Discontinuous phase transition in chemotactic aggregation with density-dependent pressure

TL;DR

This paper models chemotactic aggregation in organisms with density-dependent pressure, revealing a discontinuous phase transition to boundary aggregation through analysis of a PDE system and Lyapunov functional.

Contribution

It introduces a PDE model incorporating density-dependent pressure and demonstrates a discontinuous phase transition in chemotactic aggregation.

Findings

Aggregation occurs discontinuously near the boundary.

Lyapunov functional analysis reveals phase transition nature.

Model links chemotaxis with thermodynamic concepts.

Abstract

Many small organisms such as bacteria can attract each other by depositing chemical attractants. At the same time, they exert repulsive force on each other when crowded, which can be modeled by effective pressure as an increasing function of the organisms' density. As the chemical attraction becomes strong compared to the effective pressure, the system will undergo a phase transition from homogeneous distribution to aggregation. In this work, we describe the interplay of organisms and chemicals on a two-dimensional disk with a set of partial differential equations of the Patlak-Keller-Segel type. By analyzing its Lyapunov functional, we show that the aggregation transition occurs discontinuously, forming an aggregate near the boundary of the disk. The result can be interpreted within a thermodynamic framework by identifying the Lyapunov functional with free energy.