Crowds reaching targets by maximizing entropy: a Clausius-Duhem inequality approach

TL;DR

This paper introduces a thermodynamics-inspired model for crowd dynamics, using entropy concepts and an inequality akin to the Second Law to analyze how crowds reach targets.

Contribution

It develops a novel continuous model incorporating entropy and a Clausius-Duhem inequality to study crowd behavior with nonlocal interactions.

Findings

Entropy increases over time in the model.

Simulations show the corrector term remains small.

Entropy converges to a limit value.

Abstract

In this paper we propose the use of concepts from thermodynamics in the study of crowd dynamics. Our continuous model consists of the continuity equation for the density of the crowd and a kinetic equation for the velocity field. The latter includes a nonlocal term that models interactions between individuals. To support our modelling assumptions, we introduce an inequality that resembles the Second Law of Thermodynamics, containing an entropy-like functional. We show that its time derivative equals a positive dissipation term minus a corrector term. The latter term should be small for the time derivative of the entropy to be positive. In case of isotropic interactions the corrector term is absent. For the anisotropic case, we support the claim that the corrector term is small by simulations for the corresponding particle system. They reveal that this term is sufficiently small for the entropy still to increase. Moreover, we show that the entropy converges in time towards a limit value.