Random moves equation Kolmogorov-1934. A unified approach for description of statistical phenomena of nature

TL;DR

This paper revisits Kolmogorov's 1934 'Random Moves' equation, demonstrating its applicability to various natural stochastic phenomena and proposing a unified theoretical framework for their statistical description.

Contribution

It extends Kolmogorov's original equation to describe diverse natural processes, linking theory with empirical data across multiple phenomena.

Findings

Equation describes turbulence, earthquakes, sea waves, cosmic rays, and floods.

Empirical data support the fundamental laws of probability theory.

Unified approach reveals parameter-dependent changes in process characteristics.

Abstract

The paper by A.N. Kolmogorov 1934 "Random Moves", hereinafter ANK34, uses a Fokker-Planck-type equation for a 6-dimensional vector with a total rather than a partial derivative with respect to time, and with a Laplacian in the space of velocities. This equation is obtained by specifying the accelerations of the particles of the ensemble by Markov processes. The fundamental solution was used by A M Obukhov in 1958 to describe a turbulent flow in the inertial interval. Already recently it was noticed that the Fokker-Planck-type equation written by Kolmogorov contains a description of the statistics of other random natural processes, earthquakes, sea waves, and others. This theory, containing the results of 1941, paved the way for more complex random systems containing enough parameters to form an external similarity parameter. This leads to a change in the characteristics of a random process, for example, to a change in the slope of the time spectrum, as in the case of earthquakes and in a number of other processes (sea waves, cosmic ray energy spectrum, flood zones during floods, etc.). A review of specific random processes studied experimentally provides a methodology for how to proceed when comparing experimental data with the ANK34 theory. Thus, empirical data illustrate the validity of the fundamental laws of probability theory.