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sumption, i.e., of the assumption of the independence of the velocities of two mole-
cules which are going to collide. He used both a dynamical and a statistical method.
However, Einstein strongly disagreed with Boltzmann's statistical method, arguing
that a statistical description of a system should be based on the dynamics of the sys-
tem. This opened the way, especially for complex systems, for other than Boltzmann
statistics. It seems that perhaps a combination of dynamics and statistics is neces-
sary to describe systems with complicated dynamics (Cohen 2005 ). Sornette ( 2009 )
discusses the ubiquity of observed power law distributions in complex systems as
follows. The extension of Boltzmann's distribution to out-of-equilibrium systems
is the subject of intense scrutiny. In the quest to characterize complex systems, two
distributions have played a leading role: the normal (or Gaussian) distribution and
the power law distribution. Power laws obey the symmetry of scale invariance.
Power law distributions and more generally regularly varying distributions remain
robust functional forms under a large number of operations, such as linear combina-
tions, products, minima, maxima, order statistics, powers, which may also explain
their ubiquity and attractiveness. Research on the origins of power law relations,
and efforts to observe and validate them in the real world, is extremely active in
many fields of modern science, including physics, geophysics, biology, medical
sciences, computer science, linguistics, sociology, economics and more. Power law
distributions incarnate the notion that extreme events are not exceptional. Instead,
extreme events should be considered as rather frequent and part of the same organi-
zation as the other events (Sornette 2009 ).
In the following, it is shown that the general systems theory concepts are equiv-
alent to Boltzmann's postulates and the Boltzmann distribution with the inverse
power law expressed as a function of the golden mean is the universal probability
distribution function for the observed fractal fluctuations which corresponds closely
to statistical normal distribution for moderate amplitude fluctuations and exhibit a
fat long tail for hazardous extreme events in dynamical systems.
For any system large or small in thermal equilibrium at temperature T , the prob-
E
K B , where
K B is Boltzmann's constant . This is called the Boltzmann distribution for molecular
energies and may be written as follows:
T
ability P of being in a particular state at energy E is proportional to e
E
K B .
(1.38)
T
Pe
The basic assumption that the space-time average of a uniform distribution of pri-
mary small scale eddies results in the formation of large eddies is analogous to
Boltzmann's concept of equal probabilities for the microscopic components of the
system (Buchanan 2005 ). The physical concepts of the general systems theory en-
able to derive Boltzmann distribution (Eq. 1.38) as shown in the following.
The r.m.s circulation speed W of the large eddy follows a logarithmic relation-
ship with respect to the length scale ratio z equal to R/r (Eq. 1.4) as given as follows:
w
* log.
 
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