Brownian Particles with Rank‐Dependent Drifts: Out‐of‐Equilibrium Behavior. Issue 7 (11th April 2019)

• Record Type: Journal Article
• Authors: Cabezas, M.; Dembo, A.; Sarantsev, A.; Sidoravicius, V.
• Is Part Of: Communications on pure and applied mathematics. Volume 72:Issue 7(2019:Jul.)
• Page Start: 1424
• Page End: 1458
• Publication Date: 2019-04-11
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.21825
• Languages: English
• ISSNs: 0010-3640

Abstract

Abstract: We study the long‐range asymptotic behavior for an out‐of‐equilibrium, countable, one‐dimensional system of Brownian particles interacting through their rank‐dependent drifts. Focusing on the semi‐infinite case, where only the leftmost particle gets a constant drift to the right, we derive and solve the corresponding one‐sided Stefan (free‐boundary) equations. Via this solution we explicitly determine the limiting particle‐density profile as well as the asymptotic trajectory of the leftmost particle. While doing so we further establish stochastic domination and convergence to equilibrium results for the vector of relative spacings among the leading particles. © 2019 Wiley Periodicals, Inc.