6—
On the Ergodic Behavior of Dynamical Systems:
(LA-2055, May 10, 1955)

This is a lecture which is included in a series of lectures on the physics of ionized gases given in 1955. It presents ideas and remarks on general properties of ionized gases connected with the "Sherwood" project. "Sherwood" was one of the early attempts to use fusion for the peaceful production of energy by the confinement of thermonuclear reactions. (Author's note.)

The purpose of this lecture is to review the present status of the socalled ergodic hypothesis, summarize the mathematical results of the last twenty years or so, and indicate briefly the nature of the difficulties that still remain in applying the general theorems to specific physical situations.

As has been pointed out in previous lectures the ergodic hypothesis can serve as a fundamental point on which to base the entire structure of statistical mechanics. (See, e.g., the derivation of the H-theorem from the ergodicity assumption by ter Haar.) My own feeling, to anticipate the conclusions of this lecture, is that the mathematical work of the last twenty years has brought complete rigor to only a small part of the theory of statistical mechanics, "the equivalent of the first twenty pages or so of a standard book on the subject." One could say here that, as is often the case, the mathematicians know a great deal about very little and the physicists very little about a great deal. The ergodic theorem itself and the subsequent proof of existence and more: the prevalence of ergodic transformations among all volume or measure preserving flows assert, roughly speaking, the legitimacy of assumptions one makes in physical theories about the limiting or equilibrium states of physical systems. The next question of importance

equally essential for our understanding of statistical mechanics, concerns the rate of approach to the equilibrium, if this approach, indeed, does take place. This problem seems much more difficult and only very incomplete results exist. In the second half of this lecture, I shall mention some recent results obtained with Fermi and Pasta on the would-be approach to equilibrium in certain simple nonlinear systems. These were obtained on the calculating machine here in Los Alamos.

To start with, we have a space located in a Euclidean space E of 6n-dimensions. This space is the phase of a dynamical system which we shall assume for most of the talk to be conservative. The Hamilton equations define a flow in the space E. This flow preserves the volume in the space E. This is the theorem of Liouville and it follows directly from the Hamilton equations. The measure or volume in the space is preserved exactly, not only infinitesimally to the first order, but, of course, for volumes of any finite size. I shall not discuss here the definition of the volume in the space E. It is obtained from the ordinary Euclidean volume in the 6n-dimensional space by the most elementary geometrical considerations. The space E represents the entire available phase space of the dynamical systems. It is divided onto a one-parameter family of subspaces Ek, corresponding to fixed values k of the entire energy of the physical systems. Each of these subspaces, of one less dimension than E, undergoes a flow into itself. Each of these spaces has its own volume and this volume is also preserved under the dynamical flow. It is important to note that, in special cases, each of these spaces Ek may again be decomposed into a family of subspaces of still lower dimensions, each of which flows into itself. Indeed, if there are further integrals of the given dynamical problem in addition to the integral of energy, we shall obtain such further decompositions.

The ergodic theory deals with the asymptotic properties of a flow (volume-preserving in such spaces). One is interested in the behavior of trajectories of single points under the given flow. A single point represents one possible initial condition of the dynamical system. That is to say, the position of the n-particles at time t = 0 and all the velocities at this time. As time proceeds, this representative point will describe a line in the phase space. One is interested in how this line behaves in the given space Ek (or in case of existence of further integrals, in the "irreducible" subspace of it, Ek).

It is simpler if only for typographical reasons to consider, instead of a continuous flow in phase space, one mapping and a discrete sequence of time intervals. That is to say, instead of the family of transformations TA where A is any real number, we consider Tⁿ which means we look at the flow at intervals of one "second" each. Of course, we have in both cases either T^X+ L =TA(T/) or Tm+n =Tm(Tn). These relations

merely express that we have a one-parameter group flow or a sequence of powers of one transformation. All the mathematical results proved so far are valid in both formulations. We shall deal with the discrete one here.

