Tagged particle diffusion in one-dimensional systems with Hamiltonian dynamics - II

TL;DR

This paper investigates the temporal correlation functions of a tagged particle in various one-dimensional Hamiltonian systems, revealing normal diffusion behavior and system-specific finite size effects through numerical simulations.

Contribution

It provides a comparative analysis of different one-dimensional systems, demonstrating how their correlation functions and diffusion properties behave over time and with system size.

Findings

Tagged particles exhibit normal diffusion asymptotically.

Diffusion constants converge in the hard particle gas, consistent with mode coupling theories.

Behavior varies with density in Lennard-Jones gases, resembling different models at low and high densities.

Abstract

We study various temporal correlation functions of a tagged particle in one-dimensional systems of interacting point particles evolving with Hamiltonian dynamics. Initial conditions of the particles are chosen from the canonical thermal distribution. The correlation functions are studied in finite systems, and their forms examined at short and long times. Various one-dimensional systems are studied. Results of numerical simulations for the Fermi-Pasta-Ulam chain are qualitatively similar to results for the harmonic chain, and agree unexpectedly well with a simple description in terms of linearized equations for damped fluctuating sound waves. Simulation results for the alternate mass hard particle gas reveal that - in contradiction to our earlier results [1] with smaller system sizes - the diffusion constant slowly converges to a constant value, in a manner consistent with mode coupling theories. Our simulations also show that the behaviour of the Lennard-Jones gas depends on its density. At low densities, it behaves like a hard-particle gas, and at high densities like an anharmonic chain. In all the systems studied, the tagged particle was found to show normal diffusion asymptotically, with convergence times depending on the system under study. Finite size effects show up at time scales larger than sound traversal times, their nature being system-specific.