Methodological notes

A N Kolmogorov's 1934 paper is the basis for explaining the statistics of natural phenomena of the macrocosm

A M Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Pyzhevskii per. 3, Moscow, 109017, Russian Federation

The 1934 paper by A N Kolmogorov [1], "Random Motions," 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. The diffusion coefficient in this case is ε, the rate of energy generation/dissipation. The equation is obtained by specifying the accelerations of the particles of the ensemble by Markov processes, i.e., random processes δ-correlated in time and with each other. The fundamental solution of this equation was already indicated in [1] and was used by A M Obukhov [2] in 1958 to describe a turbulent flow in the inertial interval [3]. It was only recently [4, 5] noticed that the Fokker—Planck-type equation written by Kolmogorov in [1] contains a description of the statistics of other random natural processes, earthquakes, sea waves, and others [5]. This equation, by a change of variables with scales for velocities and for coordinates, is reduced to a self-similar form that does not explicitly contain the diffusion coefficient [6]. Numerical calculations confirm the presence of such scales in systems with the number N of events, in ensembles starting from N=10. For N = 100, these scales almost exactly coincide with the ANK34 theory. This theory, in principle, containing the results of 1941, paved the way for more complex random systems with 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, inundation 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. The article is an abridged version of the author's monograph [5], where for the first time the ideas of ANK34 were used to explain in a probabilistic sense many experimental patterns that have been considered by pure empiricism for decades.

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Fulltext is also available at DOI: 10.3367/UFNe.2023.05.039355
Keywords: random motions, Fokker—Planck—Kolmogorov equation, second moments of probability distribution for coordinates and velocities, theory of similarity and dimensions, statistical laws of nature
PACS: 05.40.−a, 91.30.−f, 92.60.−e (all)
DOI: 10.3367/UFNe.2023.05.039355
Citation: Golitsyn G S "A N Kolmogorov's 1934 paper is the basis for explaining the statistics of natural phenomena of the macrocosm" Phys. Usp. 67 80–90 (2024)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks
	author = {G. S. Golitsyn},
	title = {A N Kolmogorov's 1934 paper is the basis for explaining the statistics of natural phenomena of the macrocosm},
	publisher = {Physics-Uspekhi},
	year = {2024},
	journal = {Phys. Usp.},
	volume = {67},
	number = {1},
	pages = {80-90},
	url = {},
	doi = {10.3367/UFNe.2023.05.039355}

Received: 16th, February 2023, 11th, May 2023

Оригинал: Голицын Г С «Работа А.Н. Колмогорова 1934 г. — основа для объяснения статистики природных явлений макромира» УФН 194 86–96 (2024); DOI: 10.3367/UFNr.2023.05.039355

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