Fokker-Planck equation for Boltzmann-type and active particles: transfer probability approach

Area of Science:

• Statistical Physics
• Plasma Physics
• Complex Systems

Background:

• Brownian motion is fundamental to understanding particle dynamics in various media.
• Existing models often simplify friction and diffusion, limiting applicability to complex systems.
• The probability transition (PT) approach offers a universal framework for describing particle interactions.

Purpose of the Study:

• To investigate a Fokker-Planck equation with velocity-dependent coefficients using the PT approach.
• To analyze friction and diffusion in isotropic systems, including dusty plasmas and active particles.
• To examine the validity of Einstein's relation under non-ideal conditions.

Main Methods:

• Developing a Fokker-Planck equation with velocity-dependent coefficients.
• Applying the probability transition (PT) approach for a self-consistent description.
• Calculating specific PT forms for Boltzmann-type and absorption-type collisions.
• Investigating generalized friction forces in dusty plasmas.
• Constructing PT for active particle motion models.

Main Results:

• Demonstrated friction coefficient renormalization in 2D and 3D systems due to diffusion's tensorial nature.
• Identified violation of Einstein's relation for high-velocity Boltzmann-type collisions and all absorption-type collisions.
• Established relationships between generalized friction, collecting friction, and drag forces in dusty plasmas.
• Derived coefficients for Fokker-Planck equations describing active particle motion.
• Found stationary solutions characteristic of self-organized molecular machines.

Conclusions:

• The PT approach provides a universal and self-consistent method for describing friction and diffusion.
• Einstein's relation has limited validity, especially in systems with velocity-dependent coefficients and non-Boltzmann collisions.
• The framework is applicable to diverse systems, including dusty plasmas and active particles, offering insights into complex dynamics.