Tagged particle diffusion in one-dimensional gas with Hamiltonian dynamics

TL;DR

This paper investigates the dynamics of a one-dimensional gas of particles, analyzing tagged particle correlations and diffusion behavior under Hamiltonian evolution, revealing diffusive and sub-diffusive regimes depending on particle masses.

Contribution

It provides analytic results for equal-mass particles and numerical evidence for sub-diffusive behavior in unequal-mass cases, advancing understanding of particle diffusion in 1D Hamiltonian systems.

Findings

Analytic velocity auto-correlation for equal masses

Sub-diffusive mean square displacement for unequal masses

Damped oscillations in correlation functions at long times

Abstract

We consider a one-dimensional gas of hard point particles in a finite box that are in thermal equilibrium and evolving under Hamiltonian dynamics. Tagged particle correlation functions of the middle particle are studied. For the special case where all particles have the same mass, we obtain analytic results for the velocity auto-correlation function in the short time diffusive regime and the long time approach to the saturation value when finite-size effects become relevant. In the case where the masses are unequal, numerical simulations indicate sub-diffusive behaviour with mean square displacement of the tagged particle growing as t/ln(t) with time t. Also various correlation functions, involving the velocity and position of the tagged particle, show damped oscillations at long times that are absent for the equal mass case.