General Systems Theory Concepts in Atmospheric Flows - Rain Formation in Warm Clouds: General Systems Theory - page 26

the macroscopic state of the system and corresponds to the “degree of disorder” or
“missing information” (Chakrabarti and De 2000 ). For a probability distribution
among a discrete set of states the generalized entropy for a system out of equilib-
rium is given as (Chakrabarti and De 2000 ; Salingaros and West 1999 ; Beck 2009 ;
Sethna 2009 ).

σ

∑ 1

S =− P ln P .

(1.41)

j j

j =

In Eq. (1.41), P j is the probability for the j th stage of eddy growth in this study, σ is
the length step growth which is equal to the normalized deviation and the entropy S
represents the “missing information” regarding the probabilities. Maximum entropy
S signifies minimum preferred states associated with scale-free probabilities.
The validity of the probability distribution P (Eqs. 1.16 and 1.20) is now checked
by applying the concept of maximum entropy principle (Kaniadakis 2009 ). Substi-
tuting for log P j (Eq. 1.40) and for the probability P j in terms of the golden mean τ
derived earlier (Eqs. 1.16 and 1.20) the entropy S is expressed as follows:

σ σ σ

∑ ∑

2

∑

− 42 σ

S =− P log P = P = ( τ )

j j j

(1.42)

j = 1 j = 1 j = 1

σ

∑

− 8 σ

S = τ ≈ 1 forlarge σ.

j = 1

In Eq. (1.42), S is equal to the square of the cumulative probability density distri-
bution and it increases with increase in σ, i.e., the progressive growth of the eddy
continuum and approaches 1 for large σ. According to the second law of thermody-
namics, increase in entropy signifies approach of dynamic equilibrium conditions
with scale-free characteristic of fractal fluctuations and hence the probability distri-
bution P (Eqs. 1.16 and 1.20) is the correct analytic expression quantifying the eddy
growth processes visualized in the general systems theory.
Paltridge ( 2009 ) states that the principle of maximum entropy production (MEP)
is the subject of considerable academic study, but is yet to become remarkable for
its practical applications. The ability of a system to dissipate energy and to produce
entropy “ought to be” some increasing function of the system's structural com-
plexity. It would be nice if there were some general rule to the effect that, in any
given complex system, the steady state which produces entropy at the maximum
rate would at the same time be the steady state of maximum order and minimum
entropy (Paltridge 2009 ).
Computer simulations by Damasceno, Engel, and Glotzer ( 2012 ) show that the
property entropy, a tendency generally described as ‟disorder” can nudge particles to
form organized structures. By analyzing the shapes of the particles beforehand, they
can even predict what kinds of structures will form (University of Michigan 2012 ).