What are the justifying foundations of statistical mechanics without appealing to the ergodic hypothesis?

Question

This question was listed as one of the questions in the proposal (see here), and I didn't know the answer. I don't know the ethics on blatantly stealing such a question, so if it should be deleted or be changed to CW then I'll let the mods change it.

Most foundations of statistical mechanics appeal to the ergodic hypothesis. However, this is a fairly strong assumption from a mathematical perspective. There are a number of results frequently used in statistical mechanics that are based on Ergodic theory. In every statistical mechanics class I've taken and nearly every book I've read, the assumption was made based solely on the justification that without it calculations become virtually impossible.

Hence, I was surprised to see that it is claimed (in the first link) that the ergodic hypothesis is "absolutely unnecessary". The question is fairly self-explanatory, but for a full answer I'd be looking for a reference containing development of statistical mechanics without appealing to the ergodic hypothesis, and in particular some discussion about what assuming the ergodic hypothesis does give you over other foundational schemes.

Answer 1

The ergodic hypothesis is not part of the foundations of statistical mechanics. In fact, it only becomes relevant when you want to use statistical mechanics to make statements about time averages. Without the ergodic hypothesis statistical mechanics makes statements about ensembles, not about one particular system.

To understand this answer you have to understand what a physicist means by an ensemble. It is the same thing as what a mathematician calls a probability space. The “Statistical ensemble” wikipedia article explains the concept quite well. It even has a paragraph explaining the role of the ergodic hypothesis.

The reason why some authors make it look as if the ergodic hypothesis was central to statistical mechanics is that they want to give you a justification for why they are so interested in the microcanonical ensemble. And the reason they give is that the ergodic hypothesis holds for that ensemble when you have a system for which the time it spends in a particular region of the accessible phase space is proportional to the volume of that region. But that is not central to statistical mechanics. Statistical mechanics can be done with other ensembles and furthermore there are other ways to justify the canonical ensemble, for example it is the ensemble that maximises entropy.

A physical theory is only useful if it can be compared to experiments. Statistical mechanics without the ergodic hypothesis, which makes statements only about ensembles, is only useful if you can make measurements on the ensemble. This means that it must be possible to repeat an experiment again and again and the frequency of getting particular members of the ensemble should be determined by the probability distribution of the ensemble that you used as the starting point of your statistical mechanics calculations.

Sometimes however you can only experiment on one single sample from the ensemble. In that case statistical mechanics without an ergodic hypothesis is not very useful because, while it can tell you what a typical sample from the ensemble would look like, you do not know whether your particular sample is typical. This is where the ergodic hypothesis helps. It states that the time average taken in any particular sample is equal to the ensemble average. Statistical mechanics allows you to calculate the ensemble average. If you can make measurements on your one sample over a sufficiently long time you can take the average and compare it to the predicted ensemble average and hence test the theory.

So in many practial applications of statistical mechanics, the ergodic hypothesis is very important, but it is not fundamental to statistical mechanics, only to its application to certain sorts of experiments.

In this answer I took the ergodic hypothesis to be the statement that ensemble averages are equal to time averages. To add to the confusion, some people say that the ergodic hypothesis is the statement that the time a system spends in a region of phase space is proportional to the volume of that region. These two are the same when the ensemble chosen is the microcanonical ensemble.

So, to summarise: the ergodic hypothesis is used in two places:

1. To justify the use of the microcanonical ensemble.
2. To make predictions about the time average of observables.

Neither is central to statistical mechanics, as 1) statistical mechanics can and is done for other ensembles (for example those determined by stochastic processes) and 2) often one does experiments with many samples from the ensemble rather than with time averages of a single sample.

Answer 2

As for references to other approaches to the foundations of Statistical Physics, you can have a look at the classical paper by Jaynes; see also, e.g., this paper (in particular section 2.3) where he discusses the irrelevance of ergodic-type hypotheses as a foundation of equilibrium statistical mechanics. Of course, Jaynes' approach also suffers from a number of deficiencies, and I think that one can safely says that the foundational problem in equilibrium statistical mechanics is still widely open.

You may also find it interesting to look at this paper by Uffink, where most of the modern (and ancient) approaches to this problem are described, together with their respective shortcomings. This will provide you with many more recent references.

