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Maxwell on Chaos

Brian R. Hunt and James A. Yorke*

James Clerk Maxwell (1831- 1879) is perhaps best remembered for his contributions to the field of electromagnetism , but his work on the kinetic theory of gases was of great importance as well. He lived at a time when the intellectual world was, for compelling reasons, enthralled with the concept of. a predictable, clocklike universe, yet he had the independence of mind to take issue with this paradigm, both in his scientific work and in his more philosophical writings. It is our view that Maxwell was the first person to understand chaos-that is, he recognized the existence and importance of systems with "sensitive dependence to initial data. " We will let Maxwell explain his understanding in his own words.

In deriving his well-known distribution law for the velocities of gas molecules, Maxwell found that sensitivity to initial data in the interactions between gas molecules was essential in producing statistical regularity. One com mon view of statistical mechanics is that the complex behavior arises from the large number of particles involved: Conversely, Maxwell's line of thought begins by consid­ering the interaction of only two particles. In his ground-breaking paper of 186cf. Maxwell begins by posing the following proposi­tion .'

COlllillued on p. 3

" Inslitule fo r Physical Science and Technology. UniversilY of Maryland. College Park. MD 20742·2431

The aUlhors would appreciale receiving olher references for early views of chaos. ' and. in panicular. addilional references to Maxwell's writings. They can be reached by eleclronic mail aliwIII@ipSI.umd.edu.

James Clerk' Maxwell. Courtesy of American Institute of Physics Emilio Segre Visual Archives.

PUBLISHED BY SPRINGER-VERLAG VOLUME 3, NUMBER 1 1993

Articles in this issue ...

Scientific Article

Maxwell on Chaos Brian R. Hunt and James A. Yorke

Feature Article

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5 SERC Research Funding in the UK: Applied Nonlinear Mathematics John Brindley

Complete contents on p. 2

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Hunt and Yorke continued from p. 1

Two spheres moving in [specified] opposite directions with velocities inversely as their masses strike one another .... [FJind the probability of the direction of the velocity after impact lying between given limits.

Thus, Maxwell is picturing two balls viewed from the frame in which their total momentum is zero. Given the initial position and velocity of one of the balls, there is a cylinder which must contain the initial position of the other ball in order for there to be a collision. Assuming the initial position of the second ball to be uniformly distributed in a circular cross-section of this cylinder, Maxwell's remarkable conclusion about the velocities after im.,pact is that "all directions of rebound are equally likely. ,, -

Maxwell includes the above proposition despite the fact that his later quantitative derivations do not depend on it; this underscores its conceptual importance in his thinking. While Maxwell was trying to give a plausibility argument that the distribution ofveloci­ties in a gas is independent of direction, it made him aware of simple systems with sensitivity to initial data. Since Maxwell does not discuss simple systems except in passing. we cannot be sure of his thoughts, but we feel he would would understand sensitivity to initial data in some simple systems such as 2- and 3-dimensional billiards.

Maxwell follows his analysis of the two-particle interaction by asserting that any initial configuration of molecules should relax quickly to a standard distribution of velocities.3

If a great many spherical particles were in motion in a perfectly elastic vessel. collisions would take place among the particles, and their velocities would be altered at every collision; so that after a certain time the vis viva4 will be divided among the particles according to some regular law, the average number of particles whose velocity lies between certain limits being ascertainable. though the velocity of each particle changes at every collision.

Maxwell later summarizes his derivations thus:5

We have shewn. in the first part of this paper. that the motions of a system of many·small elastic particles are of two kinds: one. a general motion of translation of the whole system, which may be called the motions in mass; and the other a motion of agitation. or molecular motion, in virtue of which velocities in all directions are distributed among the particles' according to a certain law. In the cases we are considering, the collisions are so frequent that the law of distribution of the molecular velocities. if disturbed in any way, will be re-estab­lished in an inappreciably short time; so that the motion will always consist of this definite motion of agitation. combined with the general motion of translation.

Maxwell's views on sensitive dependence to initial data occur most clearly in his nonscientific writings. In 1873 he delivered an essay at Cambridge University concerning the debate between determinism and free wil1.6

When the state of things is such that an infinitely small7

variation of the present state will alter onlY by an infmitely small quantity the state at some future time, the condition of the system, whether at rest or in motion. is' said to be stable; but when an infinitely small variation in the present state may bring about a finite difference in the state of the system in a finite time. the condition of the system is said to be unstable.

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It is manifest that the existence of unstable conditions renders impossible the prediction of future events, if our knowledge of the present state is only approximate. and not accurate.

The above view was later echoed by Poincare.8

If we knew exactly the laws of nature and the sitUation of the universe at the initial moment. we could predict exactly the situation of that same universe at a succeeding moment. But. even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible ....

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Later in his essay Maxwell offers his view of nature as being guided qualitatively by the same mechanism that we now know produces chaos in the forced double- well Duffi ng equation, in which the trajectory is frequently found near the peak berween two potential valleys.9

No one, I suppose. would assign to free will a more than infinitesimal range. No leopard can change his spots. nor can anyone by merely wishing it. or, as some say, willing it. introduce discontinuity into his course of existence .... In the course of this our mortal life we more or less frequently find ourselves in a physical or moral watershed. where an imper­ceptible deviation is sufficient to determine into which of two valleys we shall descend.

