 |
| The End of Certainty by Ilya Prigogine |
Briefly put, Prigogine asserts that many systems which are subject to change in the physical world naturally exhibit resonances of a mathematical variety originally discovered by Henri Poincaré. Poincaré resonances exist at points on a graph where equations describing dynamical systems — systems in the physical world made of objects in motion — intersect one another. At these points of intersection, crucial denominators become vanishingly small, such that algebraic divisions have undefined results. The customary approach to modeling dynamical systems, based on the idea that particles trace out hard-and-fast trajectories, breaks down.
Prigogine saves the day by substituting for the classical model a model based on probability distributions. His probability distributions substitute a "cloud" of system states for each single point ostensibly designating the state with certainty. They replace every simple, hard-and-fast trajectory with an expanding vapor trail of system pathways.
One of the desirable results of Prigogine's strategy is that the "instabilities" typically encountered by dynamical systems under particular sets of circumstances turn out to be related to and engendered by Poincaré resonances.
Basically, resonances are where waves — sound waves, light waves, mathematical waves — reinforce one another. When waveforms are coupled together, the amplitude (i.e., the height) of the combined waves can under the right circumstances be the (possibly very large) sum of the amplitudes of the individual waveforms. By envisioning certain mathematical constructs — "plane waves" and related "wave vectors" — as the basic stuff of which systems are built when they undergo motion and change, Prigogine is able to show that what we think of as particles occupying specific points in space and moving along specific trajectories in time are really made of these wavelike mathematical entities.
Such a wave-based mathematical model of the physical world achieves Prigogine's prime objective of introducing a directional "arrow" of time into the basic equations of physics, such that the clock of the universe cannot run backward after all.
Furthermore, the Prigogine model spreads out from the description of classical dynamical systems to include thermodynamic systems, chaotic systems, "dissipative" systems, and subatomic systems subject to the laws of quantum mechanics.
In traditional interpretations of quantum theory, subatomic particles such as electrons are never assigned trajectories per se. Instead, they have "wave functions." Different from the similarly named "plane waves" Prigogine alludes to, these attributes of the tiniest bits of matter are usually modeled using an equation named after its discoverer, Erwin Schrödinger.
Schrödinger's equation, notably time-reversible or time-symmetrical, does not allow for a directional flow of time. The irreversibility of time which we attribute to events (quantum, or at macroscopic scales larger than the subatomic) occurs, in traditional theoretical interpretations of quantum mechanics, as a result of external observation of those quantum events. Under observation, and only then, the supposed wave functions of quantum particles are said to "break down," creating a measure of scientific certainty about such things as the particles' velocities and positions where there had only been a fuzzy set of probabilities.
That reduction of the quantum wave function only under conditions of external observation is called the "quantum paradox" — for why should external observation radically affect the basic attributes of subatomic stuff?
Prigogine succeeds in sidestepping the quantum paradox by replacing specific wave functions à la Erwin Schrödinger with "ensembles" of possible states of a quantum entity, where an ensemble is represented mathematically by a probability distribution. His strategy simultaneously brings into the picture an irreversible arrow of time and removes from the quantum picture "the mysterious intervention" of an "observer" (p. 131).
When it comes to systems larger than the innards of an atom, Prigogine's selfsame model works equally well for any "nonintegrable" system — any system undergoing motion and change, whose overall behavior cannot be reduced to the sum of the behaviors of the individual pieces. And, for systems that are simple enough to be integrable, Prigogine's model can easily be transformed into the description given by classical Newtonian dynamics, whose equations turn out to be but a special case of Prigogine's more general ones.
Among the nonintegrable systems Prigogine discusses are systems in "deterministic chaos," a hot topic in physics over the last few decades. Also covered are thermodynamic systems, in which entropy and disorder increase over time. Meanwhile, one of the centerpieces of Prigogine's view concerns "dissipative" systems. Such systems hold themselves far from thermodynamic equilibrium by taking in and dispelling energy as well as matter over time. They can be merely physical or chemical systems, but all organisms are also dissipative systems.
Dissipative systems are wont arrive at crossroads of instability when they have been pushed far enough away from equilibrium by their characteristic energy flows. Prigogine calls these crossroads "bifurcation points" because the system will resolve each temporary instability by branching over onto one of two (or more) new pathways exhibiting increasingly complex order and stability. Which branch is "chosen" by the system at any particular bifurcation point is, according to Prigogine, a matter of irreducible probability — a coin flip, as it were.
