Phase Transitions in a Kinetic Flocking Model of Cucker-Smale Type

We consider a collective behavior model in which individuals try to imitate each others' velocity and have a preferred speed. We show that a phase change phenomenon takes place as diffusion decreases, bringing the system from a “disordered” to an “ordered” state. This effect is related to recently noticed phenomena for the diffusive Vicsek model. We also carry out numerical simulations of the system and give further details on the phase transition

On the singularity formation and relaxation to equilibrium in 1D Fokker–Planck model with superlinear drift

We consider a class of Fokker–Planck equations with linear diffusion and superlinear drift enjoying a formal Wasserstein-like gradient flow structure with convex mobility function. In the drift-dominant regime, the equations have a finite critical mass above which the measure minimising the associated entropy functional displays a singular component. Our approach, which addresses the one-dimensional case, is based on a reformulation of the problem in terms of the pseudo-inverse distribution function. Motivated by the structure of the equation in the new variables, we establish a general framework for global-in-time existence, uniqueness and regularity of monotonic viscosity solutions to a class of nonlinear degenerate (resp. singular) parabolic equations, using as a key tool comparison principles and maximum arguments. We then focus on a specific equation and study in more detail the regularity and dynamics of solutions. In particular, blow-up behaviour, formation of condensates (i.e. Dirac measures at zero) and long-time asymptotics are investigated. As a consequence, in the mass-supercritical case, solutions will blow up in L∞ in finite time and—understood in a generalised, measure sense—they will eventually have condensate. We further show that the singular part of the measure solution does in general interact with the density and that condensates can be transient. The equations considered are motivated by a model for bosons introduced by Kaniadakis and Quarati (1994), which has a similar entropy structure and a critical mass if d ≥ 3

A hybrid mass transport finite element method for Keller--Segel type systems

We propose a new splitting scheme for general reaction–taxis–diffusion systems in one spatial dimension capable to deal with simultaneous concentrated and diffusive regions as well as travelling waves and merging phenomena. The splitting scheme is based on a mass transport strategy for the cell density coupled with classical finite element approximations for the rest of the system. The built-in mass adaption of the scheme allows for an excellent performance even with respect to dedicated mesh-adapted AMR schemes in original variables

Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics

We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as t → ∞

Long-time behaviour and phase transitions for the McKean—Vlasov equation on the torus

We study the McKean-Vlasov equation ∂t% = β −1∆% + κ ∇·(%∇(W ? %)) , with periodic boundary conditions on the torus. We first study the global asymptotic stability of the homogeneous steady state. We then focus our attention on the stationary system, and prove the existence of nontrivial solutions branching from the homogeneous steady state, through possibly infinitely many bifurcations, under appropriate assumptions on the interaction potential. We also provide sufficient conditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase these results by applying them to several examples of interaction potentials such as the noisy Kuramoto model for synchronisation, the Keller–Segel model for bacterial chemotaxis, and the noisy Hegselmann–Krausse model for opinion dynamics

The ellipse law: Kirchhoff meets dislocations

In this paper we consider a nonlocal energyIαwhose kernel is obtained by addingto the Coulomb potential an anisotropic term weighted by a parameterα∈R. The caseα= 0corresponds to purely logarithmic interactions, minimised by the circle law;α= 1 correspondsto the energy of interacting dislocations, minimised by the semi-circle law. We show that forα∈(0,1) the minimiser is the normalised characteristic function of the domain enclosed bytheellipseof semi-axes√1−αand√1 +α. This result is one of the very few examples wherethe minimiser of a nonlocal anisotropic energy is explicitly computed. For the proof we borrowtechniques from fluid dynamics, in particular those related to Kirchhoff’s celebrated result thatdomains enclosed by ellipses are rotating vortex patches, calledKirchhoff ellipses

Well-balanced finite volume schemes for hydrodynamic equations with general free energy

Well-balanced and free energy dissipative first- and second-order accurate finite volume schemes are proposed for a general class of hydrodynamic systems with linear and nonlinear damping. The variation of the natural Liapunov functional of the system, given by its free energy, allows, for a characterization of the stationary states by its variation. An analog property at the discrete level enables us to preserve stationary states at machine precision while keeping the dissipation of the discrete free energy. These schemes can accurately analyse the stability properties of stationary states in challenging problems such as: phase transitions in collective behavior, generalized Euler-Poisson systems in chemotaxis and astrophysics, and models in dynamic density functional theories; having done a careful validation in a battery of relevant test cases

Monte Carlo gPC methods for diffusive kinetic flocking models with uncertainties

In this paper we introduce and discuss numerical schemes for the approximationof kinetic equations for flocking behavior with phase transitions that incorporate un-certain quantities. This class of schemes here considered make use of a Monte Carloapproach in the phase space coupled with a stochastic Galerkin expansion in the ran-dom space. The proposed methods naturally preserve the positivity of the statisticalmoments of the solution and are capable to achieve high accuracy in the random space.Several tests on a kinetic alignment model with self propulsion validate the proposedmethods both in the homogeneous and inhomogeneous setting, shading light on theinfluence of uncertainties in phase transition phenomena driven by noise such as theirsmoothing and confidence bands