(Comments on book of the same name by Ilya Prigogine.)
When physical systems are simple (e.g. composed of only a few particles, preferably only 2) and lack instabilities, they can be described (as Newton did) in terms of particle trajectories obeying the laws of motion. Such systems are deterministic (if we know the initial conditions and the laws of motion, the future can be predicted with certainty); they are also “integrable” (his word — I think it may mean “decomposable”); and in addition they are time-reversible (they can be run forward or backward; there is no difference between the past and the future). Quantum systems are also time-reversible; the Schrodinger equation has no arrow of time. Yet in ordinary experience, time flows inexorably in one direction like a river.
However, as the number of particles increases (even a 3-body problem, such as the Sun, the Earth, and the Moon, is already difficult), instability appears because of Poincare resonances. Interactions among many particles are persistent and no longer transitory. When the number of particles becomes of the order of 1023 (Avogadro’s number, the number of molecules in one litre of gas at standard temperature and pressure), we hit the “thermodynamic limit”. We then have to abandon explanation in terms of individual particle trajectories and deal with ensembles, as in statistical mechanics. We go from micro-scale explanation to macro-scale explanation. Such complex large systems are no longer time-reversible, because the random ensembles are much more numerous than the ordered ensembles, and if we only count the variously disordered and ordered states, the system tends toward greater disorder, i.e. entropy increases. Thus time acquires an arrow, and the past differs from the future, because the future has more entropy than the past. In principle the process could be reversed to produce order in the future, but this is extremely improbable.
Even this “time-reversibility in principle” is abolished, according to Prigogine, if the system is far from equilibrium and open to the flow of matter and energy. Such systems (dissipative structures) undergo fluctuations and bifurcate repeatedly to higher forms of order (or break down instead). They become holistic, nonintegrable, nondecomposable – i.e. the whole is more than the sum of the parts, because of the persistent and frequent interactions. Time symmetry then breaks completely, because these systems evolve in unpredictable ways. (I.e. if the tape was wound back and re-run, it would produce a different result, as Gould said about evolution.) Certainty is replaced by possibility or probability, or what Gould called “contingency”. These systems evolve; they have a history, consisting of events that could have been otherwise. Determinism is gone; novelty can be produced; emergence of new phenomena becomes possible and is manifested. All this is the result of instability which may lead to chaos, out of which new order may emerge.
Nature often exhibits such instabilities, leading to phenomena such as turbulence in weather systems, in evolution, and in history. This is because initial conditions can never be established with complete accuracy, since an infinite number of irrational numbers exist between any “adjacent” fractions, and natural systems tend to diverge even if the initial conditions differ only very slightly.
Prigogine goes on to show that this (the creation of the arrow of time) applies not only to Newtonian physics, but also to quantum mechanics and to relativity theory. In the latter, time may be “spatialized” by multiplying it by the imaginary unit i, making it into “real time” (??), but this does not really happen. The space-time framework is rotated in the Lorentz transformation, but time remains different from the 3 space dimensions. But instead of discussing these aspects, I now turn to some of my own analogies.
Complex systems become holistic and non-decomposable because of frequent and persistent interactions among particles. This means that the interactions become more important than the particles; that structure predominates over matter. (Aristotle’s formal cause applies rather than his material cause.) This is very evident in living bodies, where materials and energy come and go, but the form or structure is conserved. In a way, there is conservation of information and non-conservation of matter and energy – a complete reversal of what happens in closed systems.
It is also noteworthy that, in Robert Axelrod’s “Evolution of Cooperation” experiments with Prisoner’s Dilemma games, tit-for-tat cooperators win over other strategies (especially the nasty “meanies”) when the programs meet each other again and again, thus giving rewards to each other. Interactions among people, which Axelrod’s tournaments were meant to simulate, also become cooperative because of reciprocity, but only if the partners expect to interact repeatedly. As in a net, the links become more important than the nodes.
The structure of frequent interactions is the essential web that knits together both the natural and the social world.