Entropy decrease and emergence of order in collective dynamics

TL;DR

This paper investigates the hydrodynamic behavior of collective dynamics driven by velocity alignment, demonstrating entropy decrease and global regularity for a broad class of initial conditions across multiple dimensions.

Contribution

It establishes the global regularity of Euler alignment systems with entropy considerations and proves existence beyond two dimensions under a critical threshold.

Findings

Entropy decreases over time leading to mono-kinetic states

Global regularity persists for large classes of initial conditions

Results apply in any number of spatial dimensions

Abstract

We study the hydrodynamic description of collective dynamics driven by velocity {\it alignment}. It is known that such Euler alignment systems must flock towards a limiting ``flocking'' velocity, provided their solutions remain globally smooth. To address this question of global existence we proceed in two steps. (i) Entropy and closure. The system lacks a closure, reflecting lack of detailed energy balance in collective dynamics. We discuss the decrease of entropy and the asymptotic behavior towards a mono-kinetic closure; and (ii) Mono-kinetic closure. We prove that global regularity persists for all time for a large class of initial conditions satisfying a critical threshold condition, which is intimately linked to the decrease of entropy. The result applies in any number of spatial dimensions, thus addressing the open question of existence beyond two dimensions.