Abstract
In this work we derive and analyse coarse-grained descriptions of self-propelled particles with selective attraction–repulsion interaction, where individuals may respond differently to their neighbours depending on their relative state of motion (approach versus movement away). Based on the formulation of a nonlinear Fokker–Planck equation, we derive a kinetic description of the system dynamics in terms of equations for the Fourier modes of the one-particle density function. This approach allows effective numerical investigation of the stability of solutions of the nonlinear Fokker–Planck equation. Further on, we also derive a hydrodynamic theory by performing a closure at the level of the second Fourier mode of the one-particle density function. We show that the general form of equations is in agreement with the theory formulated by Toner and Tu. The stability of spatially homogeneous solutions is analysed and the range of validity of the hydrodynamic equations is quantified. Finally, we compare our analytical predictions on the stability of the homogeneous solutions with results of individual-based simulations. They show good agreement for sufficiently large densities and non-negligible short-ranged repulsion. The results of the kinetic theory for weak short-ranged repulsion reveal the existence of a previously unknown phase of the model consisting of dense, nematically aligned filaments, which cannot be accounted for by the present hydrodynamics theory of the Toner and Tu type for polar active matter.
Highlights
In recent decades, there has been an increased research focus on far-from-equilibrium systems in biology and physics which is referred to as “active matter”
Based on the formulation of a nonlinear FokkerPlanck equation, we derive a kinetic description of the system dynamics in terms of equations for the Fourier modes of the one-particle density function
The Fokker-Planck equation (FPE) (3.6), which was derived from the linear FPE (3.3) governing the dynamics of PN, is nonlinear since the force terms Fφ depend on P (r, φ, t)
Summary
There has been an increased research focus on far-from-equilibrium systems in biology and physics which is referred to as “active matter”. Toner and Tu made a seminal contribution by formulating the hydrodynamic equations of polar active matter at largest relevant length and time scales purely based on symmetry arguments [13,14] The analysis of these generic equations, as well as their counterparts for nematic order, improved our understanding of the fundamental properties of active matter, such as the existence of long range order or giant number fluctuations [15, 16]. The direct derivation of a hydrodynamic theory of the Toner and Tu type from microscopic models of active matter was a long standing problem Recently, such a link between microscopic parameters determining the dynamics of individual active units and parameters governing the macroscopic flow of active matter was established by formulating kinetic equations for minimal models of selfpropelled particles with velocity-alignment [17,18,19,20] and self-propelled aligning rods [21]. We compare the results of the kinetic and hydrodynamic theory with direct numerical simulations of the individual-based model
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
