Abstract
We investigate a one-dimensional, many-body system consisting of particles interacting via repulsive, short-range forces, and moving in an overdamped regime under the effect of a drag force that depends on direction. That is, particles moving to the right do not experience the same drag as those moving to the left. The dynamics of the system, effectively described by a non-linear, Fokker–Planck equation, exhibits peculiar features related to the way in which the drag force depends on velocity. The evolution equation satisfies an H-theorem involving the Sq nonadditive entropy, and admits particular, exact, time-dependent solutions closely related, but not identical, to the q-Gaussian densities. The departure from the canonical, q-Gaussian shape is related to the fact that in one spatial dimension, in contrast to what occurs in two or more spatial dimensions, the drag’s dependence on direction entails that its dependence on velocity is necessarily (and severely) non-linear. The results reported here provide further evidence of the deep connections between overdamped, many-body systems, non-linear Fokker–Planck equations, and the Sq-thermostatistics.
Highlights
The robust character of the stationary densities may be of considerable significance for the Sq -thermostatistical theory, since some of its most important applications are based on the maximum Sq -entropy, stationary densities of systems described by Non-linear Fokker–Planck equations (NLFPEs)
We have investigated one-dimensional systems of confined particles with short-range, repulsive interactions, that perform over-damped motion, under direction-dependent drag forces
Particles moving to the right experience drag forces of different strength than those moving to the left
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Non-linear Fokker–Planck equations (NLFPEs) [1] are nowadays recognized as valuable tools for understanding diverse aspects of the dynamics of complex systems. They proved to be useful for the study of type-II superconductors [2,3], granular media [4], and self-gravitating systems [5,6]. The non-linear, power-law, Fokker–Planck equations admit q-Gaussian solutions, which are densities optimizing the Sq entropies under simple constraints [16,17]. We prove that the NLFPE admits particular time-dependent solutions that have a form akin to q-Gaussians
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