Dynamics in a Kinetic Model of Oriented Particles with Phase Transition

Abstract

Motivated by a phenomenon of phase transition in a model of alignment of self-propelled particles, we obtain a kinetic mean-field equation which is nothing else than the Doi equation (also called Smoluchowski equation) with dipolar potential.In a self-contained article, using only basic tools, we analyze the dynamics of this equation in any dimension. We first prove global well-posedness of this equation, starting with an initial condition in any Sobolev space. We then compute all possible steady-states. There is a threshold for the noise parameter: over this threshold, the only equilibrium is the uniform distribution, and under this threshold, there is also a family of non-isotropic equilibria.We give a rigorous prove of convergence of the solution to a steady-state as time goes to infinity. In particular we show that in the supercritical case, the only initial conditions leading to the uniform distribution in large time are those with vanishing momentum. For any positive value of the noise parameter, and any initial condition, we give rates of convergence towards equilibrium, exponentially for both supercritical and subcritical cases and algebraically for the critical case.

Partial differential equation models in the socio-economic sciences

“…[15,16]) and better understand phase transitions, i.e. critical values of certain parameters needed to successfully form flocks [17][18][19]. In this issue, Tadmor & Tan [20] follow those lines by considering a hydrodynamic model with non-local alignment interaction.…”

Large-Scale Dynamics of Mean-Field Games Driven by Local Nash Equilibria

“…Therefore, F is Liapounov functional for this dynamic and D is the free-energy dissipation term. By fine analysis of D, it is possible in some cases to deduce decay rates from this kind of estimate [14,32]. Equilibria given by (3.11) are critical points of F subject to the constraint f dy = 1 and the chemical potential µ is the Lagrange multiplier of this constraint in this optimization problem.…”

“…M κΩ (y) ≈ 1). On the other-hand, when c(κ) → 1, which happens when κ → ∞, then M κΩ (y) → δ Ω (y) (see details in [8,14]). Now, we look at the solutions of the compatibility condition (5.20).…”

“…Now, we look at the solutions of the compatibility condition (5.20). This analysis has been performed in [8,14]. We only summarize the final results in the following.…”

“…We only summarize the final results in the following. We refer to [8,14] for the precise mathematical statement of the stability result, as well as for rate estimates of convergence to the equilibria in the homogeneous configuration case.…”