Finally, if you want a mathematically more thorough discussion of the role of ergodicity (properly interpreted) in the foundations of statistical mechanics, you should have a look at Gallavotti's Statistical Mechanics - short treatise, Springer-Verlag (1999), in particular Chapters I, II and IX.

EDIT (June 22 2012): I just remembered about this paper by Bricmont that I read long ago. It's quite interesting and a pleasant read (like most of what he writes): Bayes, Boltzmann and Bohm: Probabilities in Physics.

Answer 3

I searched for "mixing" and didn't find it in other answers. But this is the key. Ergodicity is largely irrelevant, but mixing is the property that makes equilibrium statistical physics tick for many-particle systems. See, e.g., Sklar's Physics and Chance or Jaynes' papers on statistical physics.

The chaotic hypothesis of Gallavotti and Cohen basically suggests that the same holds true for NESSs.

Answer 4

I have recently published an important paper, Some special cases of Khintchine's conjectures in statistical mechanics: approximate ergodicity of the auto-correlation function of an assembly of linearly coupled oscillators. REVISTA INVESTIGACIÓN OPERACIONAL VOL. 33, NO. 3, 99-113, 2012 http://rev-inv-ope.univ-paris1.fr/files/33212/33212-01.pdf which advances the state of knowledge as to the answer to this question.

In a nutshell: one needs to justify the conclusion of the ergodic hypothesis, without assuming the ergodic hypothesis itself. The desirability of doing this has been realised for a long time, but rogorous progress has been slow. Terminology: the erdodic hypothesis is that every path wanders through (or at least near) every point. This hypothesis is almost never true. The conclusion of the ergodic hypothesis: almost always, infinite time averages of an observable over a trajectory are (at least approximately) equal to the average of that observable over the ensemble. (Even if the ergodic hypothesis holds good, the conclusion does not follow. Sorry, but this terminology has become standard, traditional, orthodox, and it's too late to change it.) The ergodic theorem: unless there are non-trivial distinct invariant subspaces, then the conclusions of the ergodic hypothesis hold.

Darwin (http://www-gap.dcs.st-and.ac.uk/history/Obits2/Darwin_C_G_RAS_Obituary.html) and Fowler (http://www-history.mcs.st-andrews.ac.uk/Biographies/Fowler.html), important mathematical physicists (Fowler was Darwin's student and Dirac was Fowler's), found the correct foundational justification for Stat Mech in the 1920s, and showed that it agreed with experiment in every case usually examined up to that time, and also for stellar reactions. Khintchine, the great Soviet mathematician, re-worked the details of their proofs (The Introduction to his slim book on the subject has been posted on the web at http://www-history.mcs.st-andrews.ac.uk/Extras/Khinchin_introduction.html), made them accessible to a wider audience, and has been much studied by mathematicians and philosophers of science interested in the foundations of statistical mechanics or, indeed, any scientific inference (see, for one example, http://igitur-archive.library.uu.nl/dissertations/1957294/c7.pdf and, for another example, Jan von Plato Ergodic theory and the foundations of probability, in B. Skyrms and W.L. Harper, eds, Causation, Chance and Credence. Proceedings of the Irvine Conference on Probability and Causation, vol. 1, pp. 257-277, Kluwer, Dordrecht 1988). Khintchine's work went further, and in some conjectures, he hoped that any dynamical system with a sufficiently large number of degrees of freedom would have the property that the physically interesting observables would approximately satisfy the conclusions of the ergodic theorem even though the dynamical system did not even approximately satisfy the hypotheses of the ergodic theorem. His arrest, he died in prison, interrupted the possible formation of a school to carry out his research program, but Ruelle and Lanford III made some progress.

In my paper I was able to prove Khintchine's conjectures for basically all linear classical dynamical systems. For quantum mechanics the situation is much more controversial, of course. Nevertheless Fowler actually based his theorems about Classical Statistical Mechanics on Quantum Theory, although Khintchine did the reverse: first proving the classical case and then attempting, unsuccessfully, to deal with the modifications needed for QM. In my opinion, the quantum case does not introduce anything new.

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Why measurement is modelled by an infinite time-average in Statistical Mechanics

This is the point d'appui for the ergodic theorem or its substitutes.

