Waking from Dreams of Determinism

By Hao Bai-lin

In the latter half of the 19th century, physicists had come tantalizingly close to the discovery of chaos, in its modern scientific sense. Poincaré approached it in his work on the stability of the Solar system. So too did Claude-Louis-Mari Navier and George Stokes in their research on turbulence. But Newtonian determinism still dominated the thinking of physicists. And interest in quantum mechanics and Einstein's theory of relativity tended to absorb the efforts of most physicists until recent decades. Theories of dynamical systems were left largely to the mathematicians. In his paper for the Tokyo symposium on chaos, Dr. Hao Bai-lin of the Institute of Theoretical Physics, Academia Sinica in Beijing, discussed some of the historical roots of the interplay between the two disciplines in the development of chaos theory. -Editor

In order to describe one and the same Nature, there have been two systems of description in the physical sciences: deterministic and probabilistic. Celestial mechanics, based on Newton's laws and the universal law of gravitation, had served as the touchstone for determinism. The deterministic view culminated on the night of 23 September 1846, when the German astronomer J.G. Gale discovered the planet Neptune at the first attempt, according to the theoretical prediction of the French astronomer U.-J.-J. Le Virrier. This was a triumph for the celestial determinism espoused a half-century earlier by the astronomer Pierre Laplace: physics seemed to be really able to predict the future of the universe, provided the present status was known with sufficient precision.

However, in approximately the same period, the development of thermal engines had stimulated the study of the property of gases and fluids. Here people were using macroscopic notions like pressure, temperature and volume and looking for the empirical relations between them. The laws of thermodynamics, based on an enormous collection of empirical facts, were at least as good as the laws of mechanics. The attempt to justify them by considering the elementary processes taking place in macroscopic bodies had led to the establishment of statistical mechanics.

A key point in statistical mechanics was the introduction of probabilistic elements. Traditionally, physicists preferred the deterministic to the probabilistic approach; they employed the latter only reluctantly and unwillingly. They were forced to do so when there was no better alternative, due to the complexity of the problem and the lack of knowledge or data.

Inheriting Old Tradition

The 20th century inherited this tradition. Although quantum mechanics challenged classical determinism, the probabilistic concept has only been invoked in a rather pragmatic way in interpreting and relating theoretical calculation to experimental observation. The basic equation of quantum mechanics remains linear and entirely deterministic. Consequently, determinism continued to thrive in physics, with Albert Einstein being the standard-bearer.

Although by the end of the last century, Henry Poincaré already recognized the random behaviour inherent in Newtonian mechanics, and many mathematicians complemented Poincaré's thesis, many physicists remained ignorant about these developments.

As late as 1981, A.S. Wightman complained that "... it is really a pedagogical scandal that after more than three quarters of a century the simple and enlightening ideas of Poincaré do not appear in elementary mechanics books written for physics students.... One might almost think... that there is a conspiracy against Poincaré."

Awakening from a 300-Year-Old Dream

No wonder then that mathematicians like Jim Yorke and ecologists like Robert May, among others, were destined to wake the physicists from their 300-years dream of determinism. In the beginning many physicists took chaos for randomness and turned their backs. Initially, only a bunch of not-so-orthodox physicists grasped the new development with any enthusiasm.

Once convinced of the existence of chaos, however, physicists began to see it everywhere. They discover chaotic behaviour in laboratories, in computer experiments, and in observation of natural processes. The explosion of literature on chaos was to a great extent due to the efforts of physicists. (In Zhang Shuyu's Bibliography on Chaos, published in 1991 by the World Scientific Publishing Company, more than 7,000 titles are listed, of which about 2,800 contain the word "chaos" or "chaotic" or "strange attractor.")

"Just Playing with Mathematics"?

When the physicists started to dig into chaos, they began to extend their vocabulary with terms which have been in use basically among mathematicians - terms like hyperbolicity, stable and unstable manifolds, homoclinic and hetereoclinic intersections, Smale's horseshoes and symbolic dynamics.

