General Systems Theory Concepts in Atmospheric Flows - Rain Formation in Warm Clouds: General Systems Theory - page 25

In the above equation, the variable k represents for each step of eddy growth, the
fractional volume dilution of large eddy by turbulent eddy fluctuations carried on
the large eddy envelope (Selvam 1990 ) and is given as (Eq. 1.17)

wr
WR

*

k = .

Substituting for k in Eq. (1.4), we have

Ww WR
wr

WR
r

= log z = log z

*

*

(1.39)

and

r
R

= log. z

The ratio r/R represents the fractional probability P of occurrence of small-scale
fluctuations ( r ) in the large eddy ( R ) environment. Since the scale ratio z is equal to
R / r , Eq. (1.39) may be written in terms of the probability P as follows:

r
R

R
r

1

=

= loglog z = log

( )

r R /

(1.40)

1

=−

P = log log . P

P

1.7.1 General Systems Theory and Maximum Entropy Principle

The maximum entropy principle concept of classical statistical physics is applied to
determine the fidelity of the inverse power law probability distribution P (Eqs. 1.16
and 1.20) for exact quantification of the observed space-time fractal fluctuations
of dynamical systems ranging from the microscopic dynamics of quantum systems
to macroscale real-world systems. Kaniadakis ( 2009 ) states that the correctness of
an analytic expression for a given power-law tailed distribution used to describe a
statistical system is strongly related to the validity of the generating mechanism. In
this sense the maximum entropy principle, the cornerstone of statistical physics, is a
valid and powerful tool to explore new roots in searching for generalized statistical
theories (Kaniadakis 2009 ). The concept of entropy is fundamental in the founda-
tion of statistical physics. It first appeared in thermodynamics through the second
law of thermodynamics. In SM, we are interested in the disorder in the distribution
of the system over the permissible microstates. The measure of disorder first pro-
vided by Boltzmann principle (known as Boltzmann entropy) is given by S = K B ln
M, where K B is the thermodynamic unit of measurement of entropy and is known as
Boltzmann constant. K B = 1.33×10 −16 erg/°C, called thermodynamic probability or