12.4 Self-organization

155

Therefore, although S has become more organized, overall it has generated more

disorganization than the organization created, and it is more accurate to call it a

self-disorganizing system. ²⁴ Hence, what we consider as the system should properly

be expanded to include its environment—it is evidently intimately connected with

it and without it there could be no organization (and here we see the importance

of the universe starting a low-entropy state; cf. Chap. 5). Despite its true nature as

a self-disorganizing system having been revealed, however, we can still speak of a

self-organizing part of the overall system, which consumes order (and presumably

energy) from its environment. It follows that this environment must necessarily have

a structure itself, otherwise there would be nothing to be usefully assimilated by the

self-organizing part.

The link between entropy (i.e., its inverse) and organization can be made explicit

with the help of the relative entropy (redundancy) upper RR previously defined (Eqs. 6.17

and 6.18). Self-organization implies that delta upper R divided by delta t greater than 0δR/δt > 0. Differentiating Eq. (6.18), we

obtain

StartFraction d upper R Over d t EndFraction equals StartFraction upper S left parenthesis d upper S Subscript normal m normal a normal x Baseline slash d t right parenthesis minus upper S Subscript normal m normal a normal x Baseline left parenthesis d upper S slash d t right parenthesis Over upper S Subscript normal m normal a normal x Superscript 2 Baseline EndFraction semicolon^dR

dt ^{= S(dSmax/dt) −Smax(dS/dt)}

S²max

;

(12.34)

our criterion for self-organization (thatupper RR must spontaneously increase) is then plainly

upper S StartFraction d upper S Subscript normal m normal a normal x Baseline Over d t EndFraction greater than upper S Subscript normal m normal a normal x Baseline StartFraction d upper S Over d t EndFraction periodS ^dSmax

dt

> Smax

dS

dt ^.

(12.35)

The implications of this inequality can be seen by considering two special cases:

1. The maximum possible entropy upper S Subscript normal m normal a normal xSmax is constant; therefore d upper S Subscript normal m normal a normal x Baseline slash d t equals 0dSmax/dt = 0 and

d upper S slash d t less than 0dS/dt < 0. Now, the entropy upper SS depends on the probability distribution of the

constituent parts (at least, those that are to be found in certain distinguishable

states); this distribution can be changed by rearranging the parts, which von Foer-

ster supposed could be accomplished by an “internal demon”.

2. The entropyupper SS is constant; therefored upper S slash d t equals 0dS/dt = 0 and the condition thatd upper S Subscript normal m normal a normal x Baseline slash d t greater than 0dSmax/dt >

0 must hold; that is, the maximum possible disorder must increase. This could be

accomplished, for example, by increasing the number of elements upper NN; however,

care must be taken to ensure thatupper SS then indeed remains constant, which probably

needs an “external” demon.

Inequality (12.35) shows how the labour is divided among the demons:d upper S slash d tdS/dt repre-

sents the internal demon’s efforts andupper SS is the result;d upper S Subscript normal m normal a normal x Baseline slash d tdSmax/dt represents the external

demon’s efforts and upper S Subscript normal m normal a normal xSmax is the result. There is therefore an advantage (in the sense

that labour may be spared) in coöperating—e.g., if the internal demon has worked

hard in the past, the external demon can get away with it by putting in less effort in

the present.

24 von Foerster (1960).