"Aha! You are just playing with mathematics," some skeptics say of their colleagues who have been deeply involved in this new game. But there is nothing strange in the situation. In the early 1920s, few physicists knew about matrix and eigenvalue problem, not to mention linear operators in Hilbert space - all notions that are taught to every physics-major nowadays.

The study of chaos emerged in the wake of breakthroughs in our understanding of phase transitions - changes from solid to liquid, from nonmagnet to magnet and so on - where the transformation can be sudden and discontinuous, the hallmarks of chaotic behaviour. Here the notion of scaling invariance and universality played a decisive role - the concept essentially that certain physical qualities have the same value, independent of any model or equation. Physicists have now mastered the skill of renormalization group techniques and are able to calculate various universal critical exponents from the linearized renormalization group equations. In this way, they have recognized structures known to mathematicians for many years - such as period-doubling cascades. A pioneering role in this area was played by our colleague at this symposium, Mitchell Feigenbaum.

Help to Other Scientists

Physicists have also been active in popularizing the new ideas in chaos theory for possible use by other branches of science. Take, for example, meteorology. Chaos has not diminished the forecasting power of science (as one might take literally from Lorenz's "butterfly effect"); it has rather dispelled any illusion about long-term forecasting powers. In fact, the study of chaos improves short-term predictions by incorporating dynamical aspects of the underlying processes. At the same time, it provides a better long-term forecasting of averaged quantities by making use of various invariant characteristics.

Speaking about the popularization of mathematical ideas, I would like to cite a little personal experience. We had found some rather complicated bifurcation and chaos "spectra" in the parameter space of several systems of ordinary differential equations. We were trying hard to understand the global structure of the parameter space and to find the systematics of periodic solutions which could be determined with confidence in numerical experiments. In so doing, we came across an old mathematical subject, namely symbolic dynamics, which has been in use in dynamical systems theory and ergodic theory since the 1920s. We soon realized that symbolic dynamics is simply what physicists call "coarse-graining," and it is a rigorous way to describe dynamics with finite precision. Therefore, it gets along with the spirit of physics very well. In fact, symbolic dynamics can be cast into a useful tool for practitioners in physical sciences.

The Problem of Quantum Chaos

In less than 20 years the "epidemic" of chaos has swept over almost all subjects of classical physics. Quantum mechanics, however, seems to have been more or less immune to the new disease.

The typical motion of a quantum system is not chaotic, but there may be some manifestation of classical chaos in finite time or frequency range - or, as some people put it, there may be a quantum signature in classical chaos. In this context, it might be appropriate to recall a comment made by Max Born, the German-British physicist who shared the 1954 Nobel Prize for his work on the mathematical basis of quantum mechanics. Born pointed out that, measured in its own natural units of time, the microscopic world, say, an atom, is much more long-living than the macroscopic world, e.g. the Solar system.

Having this comment in mind, we see the contrast: in the short-lived systems, we speak about infinite time limit and chaos, but in the eternal microworld there is no chaos. The implication of this contrast has yet to be fully understood.

New Physics in Chaos?

This leads us to the question: Is there new physics in chaos? The totality of our physics knowledge may be represented by the volume enclosed in a shuttle-shaped surface (see diagram this page). The two sharp tips of the shuttle represent the two generally recognized "frontiers" of physics: on the one hand the study of the microworld; on the other, the exploration of the universe at cosmological scale. Both require more and more sophisticated and expensive equipment which has made these frontiers a privilege only for a selected community of scientists. However, the majority of physicists has been working at the much wider real frontier, namely the investigation of the macroworld, where we all live and where success in basic research has, in general, a quick positive feedback to the society.

In dealing with this macroworld, we are facing the problems of complexity. Indeed, the study of chaos has opened new horizons in the exploration of complexity. One of the most instructive morals we have learned from chaos consists in the understanding that seemingly complex behaviour or spatial patterns, or the combination of both, may turn out to be the results of repeated application of simple elementary rules or actions.

Who dares say that there is no new physics in chaos? It is more tempting to say the contrary, namely, chaos not only brings about new physics, but also calls for a re-examination of the fundamental principles of physics.