Peculiarities of the dynamics of a brownian particle with random disturbances orthogonal to its speed

  • Dubko Valerii A., doobko2017@ukr.net Kyiv National University of Technologies and Design, Education and Scientific Institute of Modern Learning Technologies, Department of Higher Mathematics, 2, building 4, Nemyrovych-Danchenko Street, Kyiv 01011, Ukraine
Keywords: Langevin equation, orthogonal perturbations, diffusion approximation, wave equation, first integral

Abstract

The classical diffusion equations are based on the assumption that the velocities of a Brownian particle can take arbitrarily large values. In this article, it is shown that for solving the Langevin equations when random influences are orthogonal to the particle velocity there might exist an attractive surface for velocity, despite the fact that the Wiener process is a process that takes arbitrarily large values. Unlike the previous articles, here we construct an equation for determining the probability density of the distribution of particles in the coordinate space taking the initial direction of velocity into account. It is shown that small influences with a certain agreement of the coefficients in the initial stochastic equation lead to a description of the moving particle based on the wave equations with constant speed. The considered equations do not exhaust the class of models when the perturbations are orthogonal to the vector component of the solution. An extended class of stochastic equations with orthogonal perturbations was considered in previous works of the author, in particular, for n-dimensional processes, in connection with the development of the theory of first integrals for stochastic systems.

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How to Cite
Dubko, V. (2019) “Peculiarities of the dynamics of a brownian particle with random disturbances orthogonal to its speed”, Mathematical notes of NEFU, 26(3), pp. 31-46. doi: https://doi.org/10.25587/SVFU.2019.31.78.003.
Section
Mathematics