The so-called ergodic theorem proved by G. D. Birkhoff asserts the following: the time averages of functions f(Tn(p)) exists for almost every point p if T is a volume preserving transformation of a space on which a measure or volume is defined and f any integrable function. In particular (which is equivalent to the more general statement) it is true that if T is a transformation of the above sort eist fo fATⁱ (p) lim n-oo n i=l

exists for almost every point p. fA is the characteristic function of any set A in our phase space. That is to say, f(p) = I if p does, f(p) = 0 if p does not belong to A. The sum written above merely counts how many of the first n-iterates of the point fall into the set A. The theorem asserts the existence of a sojourn time for almost every point p in any volume A in phase space. This theorem is necessary to have in order to formulate rigorously the Boltzmann hypothesis according to which this sojourn time is equal to the relative volume of the region A in phase space. So, at least the existence of the sojourn time has been proved. (It should be pointed out here that a somewhat weaker form of the theorem which we have just stated, a so-called weak ergodic theorem, was proved several months before Birkhoff by John von Neumann. His theorem asserted the convergence of our sums in the mean and not for almost every point p, as did Birkhoff, which latter is a stronger statement. The difference, however, may be of less importance to physicists than to mathematicians.)

Birkhoff noticed also that the hypothesis of Boltzmann is equivalent to the following property of the transformation T. There is no subregion of subset E' C E which has positive measure less than the measure of the whole space and which goes into itself under the transformation T. Such transformations T are called by him metrically transitive.

One might say that the result of Birkhoff, as far as applications of the ergodic theorem are concerned, shifted the emphasis from the existence of limits to the search for transformations which would be metrically transitive and so satisfy the Boltzmann hypothesis. Only very special such transformations were known and only on very special manifolds E. For example, if E should be the circumference of a circle

in one-dimension and T a rotation of it through an angle a, irrational with respect to the length of the circle, then the famous results of H. Weyl on equipartition of numbers n, a modulo 1 (circumference was length 1) establish the metric transitivity in this case and the ergodic theorem with it.

The generalization of this to n dimensions has the form of the wellknown theorem of Kronecker about vectors of the form (kai,ka₂ , ... an), k = 1,2,..., where the a's are rationally independent of each other. Another case known was that of a flow along geodetic lines on a surface of two dimensions and of constant negative curvature. This result asserted the transitivity of such a flow, i.e., its ergodic properties, and was established by Hopf, Hedlund, and others.

In 1941, a rather general result was established by Oxtoby and the speaker,l and can be described as follows: let E be a manifold in any number of dimensions with the volume defined for its subsets. Consider all possible continuous and volume-preserving transformations of such a manifold E into itself. Most of such transformations will be metrically transitive, that is to say, satisfy Boltzmann hypothesis. The expression "most" is defined rigorously and, in particular, implies that arbitrarily near to any transformation T, given in advance, one will be able to find transformations which are metrically transitive. (Two transformations S and T are said to be within an e of each other, if for every point p, S(p) and T(p) are within the same e.) I shall not go into the technical definition of the word "most," etc., but would like to stress that the above result shows the existence of ergodic transformations on any manifold (e.g., sphere, ellipsoid, etc.), in any number of dimensions and what is more, the prevalence of such transformations among all possible ones. It is important to stress here that the transformations which are given by actual dynamical flows defined by Hamilton's equation could, nevertheless, form exceptions. An analogy: most real numbers are transcendental, but, of course, there exists a "small minority" of algebraic numbers. The question is still open whether or not the actual dynamical transformations are, in general, metrically transitive, i.e., ergodic. Our result above merely makes it very probable or, if one may say so, the probability a priori is equal 1 that most of these transformations will be ergodic. It is certainly true that every flow

may be perturbed arbitrarily little to become ergodic. The actual criteria, given a dynamical transformation, are, however, still lacking.

