Highly Nonlinear Ising Model and Social Segregation

TL;DR

This paper introduces a nonlinear modification to the Ising model to better simulate social segregation, revealing that phase separation occurs below a critical temperature, with critical points depending on the nonlinearity degree.

Contribution

It extends the Ising model with nonlinear terms to better represent social phenomena like segregation, and analyzes the impact on critical temperatures and phase behavior.

Findings

Phase separation occurs only below the Curie temperature.

Critical temperature depends on the nonlinearity degree n and magnetization m.

Results are consistent except for n=3, which violates universality.

Abstract

The usual interaction energy of the random field Ising model in statistical physics is modified by complementing the random field by added to the energy of the usual Ising model a nonlinear term S^n were S is the sum of the neighbor spins, and n=0,1,3,5,7,9,11. Within the Schelling model of urban segregation, this modification corresponds to housing prices depending on the immediate neighborhood. Simulations at different temperatures, lattice size, magnetic field, number of neighbors and different time intervals showed that results for all n are similar, expect for n=3 in violation of the universality principle and the law of corresponding states. In order to find the critical temperatures, for large n we no longer start with all spins parallel but instead with a random configuration, in order to facilitate spin flips. However, in all cases we have a Curie temperature with phase separation or long-range segregation only below this Curie temperature, and it is approximated by a simple formula: Tc is proportional to 1+m for n=1, while Tc is roughly proportional to m for n >> 1.