Constructing a solution to the characteristic equation for the Langevin diffusion model with orthogonal perturbations

TL;DR

This paper derives an equation for the characteristic function of a Brownian particle's position in a Langevin model with orthogonal perturbations, revealing wave-like properties and different diffusion behaviors under various regimes.

Contribution

It introduces a novel approach to Langevin equations with orthogonal influences, linking particle dynamics to wave properties and extending understanding of diffusion regimes.

Findings

Wave properties emerge in the particle ensemble model.

Different diffusion regimes depend on the influence strength.

Probability density transitions from wave-based to classical diffusion.

Abstract

For the Langevin model of the dynamics of a Brownian particle with perturbations orthogonal to its current velocity, in a regime when the particle velocity modulus becomes constant, an equation for the characteristic function $\psi (t,\lambda )=M\left[\exp (\lambda ,x(t))/V={\rm v}(0)\right]$ of the position $x(t)$ of the Brownian particle. The obtained results confirm the conclusion that the model of the dynamics of a Brownian particle, which constructed on the basis of an unconventional physical interpretation of the Langevin equations, i. e. stochastic equations with orthogonal influences, leads to the interpretation of an ensemble of Brownian particles as a system with wave properties. These results are consistent with the previously obtained conclusions that, with a certain agreement of the coefficients in the original stochastic equation, for small random influences and friction, the Langevin equations lead to a description of the probability density of the position of a particle based on wave equations. For large random influences and friction, the probability density is a solution to the diffusion equation, with a diffusion coefficient that is lower than in the classical diffusion model.