Statistical and Biological Physics

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Critical Assessment of the Boltzmann Approach to Active Systems

Florian Thüroff, Christoph A. Weber, Erwin Frey

In recent years, kinetic theory has gained considerable popularity to asses the large scale properties of self-propelled particle (SPP) systems. Specifically, the Boltzmann equation provides a compelling link between a microscopic, particle-level description of active systems, and the corresponding set of macroscopic equations. Yet, the applicability of Boltzmann's equation in the context of active media, as well its limitations due to the molecular chaos assumption remain largely elusive.

In this letter, we study the most pertinent features of binary interactions of rodlike SPPs by means of microscopic simulations and assess their consequences in the context of the system's large scale collective motion patterns. To this end, we set up a Boltzmann equation which is explicitly based on these scattering studies, and which is subsequently coarse grained to a hydrodynamic description of the system. This then allows us to analytically study the system's macroscopic dynamics in terms of the microscopic details of particle interactions. To asses the formation of polar patterns, we also present results from a direct numerical solution of the Boltzmann equation itself.

We find that this Boltzmann equation approach yields results which are in qualitative agreement with previous agent based simulations on polar systems, provided ordering proceeds via a gradual reduction in the spread of particle orientations. In line with previous studies, however, our results indicate that cluster formation processes might play a prominent role in the formation of collective motion patterns from purely repulsive particle interactions. These processes imply higher order correlations in particle orientations which are incompatible with the molecular chaos assumption, and, therefore, cause the Boltzmann equation approach to break down.