밀도에 의존하는 압력이 있는 주화성 계의 연구

Abstract

Many small organisms such as bacteria can attract each other by deposit-ing 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 Lyapunovfunctional, we show that the aggregation transition occurs discontinuously, form-ing 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.