On the Mixing of Diffusing Particles

TL;DR

This paper analyzes the statistical properties and first-passage kinetics of the inversion number in a system of N independent one-dimensional random walks, revealing universal scaling behavior and exact theoretical predictions validated by simulations.

Contribution

It introduces a novel analysis of the inversion number distribution and first-passage exponents in diffusing particles, connecting them with cone approximation and universal scaling functions.

Findings

Inversion number distribution is Gaussian with specific mean and variance.

Survival probability decays algebraically with a spectrum of exponents.

First-passage exponents are universal functions of a single scaled variable.

Abstract

We study how the order of N independent random walks in one dimension evolves with time. Our focus is statistical properties of the inversion number m, defined as the number of pairs that are out of sort with respect to the initial configuration. In the steady-state, the distribution of the inversion number is Gaussian with the average <m>~N^2/4 and the standard deviation sigma N^{3/2}/6. The survival probability, S_m(t), which measures the likelihood that the inversion number remains below m until time t, decays algebraically in the long-time limit, S_m t^{-beta_m}. Interestingly, there is a spectrum of N(N-1)/2 distinct exponents beta_m(N). We also find that the kinetics of first-passage in a circular cone provides a good approximation for these exponents. When N is large, the first-passage exponents are a universal function of a single scaling variable, beta_m(N)--> beta(z) with z=(m-<m>)/sigma. In the cone approximation, the scaling function is a root of a transcendental equation involving the parabolic cylinder equation, D_{2 beta}(-z)=0, and surprisingly, numerical simulations show this prediction to be exact.