N-dimensional nonlinear Fokker-Planck equation with time-dependent coefficients

Area of Science:

• Statistical physics
• Nonlinear dynamics
• Partial differential equations

Background:

• Fokker-Planck equations model complex systems, but time-dependent coefficients present analytical challenges.
• Generalized Gaussian functions and Tsallis statistics offer frameworks for non-equilibrium systems.

Purpose of the Study:

• To derive an exact solution for an N-dimensional nonlinear Fokker-Planck equation with time-dependent coefficients.
• To explore the relationship between coefficient time-dependence and the resulting diffusive behavior.
• To connect the solution to Tsallis statistics for broader applicability.

Main Methods:

• Investigated an N-dimensional nonlinear Fokker-Planck equation with time-varying coefficients.
• Utilized generalized Gaussian functions to construct an exact analytical solution.
• Analyzed drift-controlled and source terms within the equation.

Main Results:

• An exact solution was obtained, expressed via generalized Gaussian functions linked to Tsallis statistics.
• Demonstrated that varying the time dependence of coefficients generates a spectrum of diffusive processes.
• Confirmed the capability to produce both normal and anomalous diffusion by manipulating coefficient dynamics.

Conclusions:

• The study provides a novel analytical solution for a complex class of Fokker-Planck equations.
• Highlights the crucial role of time-dependent coefficients in shaping diffusion characteristics.
• Offers a framework for understanding and modeling diverse diffusive phenomena in physical systems.