Free energy of a chemotactic model with nonlinear diffusion

The Patlak-Keller-Segel equation is a canonical model of chemotaxis to describe self-organized aggregation of organisms interacting with chemical signals. We investigate a variant of this model, assuming that the organisms exert effective pressure proportional to the number density. From the resulting set of partial differential equations, we derive a Lyapunov functional that can also be regarded as the free energy of this model, and minimize it with a Monte Carlo method to detect the condition for self-organized aggregation. Focusing on radially symmetric solutions on a two-dimensional disc, we find that the chemical interaction competes with diffusion so that aggregation occurs when the relative interaction strength exceeds a certain threshold. Based on the analysis of the free-energy landscape, we argue that the transition from a homogeneous state to aggregation is abrupt yet continuous.