Masani, P., and N. Wiener, "Non-linear Prediction," in Probability and Statistics, The Harald Cramer Volume, ed. U. Grenander, Stockholm, 1959, p. 197: «As indicated by von Neumann ... in measuring a macroscopic quantity $x$ associated with a physical or biological mechanism... each reading of $x$ is actually the average over a time-interval $T$ [which] may appear short from a macroscopoic viewpoint, but it is large microscopically speaking. That the limit $\overline x$, as $T \rightarrow \infty$, of such an average exists, and in ergodic cases is independent of the microscopic state, is the content of the continuous-parameter $L_2$-Ergodic Theorem. The error involved in practice in not taking the limit is naturally to be construed as a statistical dispersion centered about $\overline x$.» Cf. also Khintchine, A., op. cit., p. 44f., «an observation which gives the measurement of a physical quantity is performed not instantaneously, but requires a certain interval of time which, no matter how small it appears to us, would, as a rule, be very large from the point of view of an observer who watches the evolution of our physical system. [...] Thus we will have to compare experimental data ... with time averages taken over very large intervals of time.» And not the instantaneous value or instantaneous state. Wiener, as quoted in Heims, op. cit., p. 138f., «every observation ... takes some finite time, thereby introducing uncertainty.»

Benatti, F. Deterministic Chaos in Infinite Quantum Systems, Berlin, 1993, Trieste Notes in Physics, p. 3, «Since characteristic times of measuring processes on macrosystems are greatly longer than those governing the underlying micro-phenomena, it is reasonable to think of the results of a measuring procedure as of time-averages evaluated along phase-trajectories corresponding to given initial conditions.» And Pauli, W., Pauli Lectures on Physics, volume 4, Statistical Mechanics, Cambridge, Mass., 1973, p. 28f., «What is observed macroscopically are time averages... »

Wiener, "Logique, Probabilite et Methode des Sciences Physiques," «Toutes les lois de probabilite connues sont de caractere asymptotique... les considerations asymptotiques n'ont d'autre but dans la Science que de permettre de connaitre les proprietes des ensembles tres nombreux en evitant de voir ces proprietes s'evanouir dans la confusion resultant de las specificite de leur infinitude. L'infini permet ainsi de considere des nombres tres grands sans avoir a tenir compte du fait que ce sont des entites distinctes.»

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Why we need to replace ensemble averages by phase averages, which can be accomplished in different ways, the traditional way is to use the ergodic hypothesis.

These quotations express the orthodox approach to Classical Stat Mech. The classical mechanics system is in a particular state, and a measurement of some property of that state is modelled by a long-time average over the trajectory of the system. We approximate this by taking the infinite time average. Our theory, however, cannot calculate this, anyway we don't even know the initial conditions of the system so we do not know which trajectory... what our theory calculates is the phase average or ensemble average. If we cannot justify some sort of approximate equality of the ensemble average with the time average, we cannot explain why the quantities our theory calculates agree with the quantities we measure.

Some people, of course, do not care. That is to be anti-foundational.

Answer 5

I do not agree with Marek's statement that ''in many practial applications of statistical mechanics, the ergodic hypothesis is very important, but it is not fundamental to statistical mechanics, only to its application to certain sorts of experiments.''

The ergodic hypothesis is nowhere needed. See Part II of my book Classical and Quantum Mechanics via Lie algebras for a treatment of statistical mechanics independent of assumptions of ergodicity or mixing, but still recovering the usual formulas of equilibrium thermodynamics.

Answer 6

You may be interested in these lectures:

Entanglement and the Foundations of Statistical Mechanics

The smallest possible thermal machines and the foundations of thermodynamics

held by Sandu Popescu at the Perimeter Institute, as well as in this paper

Entanglement and the foundations of statistical mechanics.

There is argued that:

1. "the main postulate of statistical mechanics, the equal a priori probability postulate, should be abandoned as misleading and unnecessary" (the ergodic hypothesis is one way to ensure the equal a priori probability postulate)

2. instead, it is proposed a quantum basis for statistical mechanics, based on entanglement. In the Hilbert space, it is argued, almost all states are close to the canonical distribution.

You may find in the paper some other interesting references on this subject.