Theoretical and Numerical Study of Self-Organizing Processes In a Closed System Classical Oscillator + Random Environment

TL;DR

This paper develops a theoretical framework combining stochastic differential equations and PDEs to analyze self-organizing classical oscillators in random environments, providing new insights into their equilibrium behavior and geometric properties.

Contribution

It introduces a novel mathematical approach linking Langevin-type equations with PDEs for self-organizing systems in random environments, including an algorithm for parallel modeling.

Findings

Derivation of second-order PDEs describing environmental field distributions.

Reduction of complex PDEs to two independent second-order PDEs.

Development of a parallel computational algorithm for modeling.

Abstract

A self-organizing joint system classical oscillator + random environment is considered within the framework of a complex probabilistic process that satisfies a Langevin-type stochastic differential equation. Various types of randomness generated by the environment are considered. In the limit of statistical equilibrium (SEq), second-order partial differential equations (PDE) are derived that describe the distribution of classical environmental fields. The mathematical expectation of the oscillator trajectory is constructed in the form of a functional-integral representation, which, in the SEq limit, is compactified into a two-dimensional integral representation with an integrand - the solution of the second-order complex PDE. It is proved that the complex PDE in the general case is reduced to two independent PDEs of the second-order with spatially deviating arguments. The geometric and topological features of the two-dimensional subspace on which these equations arise are studied in detail. An algorithm for parallel modeling of the problem has been developed.