Statistical Thermodynamics of Clustered Populations

TL;DR

This paper develops a thermodynamic framework for analyzing populations divided into clusters, revealing phase transition behavior analogous to physical systems and validated through analytic models and simulations.

Contribution

It introduces a novel thermodynamic theory for clustered populations, including a phase transition analogy and a rigorous mapping to physical thermodynamics.

Findings

Identification of a phase transition for giant component emergence

Analytic model predictions match Monte Carlo simulations

Establishment of a thermodynamic analogy for population clustering

Abstract

We present a thermodynamic theory for a generic population of $M$ individuals distributed into $N$ groups (clusters). We construct the ensemble of all distributions with fixed $M$ and $N$, introduce a selection functional that embodies the physics that governs the population, and obtain the distribution that emerges in the scaling limit as the most probable among all distributions consistent with the given physics. We develop the thermodynamics of the ensemble and establish a rigorous mapping to thermodynamics. We treat the emergence of a so-called "giant component" as a formal phase transition and show that the criteria for its emergence are entirely analogous to the equilibrium conditions in molecular systems. We demonstrate the theory by an analytic model and confirm the predictions by Monte Carlo simulation.