Macroscopic Limits and Phase Transition in a System of Self-propelled Particles

Abstract

We investigate systems of self-propelled particles with alignment interaction. Compared to previous work [13,19], the force acting on the particles is not normalized and this modification gives rise to phase transitions from disordered states at low density to aligned states at high densities. This model is the space inhomogeneous extension of [20] in which the existence and stability of the equilibrium states were investigated. When the density is lower than a threshold value, the dynamics is described by a non-linear diffusion equation. By contrast, when the density is larger than this threshold value, the dynamics is described by a similar hydrodynamic model for self-alignment interactions as derived in [13,19]. However, the modified normalization of the force gives rise to different convection speeds and the resulting model may lose its hyperbolicity in some regions of the state space.Acknowledgements: The authors wish to acknowledge the hospitality

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

“…with Q given by 8) and where ρ(x, t), ν x,t are related to f (t) by (4.6). Here, we have used (4.5) to replace Φ ε by Φ in (4.7), (4.8) and dropped the remaining O(ε) terms.…”

“…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.…”

“…In this case, a diffusion approximation procedure leads to a nonlinear diffusion equation for ρ. Details can be found in [8].…”