Thermal Equilibrium in D-dimensions: From Fluids and Polymers to Kinetic Wealth Exchange Models

TL;DR

This paper explores how diverse systems, from physics to social models, reach a common equilibrium described by a gamma distribution, with an effective dimension parameter influencing the outcome.

Contribution

It demonstrates the universality of the Gibbs-Boltzmann equilibrium distribution across different systems and introduces the concept of an effective dimension D affecting the equilibrium shape.

Findings

Systems relax to a gamma distribution with shape D/2.

Equilibrium distribution is universal across different models.

Effective dimension D controls the distribution shape.

Abstract

In this paper we discuss some examples of systems composed of $N$ units, which exchange a conserved quantity $x$ according to some given stochastic rule, from some standard kinetic model of condensed matter physics to the kinetic exchange models used for studying the wealth dynamics of social systems. The focus is on the similarity of the equilibrium state of the various examples considered, which all relax toward a canonical Gibbs-Boltzmann equilibrium distribution for the quantity $x$, given by a $\Gamma$-distribution with shape parameter $\alpha = D/2$, which implicitly defines an effective dimension $D$ of the system. We study various systems exploring (continuous) values of $D$ in the interval $[1,\infty)$.