Transport coefficients of self-propelled particles: Reverse perturbations and transverse current correlations

Abstract

The reverse perturbation method [Phys. Rev. E 59, 4894 (1999)] for shearing simple liquids and measuring their viscosity is extended to systems of self-propelled particles with time-discrete stochastic dynamics. For verification, this method is first applied to Multi-Particle Collision Dynamics (MPCD) [J. Chem. Phys. 110, 8605 (1999)], a momentum-conserving solvent. An extension to the Vicsek-model (VM) of self-propelled particles [Phys. Rev. Lett. 75, 1226(1995 shows a phenomenon that is similar to the skin effect of an alternating electric current: momentum that is fed into the boundaries of a layer decays mostly exponentionally towards the center of the layer. It is shown how two transport coefficients, i.e. the shear viscosity ν and the momentum amplification coefficient λ, can be obtained by fitting this decay with an analytical solution of the hydrodynamic equations for the VM. As for the MPCD case, the viscosity of the VM consists of two parts, the kinetic and the collisional viscosity. An analytical expression for the collisional part is derived by an Enskog-like kinetic theory. In the following paper (Part II), reasonable quantitative agreement between agent-based simulations and predictions by kinetic theory is observed. In Part II, transverse current correlations and a Green-Kubo relation are used to obtain λ and ν, in excellent agreement with the reverse perturbation results. PACS numbers:87.10.-e, 05.20.Dd, 64.60.Cn, 02.70.Ns I. INTRODUCTION During the past two decades, there has been a large interest in active matter systems, such as bird flocks [1], swarming bacteria [2, 3], active colloids [4, 5], microtubule mixtures [6] and actin networks [7] driven by molecular motors. These systems display interesting behaviors such as pattern formation, collective motion and non-equilibrium phase transitions [8,9]. Some of these features already occur in one of the simplest models for active matter, the Vicsek-model (VM) of self-propelled particles [10-12] and its variants [13][14][15][16][17][18][19][20]. Because of the simplicity of its interaction rules and the existence of a non-standard transition to a collective state of polar order, the VM became an archetype of active matter.Due to the many degrees of freedom, theoretical studies of active matter systems are often based on coarse-grained macroscopic transport equations for the slow variables such as density or momentum. Originally, the general forms of these equations were postulated by symmetry and renormalization group arguments, such as in the seminal Toner-Tu theory [21][22][23] for polar active matter. However, this approach leaves the coefficients of the terms in the transport equation largely undetermined. Furthermore, memory and other nonlocal terms are usually not considered, although for particular models there is evidence on their relevance [24]. These shortcomings motived many researchers to derive macroscopic transport equations directly from the microscopic interactions and to obtain explicit expressions for the occurring coefficients [2...