Universal Extremal Statistics in a Freely Expanding Jepsen Gas

TL;DR

This paper analyzes the extremal velocity dynamics of a one-dimensional Jepsen gas, revealing universal scaling laws for the leader's velocity distribution and collision statistics, with implications for biological evolution models.

Contribution

It provides an analytical characterization of the extremal dynamics in a Jepsen gas, discovering universal scaling behaviors based on the tail of the initial velocity distribution.

Findings

Velocity distribution exhibits universal scaling depending on the tail of (V)

Mean and variance of leader collisions grow logarithmically with time

Scaling functions differ from classical extreme-value distributions

Abstract

We study the extremal dynamics emerging in an out-of-equilibrium one-dimensional Jepsen gas of $(N+1)$ hard-point particles. The particles undergo binary elastic collisions, but move ballistically in-between collisions. The gas is initally uniformly distributed in a box $[-L,0]$ with the "leader" (or the rightmost particle) at X=0, and a random positive velocity, independently drawn from a distribution $\phi(V)$, is assigned to each particle. The gas expands freely at subsequent times. We compute analytically the distribution of the leader's velocity at time $t$, and also the mean and the variance of the number of collisions that are undergone by the leader up to time $t$. We show that in the thermodynamic limit and at fixed time $t\gg 1$ (the so-called "growing regime"), when interactions are strongly manifest, the velocity distribution exhibits universal scaling behavior of only three possible varieties, depending on the tail of $\phi(V)$. The associated scaling functions are novel and different from the usual extreme-value distributions of uncorrelated random variables. In this growing regime the mean and the variance of the number of collisions of the leader up to time $t$ increase logarithmically with $t$, with universal prefactors that are computed exactly. The implications of our results in the context of biological evolution modeling are pointed out.