Random Walk Laws by A.N. Kolmogorov as the Basics for Understanding Most Phenomena of the Nature

Abstract

In 1934, A.N. Kolmogorov considered the random walks of the 6D vector X, U of velocities generated by the Markov process and corresponding coordinates. The Fokker–Planck type equation using the scales of mean squares of velocity and coordinates as solutions was proposed to describe the time evolution of probability density of random process p(X, U, t). In 1959, A.M. Obukhov excluded time from these scales and obtained formulas in the form of asymptotes for the small-scale turbulence formulated in 1941. However, these results allow deeper insight into a wider range of phenomena, such as the size distribution of lithospheric plates (Bird [6]); the fetch laws describing wind wave growth (Toba [15]); and other phenomena (see specifically (Golitsyn [25]) considering the scales manifested in galaxies). Random accelerations and their integration set the velocities, which, being integrated, set the coordinates of particle ensembles. All these factors promote the energy input into the system increasing with time, whose growth rate ε is the doubled diffusion coefficient in the space of velocities, according to the Kolmogorov equation. It was found that the mean square velocity obtained upon solving the Fokker–Planck equation grows with time—theoretical physicists have long been aware of this result.