Modeling Probability Densities for Confined Interacting Particles with Drag Effects

Fokker-Planck equations featuring power-law nonlinearities in diffusion are essential for modeling complex systems in physics and other disciplines. These equations effectively represent overdamped particle systems that interact through short-range interactions and are confined by external potentials. Recent research has shown that the nonlinear Fokker-Planck equation can be integrated into a Vlasov-like mean-field framework, facilitating the inclusion of inertial effects in the dynamics. We have discovered exact time-dependent solutions in the form of $q$-Gaussians for one-dimensional systems with quadratic confining potentials. This study extends these solutions to multi-dimensional systems, identifying exact time-dependent $q$-Gaussian solutions for two and three spatial dimensions while examining their key properties. Our investigation emphasizes multi-dimensional $q$-Gaussian solutions within a Vlasov-like mean-field equation, which arises when inertial effects are incorporated into the dynamics of many-body systems described by power-law nonlinear Fokker-Planck equations. This framework describes particle systems interacting via short-range forces in an overdamped regime under an external confining potential. When inertial effects are accounted for, the nonlinear Fokker-Planck equation governing the $N$-dimensional spatial density of particles transitions to a Vlasov-like mean-field equation governing the $2N$-dimensional space-velocity distribution . Recent advancements indicate that $S_q$-thermostatistics can significantly enhance our understanding of the statistical dynamics of confined interacting particles subjected to drag forces in the overdamped limit. We anticipate that our findings will pave the way for further applications of $S_q$-thermostatistical methods to systems of confined interacting particles under drag forces. The potential future applications of our research, such as systems interacting through Yukawa potentials, which are relevant for modeling dusty and complex plasmas, as well as extending our analysis to systems experiencing nonlinear drag forces, are promising and could significantly advance the field. Additionally, we demonstrate that such solutions do not exist for certain spatial dimensions, $N>3$