Single-file dynamics with different diffusion constants

TL;DR

This paper studies the movement of a tagged particle in a one-dimensional system with particles of different diffusion constants, providing exact solutions for two particles and simulations for many, confirming the square root time scaling of displacement.

Contribution

It offers an exact solution for two particles with different diffusion constants and introduces an efficient simulation method for many particles, validating recent theoretical predictions.

Findings

Tagged particle mean square displacement scales as √time.

Prefactor depends on individual particle diffusion constants.

Simulation results agree with recent mathematical theories.

Abstract

We investigate the single-file dynamics of a tagged particle in a system consisting of N hardcore interacting particles (the particles cannot pass each other) which are diffusing in a one-dimensional system where the particles have different diffusion constants. For the two particle case an exact result for the conditional probability density function (PDF) is obtained for arbitrary initial particle positions and all times. The two-particle PDF is used to obtain the tagged particle PDF. For the general N-particle case (N large) we perform stochastic simulations using our new computationally efficient stochastic simulation technique based on the Gillespie algorithm. We find that the mean square displacement for a tagged particle scales as the square root of time (as for identical particles) for long times, with a prefactor which depends on the diffusion constants for the particles; these results are in excellent agreement with very recent analytic predictions in the mathematics literature.