General Systems Theory Concepts in Atmospheric Flows - Rain Formation in Warm Clouds: General Systems Theory - page 28

Representing the larger scale eddy energy as E

2 ∝ .

WE
2

(1.46)

The length scale ratio R 1 / R 2 therefore represents fractional probability P (Eqs. 1.16
and 1.20) of occurrence of large eddy energy E in the environment of the primary
small-scale eddy energy K B T (Eq. 1.45). The expression for P is obtained from
Eq. (1.44) as follows:

E

− K B .

(1.47) T

Pe ∝

The above is the same as Boltzmann's equation (Eq. 1.40).
The derivation of Boltzmann's equation from general systems theory concepts
visualizes the eddy energy distribution as follows: (1) the primary small-scale eddy
represents the molecules whose eddy kinetic energy is equal to K B T as in classi-
cal physics. (2) The energy pumping from the primary small-scale eddy generates
growth of progressive larger eddies (Selvam 1990 ). The r.m.s circulation speeds W
of larger eddies are smaller than that of the primary small-scale eddy (Eq. 1.1). (3)
The space-time fractal fluctuations of molecules (atoms) in an ideal gas may be
visualized to result from an eddy continuum with the eddy energy E per unit volume
relative to primary molecular kinetic energy (K B T ) decreasing progressively with
increase in eddy size.
The eddy energy probability distribution ( P ) of fractal space-time fluctua-
tions also represents the Boltzmann distribution for each stage of hierarchical eddy
growth and is given by Eqs. (1.16) and (1.20) derived earlier, namely

τ 4

P = .

The general systems theory concepts are applicable to all space-time scales ranging
from microscopic scale quantum systems to macroscale real-world systems such as
atmospheric flows.

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