General Systems Theory Concepts in Atmospheric Flows - Rain Formation in Warm Clouds: General Systems Theory - page 23

The general systems theory visualizes the self-similar fractal fluctuations to re-
sult from a hierarchy of eddies, the larger scale being the space-time average of
enclosed smaller scale eddies (Eq. 1.1) assuming constant values for the character-
istic length scale r and circulation speed w * throughout the large eddy space-time
domain. The collective behavior of the ordered hierarchical eddy ensembles is man-
ifested as the apparently irregular fractal fluctuations with long-range space-time
correlations generic to dynamical systems. The concept that aggregate averaged
eddy ensemble properties represent the eddy continuum belongs to the nineteenth-
century classical statistical physics where the study of the properties of a system is
reduced to a determination of average values of the physical quantities that charac-
terize the state of the system as a whole (Yavorsky and Detlaf 1975 ) such as gases,
e.g., the gaseous envelope of the earth, the atmosphere.
In classical statistical physics kinetic theory of ideal gases is a study of systems
consisting of a great number of molecules, which are considered as bodies having
a small size and mass (Kikoin and Kikoin 1978 ). Classical statistical methods of
investigation (Yavorsky and Detlaf 1975 ; Kikoin and Kikoin 1978 ; Dennery 1972 ;
Rosser 1985 ; Guenault 1988 ; Gupta 1990 ; Dorlas 1999 ; Chandrasekhar 2000 )are
employed to estimate average values of quantities characterizing aggregate mo-
lecular motion such as mean velocity, mean energy, etc., which determine the mac-
roscale characteristics of gases. The mean properties of ideal gases are calculated
with the following assumptions: (1) the intramolecular forces are completely absent
instead of being small; (2) the dimensions of molecules are ignored, and considered
as material points; (3) the above assumptions imply the molecules are completely
free, move rectilinearly and uniformly as if no forces act on them; and (4) the cease-
less chaotic movements of individual molecules obey Newton's laws of motion.
The Austrian physicist Ludwig Boltzmann suggested that knowing the prob-
abilities for the particles to be in any of their various possible configurations would
enable to work out the overall properties of the system. Going one step further,
he also made a bold and insightful guess about these probabilities—that any of
the many conceivable configurations for the particles would be equally probable.
Boltzmann's idea works, and has enabled physicists to make mathematical models
of thousands of real materials, from simple crystals to superconductors. It reflects
the fact that many quantities in nature—such as the velocities of molecules in a
gas—follow “normal” statistics. That is, they are closely grouped around the aver-
age value, with a “bell curve” distribution. Boltzmann's guess about equal prob-
abilities only works for systems that have settled down to equilibrium, enjoying, for
example, the same temperature throughout. The theory fails in any system where
destabilizing external sources of energy are at work, such as the haphazard motion
of turbulent fluids or the fluctuating energies of cosmic rays. These systems do not
follow normal statistics, but another pattern instead (Buchanan 2005 ).
Cohen ( 2005 ) discusses Boltzmann's equation as follows. In 1872 when
Boltzmann derived in his paper: Further studies on thermal equilibrium between
gas molecules (Boltzmann 1872 ) what we now call the Boltzmann equation, he
used, following Clausius and Maxwell, the assumption of “molecular chaos”, and
he does not seem to have realized the statistical, i.e., probabilistic nature of this as-