Clas Blomberg - Physics of life-Elsevier Science (2007)

268 Part VII. Non-linearity
flows such as in Ohm’s law, which means proportionality between a flow, the electric current
and a driving force—an electric voltage. The proportionality, in this case, the conductivity,
can be given by equilibrium features by fluctuation–dissipation relations. Periodic
processes are simply analysed by spectral composition. Any periodic force of a particular
frequency gives rise to a periodic current with the same frequency.
We can see two primary different aspects in the previous description. One is the appearance
of new kinds of structures, which may be uniform and static, also occurring in equilibrium
systems. The other aspect concerns the establishment of new kinds of temporal processes,
different from the behaviour close to equilibrium.
Another simple and also easily understandable example of a transition is provided by an
elastic rod. One has a rod of a certain length and puts a force between the ends. The rod is
elastic and with a relatively small force, one gets the linear result: the rod is pressed
together by an amount proportional to the strain force. When the force exceeds a certain
value, something new happens: the rod buckles out in a new direction. The buckling
requires a bending that necessitates a certain stress. As in the magnetic transition, there is a
broken symmetry, a direction of the buckled rod appears, which is not there in the original
statement of the problem. In this buckled solution, there is also a variation of the deviation
of the rod from its equilibrium position, which is a new and relevant feature Figure 26.1.
Elastic non-linear features are certainly relevant for understanding properties of certain
biological structures, for instance a skeleton and a tree, features that although important are
outside the themes of this book.
For the themes of this book, main sources for the non-linear special effects, and anyhow
main parts in most of the descriptions, have their basis in chemical reactions. However, the
non-linear effects are not as apparent for chemical reactions as in elasticity theory or engineering
scientific studies of electric non-linear circuits. When chemical oscillations first were
demonstrated in the Zhabotinsky–Belusov reaction (Zhabotinskii, 1967; see also e.g. Nicolis
and Prigogine, 1977), there was a general doubt that this really was an effect of the reactions.
There was a belief that chemical reaction dynamics should not lead to that kind of results.
Now, of course, we know after a number of instructive studies that this really is so, but also
that the key to understand the possibilities is the concept of “autocatalysis”, reactions where
production rates are enhanced by the product itself. This is frequent in the processes of life.
Template reproduction provided by the nuclide acids where a copy is built up after an existing
specimen as a primary example of such an autocatalytic process. Further there are many
processes where a protein shall be activated by binding some substrate in order to produce
a certain reaction, and there are frequent cases where this is accomplished by the product itself:
thus a product activates the catalytic process by which it is produced. This is of course what
is considered as “positive feedback”, a common and well-studied effect in control theory.
In this, one of the most important non-linear effects is mentioned: the generation of oscillations.
Chemical reactions which are driven by the steady supply of some substance can, under
certain circumstances lead to sustained oscillations. With given circumstances, these oscillations
have definite amplitudes and frequencies. Sometimes, the non-linear systems eventually
Figure 26.1 A rod that is pressed by two forces is first simply contracting, as in the left figure, but
at sufficiently large strain, it buckles out, perpendicular to the forces.