The specificity of the particle dynamics if random perturbations are orthogonal to its velocity

TL;DR

This paper investigates the behavior of particles under Langevin dynamics with orthogonal stochastic perturbations, revealing conditions where their position distribution approximates a wave equation solution.

Contribution

It introduces a novel analysis of Langevin equations with orthogonal noise, deriving probability density equations and linking particle distribution to wave equations under specific conditions.

Findings

Existence of attraction surfaces in the stochastic system

Derivation of probability density equations depending on initial velocity

Approximation of particle position distribution by wave equation solutions

Abstract

We explore properties the solution of Langevin equation when stochastic influence is orthogonal to velocity of a particle. Wiener's process can accept unlimited values. But for these equations, the attraction surfaces exist. For these stochastic equations we have constructed the equations for density of probability in co-ordinates space that are depending an on initial vector of the particle velocity. Then, using our earlier work and physical sense of coefficients, we build the equation for diffusion approximating of the original equation, and find its solution. It is shown that when at the certain concordance between of coefficients in initial stochastic equation, and influences aspire to the zero; the position distribution of the particle can be approximated by the solution of the wave equation with constant speed.