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A.Entropy.
This chapter requires some familiarity with the rudiments of
physics and mathematics. Readers lacking this familiarity
may skip it and proceed to "MORPHOGENESIS" on condition to
accept:
-the physical foundations of Informatics as granted and
-a loose definition of the concept of order.
In physics, the customary way to express the probability of
a given heat distribution is the ENTROPY E defined by the
BOLTZMANN expression
(1): E = k ln(S)
where k is the Boltzmanns gas constant and S is the number
of microscopic states in which the macroscopic state E may
be realized. The concept of entropy may be generalized by
extending the expression over all forms of energy. Setting
in the generalized expression k = 1/ln(2) we get:
(2): E = ln(S) / ln(2) = log2(S) (log2 meaning log base 2).
(2) allows to express the entropy of the system as the number
of digits of a binary string capable to address all possible
system's states. Each digit of this string determines a
degree of freedom of the system.
Entropy represents the part of energy of a system unfit for
doing mechanical work.
Replacing entropy E with information I we may say that the
complete set of system's states may be addressed with help
of I binary bits:
(3): I = log2(S)
(2) and (3) express the analogy between entropy and
information.
The second law of thermodynamics, the "LAW OF ENTROPY" which
may be generalized over all practical energetic systems
states that any change of the system state increases its
entropy. At the limit, for the purely theoretical reversible
system, entropy stays constant. It never decreases.
Entropy may be considered as the measure of disorder and its
inverse as measure of order. In information systems entropy
represents the degree of uncertainty of a message.
NOTE: The Law of Entropy holds for closed systems. As we
shall see in "MORPHOGENESIS", entropy may decrease in local
open systems. Such systems may get ordered and their
ordering is foundation of the cosmos, of life and of human
reason.
Analogically, as result of a communication some of the I
bits necessary to describe the S possible system states may
get corrupted in which case the amount of useful information
contained in I decreases. For each corrupted bit half of the
possible system states S is excluded from the information I.
This is the customary way of representing the analogy between
entropy and information.
Another way of presenting the analogy, which we find more
appropriate for the present essay is to consider not the
information, but the information carrier as analogous and to
say that for each corrupted bit a new must be added in order
to describe all S possible states of the system. In this
context the analogy becomes strictly isomorphic and we may
refer to I as to the ENTROPY OF INFORMATION, or shortly
ENTROPY, when no misunderstanding about the nature of the
involved system is possible.
NOTE: We shall further use the term Chaos as synonym of
Disorder which differs from the definition used in certain
Chaos Theories, where "Chaos" points to some "hidden order"
concealed by apparent disorder.
"Disorder" contains a suggestion of having emerged from some
preliminary Order by its destruction. "Chaos", on the
contrary, is, like "Order", just a system's state not
prejudging any origin.
We shall see in MORPHOGENESIS that Order may under certain
circumstances emerge from Chaos, which sounds better than
"Order emerging from Disorder". For instance discussing the
Big Bang we shall admit Chaos as the original state of this
Cosmos Model while the negative term "Disorder" could hardly
pertain to this context.
Based upon the notion of entropy we define the ORDER of
an information system:
(4): O = I1 / I
where I is systems entropy in a given situation and I1=log2(S)
is the minimum possible entropy of the system. Consequently,
the highest possible order is 1. When entropy increases order
decreases. Its value range is between 1 and 0.
After introduction of the concept of order the law of entropy
extended over information systems may be called DISORDERING
PRINCIPLE and formulated: any spontaneous change of a system
decreases its order. In the limit case of an ideal "reversible
system", the order stays constant. It never increases. (With
exception of local open systems which we mentioned above.
We shall discuss them in the next chapter "MORPHOGENESIS".)
With respect to entropy and order Informatics is analogous
to Physics. By virtue of this analogy we may consider
Informatics as a domain which may be investigated, ordered
and processed with help of rigorous scientific methods.