In what now follows, I shall discuss a number of examples of physical systems where the ergodic behavior seems a priori inevitable and which were considered during the last two years by E. Fermi, John Pasta, and the speaker. Numerical computations were performed on the MANIAC here in Los Alamos. The results were most unexpected to

all of us mainly because, as we shall see, they show a certain hesitancy on the part of these particular physical systems to behave ergodically, or, more exactly, the times taken for mixing or equipartition seem to be unduly long, if not infinite.

Before we take up these examples, we make two more remarks about the generality of the ergodic behavior of dynamical flows in phase space:

1) The subject of this seminar is mainly the study of behavior of charged particles or plasma in the presence of magnetic fields. The discussion given above applies to the case of motion of charged particles in a fixed given magnetic field if one neglects the changes in the field due to the motions of the particles themselves. A system of such particles acted upon by the external field and their mutual electrostatic interactions "should" behave ergodically independently of given constraints in form of walls, etc. The ergodic behavior has to be understood relative to the phase space after it has been reduced by a number of integrals of the motion imposed by such constraints.

2) It may be worthwhile pointing out here that the proof of the prevalence of ergodic flows among all possible volume preserving continuous flows as given in the paper quoted above, establishes more than equality of the time and space averages for most transformations. The construction used in the proof exhibits the general transformation as having a "turbulent" character. That is to say, a roughly periodic motion which does not, however, exactly close the orbits of points in phase space, but feeds a large periodic motion into smaller "rotational" motions which in turn do not have the orbits quite closed but feed still smaller rotations in turn and this process continues indefinitely. Fourier analysis in n dimensions of such a transformation which is the

general one would show the feeding of vortex-type flows into successively smaller vortices. This fact had not been noticed or exploited by the authors at the time the paper was written but may be useful in discussing a statistical mechanical type of treatment for motions of fluids; that is to say, systems with infinitely many degrees of freedom which, in general, it is believed tend to become turbulent. Parenthetically, we may add here that a statistical mechanical type of treatment of systems with infinitely many degrees of freedom is required if one wants to include the radiation effects due to the motion of charges. The magnetic field itself would have, of course, infinitely many degrees of freedom.

The problem which was studied numerically on the MANIAC is the following: we have a continuous string with ends kept fixed and with

forces acting between its elements which are nonlinear as functions of displacements. This continuous string is replaced, for the purpose of the computations, by a finite number of points, 64 in our actual work, and the equations describing its motion were: Xi = a(Xi+l + Xi- - 2xi) + 0[(Xi+l - Xi)² - (i - i-₁)² ] i = 1,2,...,64. or i = a(xi+l + Xi-i - 2xi) + 7[(i+l - xi) - (xi - xi-) ^3] i= 1,2,...,64.

d3 and y were chosen so that at the maximum displacement xi, the nonlinear term, was small, e.g., of the order of one-tenth of the linear term. The corresponding partial differential equation obtained by letting the number of particles become infinite is the usual wave equation plus nonlinear terms of a complicated nature. There seems to be, of course, very little hope for obtaining explicit solutions and what was done was to run a great number of special problems.

These were studied as follows: the initial position of the string, that is to say, the distribution of the x₁ at time t = 0 was assumed to be in a form of a single sine wave or in some other simple forms, e.g., a sum of two sine waves of low frequencies or triangular. Most problems were run under the assumption of constant masses (equal to 1, say) for each i. The position of this string and the total energy of it, kinetic plus potential, were analyzed during the course of each problem in Fourier series. That is to say, we studied the 64 Fourier coefficients Ak, the total energy residing in each mode Ek, k = 1, 2,..., 64, as functions of time. Since the problem is approximately linear, for times not very long the string vibrates periodically; the length of the numerical run of the problem was, in each case, equal to several hundred or more of what would be full vibration periods from the initial condition in the corresponding linear problem, if the nonlinear terms in the force had been neglected. Of course, each single vibration period in the calculation corresponded to a hundred or several hundred time cycles on the machine so the number of computation cycles in each problem

was several tens of thousands. The initial motivation for the problem was to observe how, in the course of time, the energy of the system, initially contained in the first or the first few modes, in time flows to other modes and to observe the rate at which equipartition of energy among all modes becomes established. This problem was to serve as

the first one of a series in a systematic investigation of the question of rates of approach to equilibrium, so important in many problems of statistical mechanics.

