theorem such a probability measure is invariant in time. Thus the only reason time averages could be different from ensemble averages would be a lack of ergodicity in the flow. In the case of a system consisting of only one species of indistinguishable particles, this potential difficulty is suppressed first by averaging over many initial conditions (so that even if the flow is not ergodic, the starting points may fall in different “ergodic” subregions) and second by measuring time-average values of macroscopic, not microscopic, variables. The chances that under these circumstances one would observe a difference between the predictions of statistical mechanics and experiment are very slim (recall the laws of large numbers), and indeed under these experimental conditions the predictions of classical statistical mechanics enjoyed great success. This explains the utter confidence of most physicists in the predictive power of statistical mechanics and their dismissal of the ergodic hypothesis as a technical, probably irrelevant detail. On the other hand, suppose one uses the theory to make predictions about a diatomic gas, which even under the most simplifying assumptions has at least two species of indistinguishable degrees of freedom, say vibrations and translations. Without invoking the ergodic hypothesis, I can think of no a priori reason for the contributions to the specific heat of these two types of motions being found equal in typical measurements. In fact, even if the ergodic hypothesis is true, it is possible that the coupling of these two types of motions is so weak that during typical times of observation they do not reach equilibrium with each other. Yet it was the assumption that the two types of motion are in equilibrium that led to the discrepancy between classical statistical mechanics and experiment. Therefore I feel that it is unjustified to rely upon the many successes of statistical mechanics to dismiss questions regarding its foundations. On the contrary, an understanding of the ergodic hypothesis and especially of the times involved for exciting certain degrees of freedom should be equally challenging for the mathematician and the physicist. Quantum Mechanics: A Case of Mistaken Identity? I would like to close this brief review of these complicated and long-standing problems with some speculations about a possible connection between the ergodic hypothesis and the necessity of using quantum mechanics at the microscopic level. First a few words about the blackbody radiation law. I have tried to emphasize the importance of measuring macroscopic averages, as well as that of particle indistinguishability, in obtaining agreement between the predictions of statistical mechanics and experiment. I think the case of the blackbody falls outside this realm. Consider a cubic lattice in D dimensions. At each site let there be a particle sitting in some anharmonic potential, attached through harmonic springs to its 2D nearest neighbors. If the boundary conditions are periodic, the system consists of identical yet distinguishable (by site coordinates) particles. We could form macroscopic quantities by averaging over the positions or velocities of all the particles in a cube of macroscopic size and expect reasonable agreement with the predictions of statistical mechanics. Alternatively we could describe the system in terms of its normal modes and attempt to verify the classical prediction, namely, the RayleighJeans energy distribution shown in Fig. 2 (that is, the equipartition of the energy among all the normal modes). Many such studies have been performed numerically, the first being the celebrated 1955 work of Fermi, Pasta, and Ulam (see "The Fermi-Pasta-Ulam Problem"). It is always found that at sufficiently low energy density, the distribution of energy among the modes of the lattice differs drastically from the statistical prediction and in fact depends upon the initial conditions. Obviously either these systems are not ergodic, or at least the times of thermalizing the different modes are much longer than a typical time of numerical integration. And no macroscopic averaging is available to save the day! It is also known that leaving the energy density fixed and refining the lattice (taking the continuum limit) increases the discrepancy (Patrascioiu, Seiler, and Stamatescu 1985). Although such results have been accumulating for over thirty years now, they are not yet understood. Some say the systems are so close to being integrable that KAM tori or very slow diffusion rates occur in the phase space. Others claim that statistical mechanics should hold only in the thermodynamic limit (which is clearly not attainable numerically). Most physicists dismiss the whole story, since they “know" that statistical mechanics works in real life. I think this is a very narrow point of view: the problem being discussed is very much like that of the blackbody radiation law, and that was one of the failures of classical statistical mechanics. Is there a good theoretical (dynamical) basis for predicting the Rayleigh-Jeans distribution in classical physics, as the standard textbooks claim? Or are we pushing the statistical predictions in a domain for which there is no reason to expect them to hold? In “Does Equipartition of Energy Occur in Nonlinear Continuous Systems?” I describe some numerical experiments I have performed to test the validity of the statistical-mechanics