Enlarging Knowledge: The Road to Universality

By Mitchell J. Feigenbaum

In his pioneering work on chaos, Mitchell Feigenbaum was fascinated with the technique in physics known as "renormalization," which basically involves looking at tiny pieces of the whole structure and enlarging them. To renormalize a circle, for example, one isolates progressively smaller and smaller bits of the circle's arc. What emerges is a sequence of curves, each approaching more closely a straight line. (Consider how "flat" the round earth appears to a lone human being standing on its surface.) Renormalization turns the curve into a line - the line is thus a kind of "universal attractor."

This led Feigenbaum to begin to understand the "universality" of certain features of chaos. Seemingly unrelated transitions - the boiling of liquids or the magnetization of metals - could be governed by the same rules. His discovery implied that the equations in nonlinear systems could be besides the point: when order emerged, it seemed to have forgotten what the original equation was. Dr. Feigenbaum, who is Toyota Professor at The Rockefeller University in New York City, discussed some of the implications of his work at the Tokyo symposium. - Editor

I find my assigned title - "The Impact of Chaos on Physics" - somewhat constricting, and so I am going to immediately broaden it. It is difficult in certain instances to disentangle what is physics and what is mathematics, and so I shall largely ignore the distinction.

This blurring is, after all, not so surprising. It was inevitable, for example, that the creators of mechanics and the calculus were one and the same people, since the science of mechanics is largely coequal to its kinematic framework, which is the calculus. If not so grand, it is equally true that the far-flung investigations pertaining to chaos are possible precisely because of their common underpinning, which is the coming to awareness of the intuitive content of certain mathematical ideas, while the delineation of these very ideas flowed from scientific thought.

Indeed if it is true that ideas of chaos can pertain, for example, to medical phenomena, then only a thoughtless and ignorant analysis would ascribe the reason to the profound sway the science of physics holds over the others. Quite to the contrary, it is the understanding of the meaning and use of certain mathematics that happens to underlie both disciplines. Nevertheless, it is also true that the ideas of chaos have enlarged the suzerainty of physics, enabling its reach to extend to various macroscopic phenomena in a manner previously not possible.

At a much less refined level, albeit an altogether pragmatic one, this blurring of boundaries does create difficulties. Specifically - and as is clear from the scope of this conference - the studies of chaos have perilously penetrated that anathema termed "interdisciplinary." It is thus not particularly surprising to discover new young investigators rejected by one department because his efforts are "too mathematical" and by another because the work fails to pass the acid test of "sufficient rigour." Should this investigator obtain fine results, both will enjoy his company. But all too often there is a failure - to put it in coarse terms - to "put one's money where one's mouth is." Perhaps, equally vulgar, because money and funding have come to matter so much, one finds the kitty again divided by entrenched, rigid camps.

So much by way of introduction.

Nonlinear vs. Linear

The subject matter of "chaos" is part of the study of nonlinear systems. Yet our most potent mathematical inheritance has pertained in general to what is "linear" - and hence leaves us unprepared to investigate, as it turns out, what concerns us the most about physics. Physics consists often of the development of approximate methods of calculation because, arbitrary amounts of numerical computation not withstanding, one is powerless to plumb the contents of the theories we believe in.

My personal involvement in what is called chaos issued from a strong frustration at this inability, together with an annoyance and distaste for the various approximate methods then in vogue in high energy physics. I found it vastly more satisfying to penetrate exactly one such problem - even if not the right one -than to continue in the other mode of uncertainty and approximation. Thus, for myself, chaos emerged as one (highly interesting) manifestation of nonlinearity, whereas the attempt to erect true understandings of the mathematics that appears in our theorizing is my stronger goal.

The Routes to Turbulence

Let me now mention several specific ways in which the daily investigations of physics have been modified by the advent of chaos. First of all, the subject of the onset of turbulence in fluids has been drastically altered and, in a real sense, a new subject created. In particular, there now exists the idea of "routes to turbulence" or "regimes" in which the outcome of meticulously performed experiments in all details can be predicted. The only previously offered explanations - prior to, say, ten years ago - have been ruled out as incorrect in almost all cases. Moreover, the methodology with which an experimenter gathers his data and massages it to garner its contents has been radically extended.

