Universality of order statistics for Brownian reshuffling

TL;DR

This paper demonstrates that the order statistics of particles undergoing Brownian motion in certain potentials are universal in the stationary state, with leader reshuffling timescales depending on the potential's exponent, and provides explicit reshuffling probabilities.

Contribution

It establishes the universality of order statistics for Brownian particles in asymmetric potentials and derives explicit reshuffling probability generating functions.

Findings

Order statistics are universal and independent of potential exponent in the stationary state.

Leader reshuffling timescale scales as a power of the logarithm of the population size.

The overlap coefficient of leader lists follows a universal erfc function over scaled time.

Abstract

We discuss the order statistics of the particle positions of a gas of $N$ identical independent particles performing Brownian motion in one dimension in a potential that asymptotically behaves like $V(x) \sim x^\gamma$ for $x\rightarrow+\infty$, with a positive power $\gamma>0$. We show that in the stationary state, the order statistics that describe how the leaders are reshuffled are universal and independent of $\gamma$. What depends on $\gamma$ is the timescale of the leaders' reshuffling, which scales as a power of the logarithm of the population size: $t \sim (\ln N)^\frac{2(1-\gamma)}{\gamma} \tau$, where $\tau$ is of order one. We derive the probability that the particle which has the $k$th largest value of $x$ at some time $t_1$ will have the $j$th largest value at time $t_2=t_1+t$ in the form of an explicit expression for the generating function for the reshuffling probabilities for all $k\ge 1$ and $j\ge 1$. The generating function, expressed in scaled time $\tau$, is independent of $\gamma$. In particular, we show that the average percentage overlap coefficient of leader lists takes the universal, $\gamma$-independent form ${\rm erfc}(\sqrt{\tau})$ for long lists.