On Stochastic Deformations of Dynamical Systems

DOI

10.1142/S1402925110000805How to use a DOI?

Keywords

Stochastic equations; Gibbs distribution

Abstract

I discuss the connection of the three different questions: The existence of the Gibbs steady state distributions for the stochastic differential equations, the notion and the existence of the conservation laws for such equations, and the convergence of the smooth random perturbations of dynamical systems to stochastic differential equations in the Ito sense. I show that in all cases one needs to include some additional term in the standard form of stochastic equation. I call such approach to describing the influence of the noise on the dynamical systems the “stochastic deformation” to distinguish it from the conventional “stochastic perturbation”. I also discuss some consequences of this approach, in particular, a connection between the noise intensity and the temperature. This connection is known in physics (for the case of linear system of differential equations) as “fluctuation-dissipation theorem” (L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 9, Statistical Physics. Part 2). In conclusion, I present an interesting physical example of the dynamics of magnetic dipole in a random magnetic field.

Copyright

© 2010 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Ilya Shereshevskii
PY  - 2021
DA  - 2021/01/07
TI  - On Stochastic Deformations of Dynamical Systems
JO  - Journal of Nonlinear Mathematical Physics
SP  - 71
EP  - 85
VL  - 17
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925110000805
DO  - 10.1142/S1402925110000805
ID  - Shereshevskii2021
ER  -