Abstract
We study the McKean–Vlasov equation ∂tϱ=β-1Δϱ+κ∇·(ϱ∇(W⋆ϱ)),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\partial _t \\varrho = \\beta ^{-1} \\Delta \\varrho + \\kappa {{\\,\\mathrm{\\nabla \\cdot }\\,}}(\\varrho \\nabla (W \\star \\varrho )), \\end{aligned}$$\\end{document}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.
Highlights
Systems of interacting particles arise in a myriad of applications ranging from opinion dynamics [41], granular materials [6,11,25] and mathematical biology [8,47] to statistical mechanics [50], galactic dynamics [18], droplet growth [29], plasma physics [14], and synchronisation [48]
It is clear that what we have described is a set of interacting overdamped Langevin equations
We extend the L2-decay results of [21] to arbitrary dimensions and arbitrary sufficiently nice interactions and provide sufficient conditions for convergence to equilibrium in relative entropy
Summary
Systems of interacting particles arise in a myriad of applications ranging from opinion dynamics [41], granular materials [6,11,25] and mathematical biology [8,47] to statistical mechanics [50], galactic dynamics [18], droplet growth [29], plasma physics [14], and synchronisation [48]. The rest of the paper is devoted to the analysis of the properties of non-trivial stationary states of the Mckean–Vlasov system, that is, nontrivial solutions of β−1 + κ∇ · ( ∇W ) = 0 Previous results in this direction include those by Tamura [66], who provided some criteria for the existence of local bifurcations on the whole space by using tools from nonlinear functional analysis, in particular, the Crandall–Rabinowitz theorem. His analysis depends crucially on the unphysical assumption that the interaction potential is an odd function. We extend the results of [27] and provide additional criteria for the existence of continuous and discontinuous phase transitions
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
