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elements that make up the system. These may include individual molecules in a
Belousov-Zhabotinsky chemical reaction-diffusion system, individuals in a stock
of fish, and individual people in a crowd gathered on the square. These components
must interact with each other, i.e., the system must be dynamic, functioning on the
basis of the dynamic mechanisms.
2. An important feature of self-organizing processes is that they are carried out in
open systems.
In a thermodynamically closed system, the evolution in time leads to a state of
equilibrium to which the maximum entropy value of the system corresponds. And,
according to Boltzmann, this state is characterized by the maximum degree of
randomness.
In open systems, two variants of evolutionary processes are possible:
Time evolution toward the equilibrium state (in general, this can also be the
evolution toward a nonequilibrium but steady state)
Evolution through a sequence of stationary states, with a change of stationary
states due to the slow change in the so-called control parameters (e.g., ambient
temperature during the formation of Benard cells)
The well-known Russian physicist Yu. L. Klimontovich cited as a good example
the evolution theory of Charles Darwin. It is based on the principle of natural
selection. Evolution can lead either to degradation or to be a process of self-
organization, in which more complex and more sophisticated structures emerge.
Self-organization is therefore not the only result of evolution. “Inner striving” for
self-organization is inherent to neither physical nor even biological systems. An
alternative way may be degradation, a physical example of which is the temporal
evolution of a closed system toward equilibrium. Thus, self-organization is only
one of the possible paths of evolution. In order to understand which path will be
adopted by a developing system, a criterion for self-organization is required.
A number of systems are known for which such criterion is apparent. Thus, in the
case of Belousov-Zhabotinsky chemical reaction-diffusion systems, the initial
state corresponds to the uniform, i.e., chaotic, distribution of molecular components
of the medium. The order, i.e., self-organization, corresponds to the formation of
dissipative structures. It would seem that self-organization must correspond to the
maximum degree of order.
But in a general case, the situation becomes much more complicated.
Yu. L. Klimontovich refers to the human body as an example. Its stationary state
corresponds to a certain degree of randomness, because the equilibrium state
(complete randomness) differs in principle from the state of life. Namely, it must
be regarded as an ordered state. Thus, a certain standard of chaos must correspond
to the order, and deviations from it disrupt vital functions, i.e., the degree of order in
the system.
The ground for understanding the spontaneous emergence of order was laid by
the great mathematician of the last century, Alan Turing, in his paper “The
chemical basis of morphogenesis.” He showed that nonlinear dynamic mechanisms
in an initially homogeneous medium give rise to ordered structure. Somewhat later.
