Overdamped dynamics of particles with repulsive power-law interactions

Abstract

We investigate the dynamics of overdamped D-dimensional systems of particles repulsively interacting through short-ranged power-law potentials, V (r) ∼ r −λ (λ/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 − λ/D. We also calculate the velocity distributions P (vx, 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.

Nonlinear Fokker–Planck Equation Approach to Systems of Interacting Particles: Thermostatistical Features Related to the Range of the Interactions

“…Now, we briefly review the application of the power-law nonlinear Fokker-Planck equation to the thermostatistics of systems of confined particles interacting through short-range forces and moving in the overdamped regime [20][21][22][23][24][25]. We pay special attention to the kind of inter-particle forces for which this type of treatment can be implemented.…”

“…In those situations, the interaction between the particles can be modeled by a nonlinear power-law diffusion term, as in (9), but with exponents different than 2. These scenarios were discussed in [22] for particles interacting through a force obeying…”

“…Another of the most fruitful applications of the S q -thermostatistics to physics (and other scientific disciplines) is based on its connection with nonlinear diffusion and Fokker-Planck equations, which was first pointed out in [13]. Nonlinear Fokker-Planck equations govern the behavior of a wide family of systems in physics, biology, and other branches of science [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. Nonlinear diffusion processes [29,30] are important in various fields, such as in the study of the spread of biological populations [31,32].…”

“…They have also been used to model the spread of energy in nonlinear lattices [33]. The nonlinear diffusion and Fokker-Planck equations are also closely related to nonlinear versions of the Schroedinger, Dirac, and Klein-Gordon equations [34] admitting complex, q-plane wave, soliton-like, analytical solutions that comply with the Einstein-Planck-de Broglie relations [34], and to other nonlinear wave equations related to the S q -thermostatistics, having either real q-Gaussian solutions or exponential plane wave solutions modulated by q-Gaussians [35].An important class of systems described by nonlinear Fokker-Planck equations is given by confined many-body systems with short-range interactions, in the regime of overdamped motion [20][21][22][23][24][25]. This class of systems includes, as a prominent and interesting example, systems of interacting vortices in type-II superconductors [20,21].…”

“…An important class of systems described by nonlinear Fokker-Planck equations is given by confined many-body systems with short-range interactions, in the regime of overdamped motion [20][21][22][23][24][25]. This class of systems includes, as a prominent and interesting example, systems of interacting vortices in type-II superconductors [20,21].…”