Although Prigogine does not make this point explicitly, it looks to me as though he feels Poincaré resonances engender the "fluctuations" or "perturbations" which he says are associated with an evolving dissipative system's movement over onto one new branch or another at any given crossroads. These resonances seem to be intimately associated with the bifurcation points' fundamental character as instabilities. But Prigogine does not say how the resonances "decide" which of two possible paths the system will actually take.
Specifically, he does not say how a system undergoing characteristic junctures of instability makes its "choice" among alternate pathways in response to the minute "fluctuations" or "perturbations" that occur at a critical moment and that the reader must assume are associated with Poincaré resonances.
So, the reader is entitled to ask, if the resonances themselves do not "decide" which of two possible paths the system will actually take, what does? Prigogine seems to be content to allow that question to remain unanswered, except by reference to the influence of his "fluctuations" or "perturbations."
As for me, I consider "fluctuations" or "resonances," as an answer to the "what decides" question, to be just as mysterious as Prigogine calls the deciding influence of an outside observer, in discussing the paradoxes of quantum theory.
It looks to me as if there are two principled responses to this ambiguity. The first is the approach which Prigogine himself appears to take. He categorizes his probabilistic description of dynamical systems at various crossroads of mathematical instability as "primitive" and "irreducible."
You can't account for the primitive or irreducible elements that are necessarily found in every scientific theory. For example, the constant speed of light is a primitive, irreducible element of Einstein's theory of relativity. There is no way to explain why light alone should have an absolute velocity that is not relative to that of other moving entities. But when you predicate relativity theory on such an unexplained assumption, the theory then goes on to provide you with excellent explanations for, and reliable predictions about, many other phenomena.
Prigogine takes a like attitude toward the probabilistic-but-not-otherwise-predictable behavior of systems subject to Poincaré resonances. His main reason: only by taking the probabilistic uncertainty of the system as a given can you introduce the directional flow of time into a mathematical model of the physical world.
Put a different way, Prigogine settles for a theory that is stunningly powerful from a descriptive point of view, in that a single mathematical model adequately describes the behavior of systems from multiple disciplines: classical dynamics, thermodynamics, chaos theory, quantum mechanics, and the theory of dissipative, self-organizing systems.
Yet the model which describes the behavior of these systems does not explain it. It does not explain, in effect, why the probabilistic coin flip produces tails, not heads.
In fact, it undermines one of the assumptions classical physics traditionally made about coin flips and the like: if we could only ascertain the forces acting on the coin down to the nth degree of accuracy, we would be able to know in advance which coin flips would produce heads and which tails. The assumption was that coin flips and all other seemingly random events in the physical world were actually deterministic. Randomness was thought to be an illusion based on ignorance — as was the apparently unidirectional flow of time.
In undermining that assumption about the arrow of time being a bogus perception on our part, Prigogine likewise explodes the parallel assumption that coin flips are deterministic events. According to him, crucial events in the evolution of "nonintegrable" systems subject to Poincaré resonances are truly inexplicable except by reference to the laws of chance.
The other principled way to treat the "what decides" question is to decline to settle for the supposition that the choices made by systems at key crossroads of instability that stem from to Poincaré resonances have no further explanation outside the laws of chance.
What if there is a God behind these chance events?
According to Prigogine, if I read him aright, bifurcation points of dynamical instability match up with regions of "phase space" wherein chaos prevails (see p. 41). "Phase space" is an abstract realm made up of all the possible points on a hypothetical graph plotting the conceivable states of a system. These conceivable states in phase space are defined in terms of the instantaneous locations and movements of its constituent particles.
Under classical assumptions, at every individual moment the system as a whole occupies a single point in phase space. Since the system is "dynamical," it is subject to change over time. As it changes, in the classical view it traces out a "nice" (i.e., deterministic) trajectory, which amounts to a single, crisp line or curve through phase space.
In Prigogine's revision of the classical view, that single point in phase space has to be replaced by a "cloud" of possible points describing the system state probabilistically. The classical trajectory becomes an "ensemble" of possible trajectories through phase space. The "ensemble" can under some circumstances be viewed as a "random" trajectory, rather than a "nice," deterministic one. Under these circumstances, the system is chaotic. "Random" trajectories "wander erratically through regions of phase space" (p. 41). The future states of the system become exponentially less predictable as time marches on.
According to Prigogine, chaos, even when it is not "fully developed," can be associated with points in phase space that are "characterized by resonances" (p. 41). It is not clear to me exactly how closely the behavior of dissipative systems at resonance-engendered bifurcation points matches the "randomness" of chaos, but I assume the path the dissipative system "chooses" at such points of instability equates to a reemergence of order out of chaos.
Metaphorically, then, the choice could represent what naturally happens when God "stills the seas" of chaos.
No comments:
Post a Comment