Overdamped dynamics of particles with repulsive power-law interactions

TL;DR

This paper studies overdamped particles with power-law repulsive interactions, showing their dynamics follow a non-linear diffusion equation and their distributions are well-described by $q$-Gaussians, confirming a generalized thermodynamics approach.

Contribution

It demonstrates that overdamped power-law interacting particles obey a non-linear diffusion equation and their distributions follow $q$-Gaussians, linking microscopic dynamics to $q$-generalized entropy.

Findings

Numerical simulations confirm the non-linear diffusion model predictions.

Particle distributions follow $q$-Gaussian forms with $q=1-rac{ ext{interaction exponent}}{ ext{dimension}}$.

Velocity distributions also follow the same $q$-Gaussian form, especially in 1D.

Abstract

We investigate the dynamics of overdamped $D$-dimensional systems of particles repulsively interacting through short-ranged power-law potentials, $V(r)\sim r^{-\lambda}\;(\lambda/D>1)$. We show that such systems obey a non-linear diffusion equation, and that their stationary state extremizes a $q$-generalized nonadditive entropy. Here we focus on the dynamical evolution of these systems. Our first-principle $D=1,2$ many-body numerical simulations (based on Newton's law) confirm the predictions obtained from the time-dependent solution of the non-linear diffusion equation, and show that the one-particle space-distribution $P(x,t)$ appears to follow a compact-support $q$-Gaussian form, with $q=1-\lambda/D$. We also calculate the velocity distributions $P(v_x,t)$ and, interestingly enough, they follow the same $q$-Gaussian form (apparently precisely for $D=1$, and nearly so for $D=2$). The satisfactory match between the continuum description and the molecular dynamics simulations in a more general, time-dependent, framework neatly confirms the idea that the present dissipative systems indeed represent suitable applications of the $q$-generalized thermostatistical theory.