predictions for a one-dimensional version of the blackbody problem and for the specific heats of systems with more than one species of degrees of freedom. Notably I found that, over the times of observation available in computer experiments, the systems failed to fulfill the ordinary expectations of an equipartition of energy. The same discrepancy has been found in many other numerical experiments. It is well known that the resolution of the above-mentioned experimental difficulties of statistical mechanics (specific heats and blackbody radiation) was found in abandoning the classical approach to physics in favor of the quantum one. As mentioned in the introduction, this revolution has had an unqualified experimental success, although it has raised serious epistemological questions, which continue to haunt us more than sixty years after the advent of the quantum theory. I would like to give a brief outline of a heresy that I have advocated for a few years now (Patrascioiu 1983), one directly connected to the ergodic hypothesis. As I mentioned earlier, if one contemplates a dynamical basis for statistical mechanics, one is faced with a real dilemma. The accepted formulation of the electromagnetic and the gravitational interactions demands that, in essence, everything in the universe interact with everything else. (This is so because of the long-range nature of these interactions.) In fact, the notion of an isolated object (or even system) is clearly an abstraction without any a priori physical basis, since ultimately everything is coupled to everything else through the electromagnetic and gravitational fields. All we can hope is either that the ergodic hypothesis is strictly false or that the times needed to excite certain degrees of freedom are so large that we can ignore them under some circumstances. In either case certain prejudices that have been passed from generation to generation should be abandoned and their bases be opened for investigation. For instance, in the absence of a dynamical calculation, there is no basis to claim that Planck's distribution for blackbody radiation is irreconcilable with classical electromagnetism. (In fact, the distribution found numerically and shown in Fig. 2 of the sidebar very much resembles Planck's law.) continued on page 278 Does Equipartition of Energy Occur in Nonlinear Continuous Systems? The celebrated work of Fermi, Pasta, and Ulam was the first of numerous attempts to study the distribution of energy in nonlinear continuous media. These attempts have all been indirect in that the systems are simulated by lattices of particles interacting through nonlinear potentials. The results have consistently failed to support the classical point of view regarding equipartition of energy—and yet they have stirred little excitement in the physics community. Perhaps this is so for two reasons: (i) the systems analyzed may be subject to an infinite number of conservation laws (and thus may be effectively linear), so that the individual degrees of freedom are not coupled and equipartition of energy cannot occur; (ii) the results may simply be artifacts of the lattice simulations. Here I present some results from two of my own studies, the first of a onedimensional model of the blackbody problem (Adrian Patrascioiu, Physical Review Letters 50(1983): 1879) and the second of a three-dimensional system that may give insight into the specific heats of systems with two species of degrees of freedom, such as the rotations and vibrations of diatomic molecules (K. R. S. Devi and A. Patrascioiu, Physica D 11(1984): 359). In the case of blackbody radiation, the continuous medium (the electromagnetic field) is linear. Nonlinearity is introduced into the problem through the interaction of the field with the atoms in the walls of the cavity. Let us investigate a one-dimensional version of this problem, two nonlinear oscillators (particles and nonlinear springs) interacting through a linear string (Fig. 1). The string represents the electromagnetic field, and the oscillators represent the atoms. This model has the advantage that the string can be treated exactly so that no spatial lattice is needed. The string and the particles move in the z direction only. The equation of motion for the string is 222(x.0) a z(x, t) = 0, for x # #1, (1) at2 Әr 2 UNEQUAL PARTITION OF ENERGY of two coupled ordinary differential equations for the functions f and g. The excitation of the string at 1 = 0 was specified by setting f (x) = a sin(wx+2/2) and g(x) = 0. The differential equations were integrated numerically, and conservation of energy was used to verify the accuracy of the calculations. I would like to emphasize what outcome one would predict by following the same line of thought used to derive the Rayleigh-Jeans formula. The system, being nonlinear and (probably?) sufficiently complicated, will wander with equal probability throughout its phase space of given total energy. Let us choose initial conditions such that the total energy is finite. If ensemble averages and time averages are equal for this microcanonical ensemble, that is, if Fig. 2. Typical results for the distribution of energy among the normal modes of the string in the one-dimensional model of the blackbody problem (see Fig. 1). The exact shape of the energy distribution depends on the values assigned to various parameters, but in all cases the distribution was similar to a Planck distribution (see