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In conclusion, the Pythagorean recombination game pushes the system toward
the maximum of entropy, because the more the game goes on, the more velocities
approximate to normal distributions, which maximize the entropy, by keeping the
initial variance (interactions casually increase and velocities depend on an increas-
ing number of small independent effects). In fact, Figs. 4.10, 4.11, 4.12, 4.13, and
4.14 show that our pool of numbers (sums of squares of velocities) distribute ac-
cording tho a
2
distribution.
This analysis shows that time's arrow is a population phenomenon strictly related
to the complexity of systems, where the statistical and informational perspective
transforms the reversibility of individual events into an irreversible population pro-
cess. If we realize that also life is a population phenomenon based on bio-molecular
dynamics, then we easily can deduce that life postulates a time arrow. However, de-
spite this point of contact between thermodynamics and life, an essential difference
is immediately evident. The thermodynamical complexity implies an irreversibility
pushing an isolated system to an equilibrium state of maximum entropy, while life
evolution generates an increasing complexity of living organisms pushing species to
evolve toward increasing biological complexity, far from entropic maxima. There-
fore, the two arrows follow opposite directions. The research for answering this kind
of questions is crucial for a clear definition of biological complexity and for under-
standing the deep reasons on which the origin of life is based [122]. However, two
important remarks are appropriate in this context: i) living organisms cannot be iso-
lated systems, ii) in order to keep their dissipative character (assimilating from the
environment and dispersing to it) they need to be far from thermodynamical equi-
librium states. This non-equilibrium situation is the great discrimination between
organic and nonorganic organizations, and introduces the time arrow opposite to the
thermodynamical arrow, which however prevails when individual deaths occur, by
restoring matter to the organic course.
Two giants of science, Galileo and Darwin are crucially involved in the scien-
tific analysis of time. The former discovered the basis for a reliable measurement of
time. The latter introduced a “historical” perspective in the scientific analysis of life.
In fact, starting from considering living organisms according to suitable time scales,
Darwin found an interpretation key of their internal organization and development
in the framework of the evolution processes. Biological evolutions are population-
based dynamics driven by environment, chance (mutation+recombination), and
heredity. Heredity is, of course, an oriented transmission of information from a past
to a future.
As we have shown in this section, Boltzmann introduced an orientation in the
time of physics, by making the first step in the conciliation of Galileo's and Darwin's
times. Certainly, other steps are necessary in this direction for a better understanding
of time and life [126].
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