Page 97 - Nanotechnology an introduction

8.2.2. Thermodynamics of Self-Organization
Consider a universe U comprising a system S and its environment E; i.e., U = S ∪ E (Figure 8.4). Self-organization (of S) implies that its entropy
spontaneously diminishes; that is,

(8.1)
Accepting the second law of thermodynamics, such a spontaneous change can only occur if, concomitantly,

(8.2)
with some kind of coupling to ensure that the overall change of entropy is greater than or equal to zero. If all processes were reversible, the two
changes could exactly balance each other, but since (inevitably, we may suppose) some of the processes involved are irreversible, overall

(8.3)
Therefore, although the system itself 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 [57]. Hence, the “system” must properly be expanded to include its environment—it is
evidently intimately connected with it; without it there could be no organization. Despite its true nature as a self-disorganizing system having been
revealed, nevertheless we can still speak of a self-organizing part S of the overall system that consumes order (and presumably energy) from its
environment. It follows that this environment must necessarily have structure itself, otherwise there would be nothing to be usefully assimilated by the
self-organizing part.

Figure 8.4 Universe U comprising system S and its environment E.
The link between entropy (i.e., its inverse) and organization can be made explicit with the help of relative entropy R (called redundancy by
Shannon), defined by

(8.4)
where S max is the maximum possible entropy. With this new quantity R, self-organization implies that δR/δt > 0. Differentiating equation (8.4), we
obtain

(8.5)

our criterion for self-organization (namely, that R must spontaneously increase) is plainly

(8.6)
The implications of this inequality can be seen by considering two special cases [57]:
1. The maximum possible entropy S max is constant; therefore dS max /dt = 0 and dS/dt < 0. Now, the entropy S 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 Foerster supposed could be accomplished by an “internal demon”.
2. The entropy S is constant; therefore dS/dt = 0 and the condition that dS max /dt > 0 must hold; that is, the maximum possible disorder must
increase. This could be accomplished, for example, by increasing the number of elements; however, care must be taken to ensure that S then
indeed remains constant, which probably needs an “external” demon.
Looking again at inequality (8.6), we see how the labor is divided among the demons: dS/dt represents the internal demon's efforts, and S is the
result; dS max /dt represents the external demon's efforts, and S max is the result. There is therefore an advantage (in the sense that labor may be
spared) in cooperating—e.g., if the internal demon has worked hard in the past, the external demon can get away with putting in a bit less effort in
the present.
These considerations imply that water is an especially good medium in which self-assembly can take place because, except near its boiling point,
it has a great deal of structure (Section 3.8) that it can sacrifice to enable ordering in S. Hence, biological self-assembly of compact protein and
nucleic acid structures takes place in an aqueous environment. Presumably thermophilic microbes that live at temperatures close to 100°C have
some difficulties on this score.

8.2.3. The “Goodness” of the Organization