Scaling and finite ensembles of particles in motion with the energy influx

On the cumulative distribution of the lithospheric plates by their areas

“…However it is always necessary to know how fast solutions for finite ensemble approach to that ones for the infinite ensemble. For this Gledzer and Golitsyn [2010] integrate equationṡ = ,˙= ,…”

“…This equation was solved by Gledzer and Golitsyn [2010] at the initial condition ( , , = 0) = ( ) ( ) i.e. with delta-like conditions.…”

“…The intensity of the convection can be described by the chief single parameter: the rate of generation of its kinetic energy per unit mass . Its mean rate is of order 10 −11 2 −3 [Golitsyn, 2007] as a drive for geodynamics and seismicity. The value of was chosen later as a governing parameter of the plate formation and and area give for the cumulative number of plates which has the dimension of frequency or inverse time…”

Kaula's rule -- a consequence of probability laws by\\ A. N. Kolmogorov and his school

“…This is an approximation of the processes where correlation time of the random forces is much shorter than the reaction time of the considered system. In 1D approximation the action of respective random accelerations with ⟨ ( 1 ) ( 2 )⟩ = 2 ( 1 − 2 ) can be described by the simplest equations [Gledzer and Golitsyn, 2010],…”

“…This solution has two scales (angular brackets mean ensemble averaging): Figure 1. Temporal behaviour of the second velocity and coordinate moments for different particle ensemble sizes as in [Gledzer and Golitsyn, 2010] (see text).…”