Long Time Tail of the Velocity Autocorrelation Function in a Two-Dimensional Moderately Dense Hard Disk Fluid

TL;DR

This study uses large-scale simulations to analyze the decay of the velocity autocorrelation function in a two-dimensional hard disk fluid, finding a decay slightly faster than the classical power law and comparing it with theoretical predictions.

Contribution

The paper provides the first large-scale simulation data showing the decay of the tail is faster than $1/t$, and compares it with mode-coupling theory predictions.

Findings

Decay slightly faster than $1/t$ in dense fluids

Simulation data supports the $1/(t\\sqrt{\ln t})$ decay prediction

Large-scale simulation with one million particles conducted

Abstract

Alder and Wainwright discovered the slow power decay $\sim t^{-d/2}$ ($d$:dimension) of the velocity autocorrelation function in moderately dense hard sphere fluids using the event-driven molecular dynamics simulations. In the two-dimensional case, the diffusion coefficient derived using the time correlation expression in linear response theory shows logarithmic divergence, which is called the ``2D long-time-tail problem''. We revisited this problem to perform a large-scale, long-time simulation with one million hard disks using a modern efficient algorithm and found that the decay of the long tail in moderately dense fluids is slightly faster than the power decay ($\sim 1/t$). We also compared our numerical data with the prediction of the self-consistent mode-coupling theory in the long time limit ($\sim 1/(t\sqrt{\ln{t}})$).