On Self-Organizing Systems and Their Environments - Understanding Understanding: Essays on Cybernetics and Cognition - page 17

Z i = a e

- i b

(vii)

The two undetermined multipliers a and b can be evaluated from eqs. (i)
and (ii):

Â

e -

i b

a = Z (viii)

Â

a i e -

i b

= N (ix)

Remembering that

d
db

 Â

- i b - i b

- e = i e

we obtain from (viii) and (ix) after some manipulation:

Z
n

( ) ª a= Z 1 - e

- 1 n

(x)

11
nn

Ê
Ë

ˆ
¯ ª

b= ln 1 + (xi)

where n , the mean cell population or density N / Z is assumed to be large in
order to obtain the simple approximations. In other words, cells are
assumed to be large enough to lodge plenty of elements.
Having determined the multipliers a and b, we have arrived at the most
probable distribution which, after eq. (vii) now reads:

Z
n

Z

= e

- in

(xii)

i

From eq. (iv) we obtain at once the maximum entropy:

H = () ln en . (xiii)

Clearly, if the elements are supposed to be able to fuse into pairs (Fig. 7 b ),
we have

( )

HZ ln ¢= en 2 . (xiv)

Equating H with H m and H ¢ with H , we have for the amount of order after

fusion:

()

Z
Z

ln
ln

en
en

1

R =- 1

) =

(xv)

( () 2 log en

2

References
1. L. Wittgenstein, Tractatus Logico-Philosophicus , paragraph 6.31, Humanities
Publishing House, New York (1956).