Numerical treatment of the Boltzmann equation for self-propelled particle systems

摘要

Kinetic theories constitute one of the most promising tools to decipher thecharacteristic spatio-temporal dynamics in systems of actively propelledparticles. In this context, the Boltzmann equation plays a pivotal role, sinceit provides a natural translation between a particle-level description of thesystem's dynamics and the corresponding hydrodynamic fields. Yet, the intricatemathematical structure of the Boltzmann equation substantially limits theprogress toward a full understanding of this equation by solely analyticalmeans. Here, we propose a general framework to numerically solve the Boltzmannequation for self-propelled particle systems in two spatial dimensions and witharbitrary boundary conditions. We discuss potential applications of thisnumerical framework to active matter systems, and use the algorithm to give adetailed analysis to a model system of self-propelled particles with polarinteractions. In accordance with previous studies, we find that spatiallyhomogeneous isotropic and broken symmetry states populate two distinct regionsin parameter space, which are separated by a narrow region of spatiallyinhomogeneous, density-segregated moving patterns. We find clear evidence thatthese three regions in parameter space are connected by first order phasetransitions, and that the transition between the spatially homogeneousisotropic and polar ordered phases bears striking similarities to liquid-gasphase transitions in equilibrium systems. Within the density segregatedparameter regime, we find a novel stable limit-cycle solution of the Boltzmannequation, which consists of parallel lanes of polar clusters moving in oppositedirections, so as to render the overall symmetry of the system's ordered statenematic, despite purely polar interactions on the level of single particles.