermodynamic limit. The approach has been illustrated with the kinetic Husimi-Temperley model (HTM) evolving under the Glauber dynamics. At finite $ N $ the known numerical results has been reproduced and extended to broader parameter ranges. In the thermodynamic limit an exact nonlinear partial differential equation of the Hamilton-Jacobi type for EH has been derived. It has been shown that the average magnetization in HTM evolves according to the conventional kinetic mean field equation.