Maxwell took a dim view of the idea of a mechanical universe in which the future is completely predictable. to

It is a metaphysical doctrine that from the same antecedents follow the same consequents . No one can gainsay this. But it is not of much use in a world like this , in which the same antecedents never again concur, and nothing ever happens twice .... The physical axiom which has a somewhat similar aspect is " That from like antecedents follow like conse­quents .· ' But here we have passed ... from absolute accuracy to a more or less rough approximation. There are certain classes of phenomena, as I have said, in which a small error in the data only introduces a small error in the result. Such are, among others, the larger phenomena of the Solar Sys­tem, and those in which the more elementary laws in Dynamics contribute the greater part of the result. The course of events in these cases is stable. There are other classes of phenomena which are more complicated, and in which cases of instability may occur, the number of such cases increasing, in an exceedingly rapid manner, as the number of variables increases.

It is our view that Maxwell was the first person to understand chaos-that is, he recognized the existence and importance of systems with "sensitive dependence to initial data."

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But prior to the advent of graphics computers, mathematicians and scientists found it difficult to apply Poincare's ideas to most of the systems of equations studied in the sciences. Thus, for a time Poincare's approach led to the cautious, rigorous study of a variety of specific sysJems with none of the bold, intuitive, philosophical generalizations that Max­well and Poincare felt were justified.

While Maxwell felt his ideas would be most applicable to systems with many variables, we believe, as stated earlier, that he also would readily have understood billiard-like problems with few degrees of freedoin. He seems to have thought in terms of individual particles, as with his well-known discussion of "Maxwell's de­mon." We also believe he would have understood the chaos of two elastically colliding hard spheres in a box, as suggested by his discussion of collisions of two bodies.

Poincan!'s .approach to chaos emphasized problems with a small number of degrees of freedom. He recognized that even in problems of celestial mechanics, the behavior 'of solutions was incredibly complicated. He investigated in particular the special three-body problem in which two massive booies revolve in a circle while the third body has zero mass. I I He showed that complex behavior (both sensitivity to initial data and infinitely many periodic orbits) oc­curred when there are nontrivial intersections of the stable and unstable manifolds of a fixed point of a map. This criterion is applicable in general, and is often used to guarantee that there are compact invariant sets exhibiting sensitivity to initial data (horse­shoes).

Poincare also understood that sensitivity to initial data applies to some systems with many degrees of freedom. In explaining that the unpredictability of the weather stems . from instabilities, Poincare asks rhetoricallyY .

Why is it that showers and even storms seem to come by chance, so that many people think it quite natural to pray for rain or fine .weather, though they would consider it ridiculous to ask for an eclipse by prayer?

Today scientists can readily compute stable and unstable mani­folds numerically with high precision to determine if they intersect, and we know they do for many systems. But prior to the advent of graphics computers, mathematicians and scientists found it difficult to apply Poincare's ideas to most of the systems of equations studied in the sciences. Thus, for a time Poincare's approach led to the cautious, rigorous study of a variety of specific systems with none of the bold, intuitive, philosophical generalizations that Maxwell and Poincare felt were justified. Was this absence due to the tendency of mathematicians not to speculate in writing? Indeed, Maxwell's and Poincare's scientific writings avoid these specula­tions.

Maxwell concludes his 1873 essay by encouraging his fellow scientists not to limit themselves to the study of nonchaotic models for physical systems. 13

If, therefore, those cultivators of physical science ... are led in pursuit of the arcana of science to the study of the singularities and instabilities, rather than the continuities and stabilities of things, the promotion of natural knowledge may tend to remove that prejudice in favour of determinism which seems to arise from assuming that the physical science of the future

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is a mere magnified image of that of the past.

When one studies the textbook solutions of Newton's equations for bounded dynamics, we see only steady state and periodic or quasiperiodic motions. One might be led to ask why life is so unlike these stable solutions in the texts. Maxwell, at least, viewed chaos as a key to life. 14

All great results produced by human endeavor depend on taking advantage of these singular states when they occur.

There is a tide in the affairs of men Which, taken at the flood, leads on to fortune.

Acknowledgments. We thank P. Holmes and S. Brush for their thoughtful comments on earlier drafts of this article and S. Brush and C. Grebogi for their help in locating some of the quotations included here.

Footnotes 1 James Clerk Maxwell. "Illustrations of the dynamical theory of gases,"

Philosophical Magazine 19 (1860); in The Scientific Papers of James Clerk Maxwell, W. D. Niven. ed., Dover, New York (1965), pp. 378-379. This article also appears as Document 3-6 in Maxwell on Molecules and Gases, Elizabeth Garber, Stephen G. Brush, and C. W. F. Everitt, eds., MIT Press. Cambridge, Mass. (1986), quotation on pp. 287-288.

2Ibid., p. 379. 3Ibid., p. 380. '1iere vis viva essentially means kinetic energy (to be precise, twice the

kinetic energy). 5Re f. I , p. 392. 6Quoted in Lewis Campbell and William Garnett, The Life of James Clerk

Maxwell, Macmillan, London (1882); reprinted by Johnson Reprint, New York (1969), p. 440.

7Maxwell often uses phrases like "infinitely small," which we interpret loosely.

8Henri Poincare, Science et methode, Flammanon, Paris (1908); transla­tion by Francis Maitland. Science and Method, Dover, New York (1952), p. 68.

9Ref. 6, p. 441. l'1bid., p. 442. . IIHenri Poincare, Sur Ie probleme des trois corps et les equations de la

dvnamique, Acta Mathematica 13 (l8W, pp. 1-270. p 0

-Ref. 8, p. 68. I~ef. 6, p. 444. 14rbid., p. 443. Maxwell quotes the last two lines from Shakespeare.

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