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Russian Meteorology and Hydrology

, Volume 43, Issue 3, pp 135–142 | Cite as

Laws of Random Walks Derived by A.N. Kolmogorov in 1934

  • G. S. Golitsyn
Article

Abstract

In the first half of the 1930s A.N. Kolmogorov was developing analytical methods for the probability theory and presented the solution of the Fokker–Planck type equation. This solution contains scales for the distribution function moments of the mean squares for velocities and relative displacements of the analyzed objects and for the mixed moments of velocities and coordinates. The exclusion of time from these moments leads to the 2/3 law for the velocity structure function and to the Richardson–Obukhov law for the eddy diffusion. The analysis of the fetch laws for wind waves demonstrates that the Kolmogorov laws are manifested in the growth of wave amplitudes and in the form of elevation spectra. These laws also work in the statistics of the planetary surface relief, in the size distribution of the lithospheric plates, in the energy spectra of cosmic rays, and in other processes. In the equation deduced in 1934, probability distribution functions are derived only under the condition of homogeneity of these functions and thereby allow describing a broad range of phenomena and processes.

Keywords

Random walks in the coordinate and velocity space turbulence laws wind wave fetch laws elevation spectra relief statistics 

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© Allerton Press, Inc. 2018

Authors and Affiliations

  1. 1.Obukhov Institute of Atmospheric PhysicsRussian Academy of SciencesMoscowRussia

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