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

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.

Keywords

• general theory of random and stochastic dynamical systems
• partial differential equations
• measure and integration
• noncommutative differential geometry
• parallel computing