Non-mean-field Vicsek-type models for collective behavior

Abstract

We consider interacting particle dynamics with Vicsek-type interactions, and their macroscopic Partial Differential Equation (PDE) limit, in the non-mean-field regime; that is, we consider the case in which each particle/agent in the system interacts only with a prescribed subset of the particles in the system (for example, those within a certain distance). In this non-mean-field regime the influence between agents (i.e. the interaction term) can be normalized either by the total number of agents in the system (global scaling) or by the number of agents with which the particle is effectively interacting (local scaling). We compare the behavior of the globally scaled and the locally scaled systems in many respects, considering for each scaling both the PDE and the corresponding particle model. In particular, we observe that both the locally and globally scaled particle system exhibit pattern formation (i.e. formation of traveling-wave-like solutions) within certain parameter regimes, and generally display similar dynamics. The same is not true of the corresponding PDE models. Indeed, while both PDE models have multiple stationary states, for the globally scaled PDE such (space-homogeneous) equilibria are unstable for certain parameter regimes, with the instability leading to traveling wave solutions, while they are always stable for the locally scaled one, which never produces traveling waves. This observation is based on a careful numerical study of the model, supported by further analysis.

Dynamic density functional theory for sedimentation processes on complex domains: Modelling, spectral elements, and control problems

“…In order to solve the DDFT model (8) and control problem (9), appropriate numerical methods have to be used that are This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.…”

“…It can be solved on complicated domains using the MultiShape package, as described in [41]. The numerical methods used to solve (8) are briefly discussed in Section III and numerical examples are presented in Section IV.…”

“…We note that the mathematical analysis of such non-linear problems is highly challenging; see, e.g., [11]. In this section, we will outline the necessary aspects of the optimal control framework for the DDFT model (8).…”

On the study of slow–fast dynamics, when the fast process has multiple invariant measures

“…If a McKean–Vlasov equation of this type was used to model the fast process (and the slow variable appeared in the noise coefficients of the fast process, a scenario which is at the moment work in progress from the second author and collaborators) then clearly the assumption that we make in this paper would not hold. However, still within the class of McKean–Vlasov dynamics, evolutions inspired by the Vicsek model have been shown to have the same number of invariant measures, irrespective of the strength of the noise—a phenomenon which goes under the name of unconditional flocking , see [12–15]. In this case, the assumption we make in this paper would be satisfied.…”

“…Assuming that Ṽ λ,x,v t has L absorbing states (which are the same, irrespective of the value of x we fix), and that, given any initial datum Ṽ λ,x,v 0 = v, there is probability one of reaching one of them, one can consider the probability q (i) (x, v) of Ṽ λ,x,v t being absorbed in e (i) . The limiting dynamics is then given by (12), where now ζ v is replaced by ζ x,v , where ζ x,v is equal to e (i) with probability q (i) (x, v). This fact is proven in Theorem 4.2.…”