Brownian Particles with Rank-Dependent Drifts: Out-of-Equilibrium Behavior

TL;DR

This paper analyzes the long-term behavior of an infinite system of Brownian particles with rank-dependent drifts, deriving explicit density profiles and particle trajectories, and establishing convergence properties.

Contribution

It introduces a novel analysis of out-of-equilibrium rank-dependent Brownian systems using Stefan equations, providing explicit solutions for density and trajectory behaviors.

Findings

Explicit limiting particle-density profile derived

Asymptotic trajectory of the leftmost particle determined

Convergence to equilibrium established for relative spacings

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.