Abstract
Kinetic theories constitute one of the most promising tools to decipher the characteristic spatiotemporal dynamics in systems of actively propelled particles. In this context, the Boltzmann equation plays a pivotal role, since it provides a natural translation between a particle-level description of the system’s dynamics and the corresponding hydrodynamic fields. Yet, the intricate mathematical structure of the Boltzmann equation substantially limits the progress toward a full understanding of this equation by solely analytical means. Here, we propose a general framework to numerically solve the Boltzmann equation for self-propelled particle systems in two spatial dimensions and with arbitrary boundary conditions. We discuss potential applications of this numerical framework to active matter systems and use the algorithm to give a detailed analysis to a model system of self-propelled particles with polar interactions. In accordance with previous studies, we find that spatially homogeneous isotropic and broken-symmetry states populate two distinct regions in parameter space, which are separated by a narrow region of spatially inhomogeneous, density-segregated moving patterns. We find clear evidence that these three regions in parameter space are connected by first-order phase transitions and that the transition between the spatially homogeneous isotropic and polar ordered phases bears striking similarities to liquid-gas phase transitions in equilibrium systems. Within the density-segregated parameter regime, we find a novel stable limit-cycle solution of the Boltzmann equation, which consists of parallel lanes of polar clusters moving in opposite directions, so as to render the overall symmetry of the system’s ordered state nematic, despite purely polar interactions on the level of single particles.5 MoreReceived 2 April 2014DOI:https://doi.org/10.1103/PhysRevX.4.041030This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical Society
Highlights
Apart from its inherent limitations, which are largely due to the assumptions of binary particle interactions and molecular chaos, the following advantages of this framework are manifest: (i) The structure of the Boltzmann equation, relating convection and collision processes on the level of the one-particle distribution function, is ideally suited to explicitly implement a microscopic picture of particle dynamics. (ii) Because of its mesoscopic character, the Boltzmann equation provides an immediate connection to the system’s hydrodynamic variables, which naturally arise in the form of the various
For sufficiently dilute systems, binary particle interactions are believed to dominate over higher-order interactions and the spatiotemporal dynamics of the oneparticle distribution function fðx; θ; tÞ can be captured by means of the Boltzmann equation [39]
We have developed a general numerical framework to solve the Boltzmann equation for twodimensional systems of self-propelled particles, which we referred to as SNAKE
Summary
Developing a deeper understanding of active matter [1,2,3,4] has been the major focus of a considerable amount of theoretical work over the last few decades [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. SNAKE provides full access to the flexibility of the Boltzmann equation in the implementation of specific model systems that are based on a concrete microscopic picture of particle dynamics This flexibility equips this approach with powerful capabilities to incorporate physical boundary conditions, which can be formulated on the basis of actual particle-wall interactions rather than on the level of macroscopic, hydrodynamic field variables. III E, we shift our focus to a deeper discussion of densitysegregated patterns and report on the emergence of previously unseen “cluster-lane patterns” that seem to occupy the same parameter region as the familiar solitary-wave patterns and to constitute a stable limit-cycle solution of the underlying Boltzmann equation These patterns consist of parallel lanes of polar clusters moving in opposite directions; see Figs.
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