The study of dynamical (oscillatory) chemical reactions has been drastically altered in the last few years. Just a decade ago, the notion that such reactions could proceed chaotically was energetically denied by some of the discipline's most distinguished researchers on grounds of "thermodynamic impossibility." The presence of chaos in these reactions has now been well observed, with theory playing some role in directing investigation.

One observes chaotic behaviour in physical systems as diverse as lasers, Josephson junctions, all sorts of fluid motions, electronic devices and their control systems, and in biological systems.... At times, these behaviours are so intrinsic and informative that researchers end up asking new questions and seeking new sorts of relations between different aspects of their data. Under these circumstances, even the mere intrusion of different modalities of relationships can provoke investigations conceivably more profitable than the associations that gave rise to them in the first place.

Impact of Chaos on Knowledge

Let me now turn to a more theoretical discussion of the impact of chaos on our knowledge and approach to general questions. Take, for example, the question of linear vs. nonlinear. Linearity means that the differential response of an object to any impressed force is independent of the state of that object, and so, mathematically described by a linear equation. The deep virtue of this circumstance is that all linear objects enjoy membership in a simple geometry of a fixed character. It is our inherited potency at visualizing and deeply comprehending this geometry that has allowed our prowess over such problems.

The situation for nonlinear problems is quite otherwise. For the purpose of physical utilization, the available mathematical knowledge is so pitifully skimpy that most problems, for which the theoretical formulation proves to be nonlinear, receive no satisfactory resolution at present. Insofar as this circumstance inheres in almost all physics investigations of present topical interest, it is almost impossible for a physicist wanting to get some results not to dabble in creating new mathematics. The study of chaos is one arena in which a predisposition to succeed in a calculation informed by some physical intuition has resulted in physicists putting forth what is apt to be mathematically interesting.

The tradition of mathematics of say the last half-century has come often to eschew calculation, perhaps through the realization that too much of what is considered is just specific detail and baggage that surrounds a much simpler or more transparent set of relations and considerations. Once rid of the baggage, wonderful and tight inner workings can emerge.

Sometimes, however, too much "baggage" is stripped away - so that what is general can allow only the mention of special circumstances which makes detailed investigation out of consideration. My own work on period doubling represents a case in point. The equations of analytic science tend to have a certain degree of smoothness - virtually all equations of physics, chemistry, etc. are so endowed. But this proves to be baggage in topological thought. Faced with what is perceived as an arbitrary detailed special case, topological methods are powerless. For this reason, for example, no idea whatsoever of the universal quantitative character of period doubling existed within the community of mathematicians - nor could it have. This is to reiterate that delineations of disciplines are certainly to a large measure the conventional impositions of a human community in pursuit of the present.

Impact of Universality

Let me conclude with some thoughts on the impact of universality on physics. The upshot of universality is that it is altogether irrelevant what the specific details of a physical system are in order to quantitatively deduce its behaviour.

That is, the predictable results do not depend upon the equations. How then is a particular microscopic theory to be held as explanatory towards and foundational for the phenomenon? And suppose the theory is to be held but the equations are too arduous to solve, whereas a vastly simpler theory produces exactly the same results.

What then is the human illumination that flows from the deeper theory? The question is truly and verifiably posed, and so requests honest thought towards a resolution. I will simply leave it here as a conundrum that should make the reader pause to reflect on what, if anything, the usual notion of "the scientific method" means.

What is unquestionably true is that the theory of the onset of fluid turbulence, under certain conditions, is now well understood and measurably confirmed, despite the fact that no one yet can solve the correct equations of fluid motion. The ideas of the theory prove to be rigorous instances of "renormalization group" notions. The theory of period doubling afforded the first serious exact example of these ideas and furnished the setting for some precise understanding. Again, the ideas of renormalization in this setting proved mathematically fecund, resulting in deeper understanding of other classical results in, for example, dynamics and probability theory.