Hydrodynamic limits for kinetic flocking models of Cucker-Smale type

Abstract

We analyse the asymptotic behavior for kinetic models describing the collective behavior of animal populations. We focus on models for self-propelled individuals, whose velocity relaxes toward the mean orientation of the neighbors. The self-propelling and friction forces together with the alignment and the noise are interpreted as a collision/interaction mechanism acting with equal strength. We show that the set of generalized collision invariants, introduced in [29], is equivalent in our setting to the more classical notion of collision invariants, i.e., the kernel of a suitably linearized collision operator. After identifying these collision invariants, we derive the fluid model, by appealing to the balances for the particle concentration and orientation. We investigate the main properties of the macroscopic model for a general potential with radial symmetry.

Monte Carlo gPC Methods for Diffusive Kinetic Flocking Models with Uncertainties

“…We introduce Stochastic Galerkin (SG) numerical methods with applications to the nonlinear Vlasov-Fokker-Planck (VFP) equation (1). We discuss the class of stochastic Galerkin (SG) methods and, in particular, we concentrate on the generalized Polynomial Chaos (gPC) decomposition [22,46,47,49].…”

“…We now approximate the limiting stochastic kinetic equation taking advantage of the particle reformulation of the problem. In fact, since the solution of the system of SDEs (10) converges in distribution to the solution of the original problem (1) for N → +∞, we can approximate the original dynamics by means of a Monte Carlo (MC) method in the phase space. The main drawback of this approach lies in the computational cost O(M 2 N 2 ), since at each time step and for each gPC projection each agent modifies its velocity in a genuine nonlinear way.…”

“…In the present paper we concentrate on a flocking model for interacting agents with self-propulsion and diffusion where the parameters are assumed to be stochastic. This model has been recently investigated in [1,6,7,10] in absence of uncertainties. The introduction of uncertain quantities points in the direction of a more realistic description of the underlying processes and helps us to compute possible deviations from the prescribed deterministic behavior.…”

Physics of growing biological tissues: the complex cross-talk between cell activity, growth and resistance

“…The macroscopic behaviour of tissues can be derived from the dynamics at the cellular level by letting intercellular distances tend to those of the tissue level. The link between the cellular and the tissue scales is thus obtained through usual hydrodynamic limit procedures [122,123]. Typically sarcoma tumour growth would be a good candidate for such treatment.…”

Fluid models with phase transition for kinetic equations in swarming

“…35, 33 and 34, see related results by different approaches. 1,19 These hydrodynamic systems are usually referred as Self-Organized Hydrodynamics (SOH). Our goal in this work is to derive the corresponding SOH system for kinetic inhomogeneous problems of Cucker-Smale type with phase transition at their corresponding homogeneous Fokker-Planck equation in contrast to Refs.…”