Overdamped dynamics of particles with repulsive power-law interactions

We investigate the dynamics of overdamped D-dimensional systems of particles repulsively interacting through short-ranged power-law potentials, V (r) similar to r(-lambda) (lambda/D > 1). We show that such systems obey a nonlinear 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 nonlinear 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 (upsilon(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.