The actual results were somewhat surprising to us since, instead of what one expected-a continual and steady flow of energy, say from the first mode to all higher modes and an asymptotically uniform approach to equipartition--something entirely different seems to happen. For example, in the problem where the initial position was a pure sine wave and all of the energy was in the first mode, the behavior of the system was the following: initially, as predicted by Rayleigh's perturbation analysis, that is to say, the infinitesimal study for short t, the other modes grow in energy one by one, the first mode feeds the second, the first and second together the third, the second feeds the fourth and so on. This, indeed, was observed but later on it was only single modes, say, mode No. 3, that continued to grow in energy systematically and for many dozens of the periods of the vibration the higher modes not growing at all. Then the energy in the third mode was dropping steadily and mode No. 5 was increasing. The first few modes, that is 1, 3, 5, 7, were exchanging energy among themselves slowly with the higher modes not obtaining any sizeable contributions and after 30,000 cycles or 300 "would be" full vibrations of the string, the system came back within one percent of the total energy, to its original shape, that is to say, a pure sine wave.

This behavior seems to be typical in other cases, too. It is the first few modes that exchange energy among themselves in a somewhat erratic fashion but it is always one or the other or very few of them that seem to predominate and, far from a tendency towards equipartition among all modes proceeding steadily, one sees an almost periodic exchange between the low modes. In the case of initially triangular shape where modes 1, 3, 5, 7, are mainly involved in the beginning, again these played among themselves and the string did not show, up to times where the computation was stopped, any tendency to become really turbulent or energy thermalized.

Another problem recently calculated was the following: instead of assuming the nonlinear term to be quadratic or cubic in displacement, we took this nonlinear part to be represented by a broken polygon imitating the shape of a cubic (and also small compared to the linear term). This was done because perhaps the quadratic or cubic form of the forces would introduce some analytic peculiarities which could possibly explain this almost periodic and non-mixing or only slowly mixing behavior, and here a non-analytic form of the force would possibly remove this special character and the unknown analytic reason for the unexpected behavior of the motion. The results were again very

similar. Starting with a single wave, only the first few Fourier modes exchanged energy among themselves. There seems to be one difference; the system does not come back to its original position, but again the first few modes only seem to exchange energy significantly. The behavior of this system thus may seem to serve as a warning against relying too much on the statistical arguments for an approach to equilibrium for systems of many degrees of freedom. These results cannot be discussed here in any detail, but a report on all the work done will be available shortly.* It was merely intended to point out the existence of cases where the estimate, a priori, based on the usual volume in phase space considerations and estimates of relaxation times seems quite inadequate. Problems which are nonlinear, but still are algebraic or "simple" in terms of the forces involved, may not be good examples of the general or random flows which Boltzmann or Gibbs had in mind but instead show an almost periodic behavior, a slow transfer of energy between the degrees of freedom. More generally, one might suspect, on certain mathematical grounds, the existence, instead of, as in the linear case, states of a system, the appearance of quasi-states between which the system oscillates. These quasi-states apparently need not form a continuum or be too dense, but may, approximately, consist of combinations of a few states in the corresponding linear problem.

Reference

1. J. C. Oxtoby and S. Ulam, "Measure-Preserving Horneomorphisms and Metrical Transitivity," Ann. Math., 40, 2, 874-920, 1941.

* This refers to the preceding report which was distributed after this lecture was delivered, but before this collection of lectures was issued. (Eds.)