On Self-Organizing Systems and Their Environments - Understanding Understanding: Essays on Cybernetics and Cognition - page 8

the condition for a system to be self-organzing expressed in terms of
entropies:

d H

t

d
d

H
t

m

(5)

H

> H

m

d

In order to see the significance of this equation let me first briefly discuss
two special cases, namely those, where in each case one of the two terms
H , H m is assumed to remain constant.

( a ) H m = const.
Let us first consider the case, where H m , the maximum possible entropy
of the system remains constant, because it is the case which is usually visu-
alized when we talk about self-organzing systems. If H m is supposed to be
constant the time derivative of H m vanishes, and we have from eq. (5):

d H

t

d
d

H
t

m =

for (6)

0 LL < 0

d

This equation simply says that, when time goes on, the entropy of the
system should decrease. We knew this already—but now we may ask, how
can this be accomplished? Since the entropy of the system is dependent
upon the probability distribution of the elements to be found in certain dis-
tinguishable states, it is clear that this probability distribution must change
such that H is reduced. We may visualize this, and how this can be accom-
plished, by paying attention to the factors which determine the probability
distribution. One of these factors could be that our elements possess certain
properties which would make it more or less likely that an element is found
to be in a certain state. Assume, for instance, the state under consideration
is “to be in a hole of a certain size.” The probability of elements with sizes
larger than the hole to be found in this state is clearly zero. Hence, if the
elements are slowly blown up like little balloons, the probability distribu-
tion will constantly change. Another factor influencing the probability dis-
tribution could be that our elements possess some other properties which
determine the conditional probabilities of an elements to be found in
certain states, given the state of other elements in this system. Again, a
change in these conditional probabilities will change the probability distri-
bution, hence the entropy of the system. Since all these changes take place
internally I'm going to make an “internal demon” responsible for these
changes. He is the one, e.g. being busy blowing up the little balloons and
thus changing the probability distribution, or shifting conditional probabil-
ities by establishing ties between elements such that H is going to decrease.
Since we have some familiarity with the task of this demon, I shall leave
him for a moment and turn now to another one, by discussing the second
special case I mentioned before, namely, where H is supposed to remain
constant.