Fig. 2 of the main text) and was never flat, as it would be if the energy were partitioned equally among all the normal modes. then the time-average kinetic energy of either particle should tend to zero for any initial conditions since the number of degrees of freedom is infinite. Over my times of observation, this did not seem to be the case! Under the assumption that the times of observation were sufficiently long, this result indicates that the microcanonial measure (Eq. 7 in the main text) is not applicable. We are left with two possibilities: (i) the motion of the system is quasiperiodic, or (ii) the phase space is broken into an infinite number of ergodic cells of finite size. I also investigated the distribution of energy among the normal modes of the string. Figure 2 shows typical results for the time-average values of the fraction of the string energy in the nth normal mode. In all the runs performed the distribution of energy of the string among its normal modes is highly peaked (like the Planck distribution) and shows no tendency to become flat. Its shape does depend on the values of the various parameters in the problem and on the initial conditions. If all the parameters are kept fixed and the total energy is increased, the peak broadens. The shape of the distribution also varies with the frequency chosen for the initial excitation of the string, remaining constant over some range of w and then jumping to a new shape. The results of this study raised naturally several questions: (i) Was the observed unequal partition of energy among the normal modes of the string (the continuous medium) related to the one-dimensional nature of the medium? (ii) The unequal partition of energy reflected in the specific heats of diatomic gases results from motions of particles (rather than motions of a field, as in the case of blackbody radiation). Can this phenomenon be reproduced in a classical dynamical system? To help answer these questions, Devi and I performed a study of a threedimensional version of the system shown in Fig. 1. This system included four particles and six strings (Fig. 3). Our results exhibited several notable features over the times of observation: (i) time averages of, for example, total energies of particles and strings seemed to reach their asymptotic values; (ii) unequal partition of energy among the normal modes of the strings persisted, and the distributions obtained were reminiscent of that given by Planck's law; and (iii) for a variety of initial conditions, the four particles did not achieve the same average kinetic energy, a situation similar to the unequal partition of energy between the vibrational and the rotational degrees of freedom of diatomic gases. The fact that we obtained these types of results using several nonlinear (spring) potentials suggests their generality. I 2(x,1) Nor is there any basis to the claim that the classical atom is inevitably unstable because of the "ultraviolet catastrophe" (escape of all of the energy into the ultraviolet modes of the electromagnetic field, as required by the equipartition-of-energy principle of classical statistical mechanics). After all, maybe classical electromagnetism leads to a nonergodic flow (if the notion of ergodicity makes sense at all for a continuous medium) or maybe the diffusion of energy to the high modes is so slow that it has not occurred appreciably in the twelve to eighteen billion years since the big bang. That such slow diffusion is not a far-fetched supposition follows from some results obtained in the last few years. Since point charges have infinite self-energies, let us spread them by introducing a charged scalar (zero-spin) field. It has been shown rigorously that, in a certain gauge (axial), the system of coupled nonlinear equations describing the interaction of the classical electromagnetic field with this classical charged field has finite-energy-density solutions for all times. Moreover, these solutions retain their initial smoothness (number of derivatives). Using this latter property one can show that after an arbitrarily long time of evolution, an infinite number of normal modes of these fields are arbitrarily close to their initial energies (Patrascioiu 1984). Whereas there is no guarantee that this model captures the true physics in the universe, it seems hard to imagine a field whose modes thermalize in a finite amount of time. So perhaps quantum mechanics is nothing more than classical statistical mechanics done the right way in a universe filled with particles interacting primarily via electromagnetic and gravitational forces. If so, its mysteries should be understandable once the complicated Brownian process produced by particles constantly absorbing and emitting radiation is mastered. While this scenario may seem far-fetched to many, I think it arises inescapably from contemplating the foundations of statistical mechanics. It does not contradict the experimentally observed violation of Bell's inequality unless the latter persists for truly space-like settings of the magnets. It has epistemological value and would, for example, allow the computation of the fine-structure constant and its variation with temperature (Patrascioiu 1981). In conclusion, I think neither physicists nor mathematicians should close the book on the venerable problem of the ergodic hypothesis, and I guess some big surprises may be in store once the problem is better understood. • |