Brownian motion of Massive Particle in a Space with Curvature and Torsion and Crystals with Defects

TL;DR

This paper develops a generalized theory of Brownian motion for massive particles in spaces with curvature and torsion, extending classical diffusion models to complex geometries like defective crystals.

Contribution

It introduces a nonholonomic mapping approach to derive Langevin, Kubo, and Fokker-Planck equations in Cartan spaces, broadening the understanding of diffusion in curved and torsioned geometries.

Findings

Derived Langevin equations in spaces with curvature and torsion.

Formulated Kubo and Fokker-Planck equations for such spaces.

Applied the theory to diffusion in crystals with defects.

Abstract

We develop a theory of Brownian motion of a massive particle, including the effects of inertia (Kramers' problem), in spaces with curvature and torsion. This is done by invoking the recently discovered generalized equivalence principle, according to which the equations of motion of a point particle in such spaces can be obtained from the Newton equation in euclidean space by means of a nonholonomic mapping. By this principle, the known Langevin equation in euclidean space goes over into the correct Langevin equation in the Cartan space. This, in turn, serves to derive the Kubo and Fokker-Planck equations satisfied by the particle distribution as a function of time in such a space. The theory can be applied to classical diffusion processes in crystals with defects.