Exact probability distribution for the two-tag displacement in single-file motion

TL;DR

This paper derives the exact probability distribution for the displacement between two tagged particles in a one-dimensional single-file system with arbitrary dynamics, providing insights into their statistical behavior and correlations.

Contribution

It introduces a mapping to a non-interacting system to compute the exact distribution for two-tagged particle displacement in single-file motion with general dynamics.

Findings

Exact distribution function derived for two-tagged particle displacement

Computed large deviation function and cumulants

Analyzed velocity auto-correlation for Hamiltonian dynamics

Abstract

We consider a gas of point particles moving on the one-dimensional line with a hard-core inter-particle interaction that prevents particle crossings --- this is usually referred to as single-file motion. The individual particle dynamics can be arbitrary and they only interact when they meet. Starting from initial conditions such that particles are uniformly distributed, we observe the displacement of a tagged particle at time $t$, with respect to the initial position of another tagged particle, such that their tags differ by $r$. For $r=0$, this is the usual well studied problem of the tagged particle motion. Using a mapping to a non-interacting particle system we compute the exact probability distribution function for the two-tagged particle displacement, for general single particle dynamics. As by-products, we compute the large deviation function, various cumulants and, for the case of Hamiltonian dynamics, the two-particle velocity auto-correlation function.