  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.
