Extended foundations of stochastic prediction

TL;DR

This paper develops a universal quantitative framework for predicting the behavior of ergodic systems undergoing bifurcations, using an extended Fokker-Planck equation and energy spectrum quantization.

Contribution

It introduces a novel extended Fokker-Planck model with time-dependent diffusion and quantized energy spectra for ergodic systems, enabling improved prediction and control.

Findings

Quantized energy spectrum of phase states exists in ergodic systems.

Extended Fokker-Planck equation with time-dependent diffusion describes system dynamics.

Energy-based control parameters can stabilize ergodic systems.

Abstract

The basic purpose of this work was to suggest universal quantitative description of ergodic system intermediate bifurcation and obligatory conditions of this transition. Conditions for existence of phase state and first order phase transition were introduced in terms of energy balance for system volume unit. Extended Fokker - Plank equation with time dependent diffusion factor was formulated. It turned out that for ergodic system with fixed boundary quantized energy spectrum of phase stable states exists. Obtained results may be applied for prediction of ergodic system behavior. If isolation condition is satisfied, phase spectrum quantization allows selecting proper control parameters for system stabilization. Information about current system coarsened energy allows predicting of future stochastic system behavior on the basis of extended Fokker - Plank model.