On Self-Organizing Systems and Their Environments - Understanding Understanding: Essays on Cybernetics and Cognition - page 16

FIGURE 7.

whence we obtain the entropy of the system for a large number of cells and
elements:

== Â HPZZ ZZ

i

ln ln ln (iv)

i

In the case of maximum entropy H we must have

d H = 0

(v)

observing also the conditions expressed in eqs. (i) and (ii). Applying the
method of the Lagrange multipliers we have from (iv) and (v) with (i) and
(ii):

Â
Â
Â

( )

ZZ
iZ
Z

ln + 1 d = 0

i i

d =

=

0
0

b

i

|

( ) d - + 1 ln a

i

multiplying with the factors indicated and summing up the three equations
we note that this sum vanishes if each term vanishes identically. Hence:

ln

Zi

i ++ -- =

1 b 1 ln a 0 (vi)

whence we obtain that distribution which maximizes H :