# Microcanonical ensemble, ergodicity and symmetry breaking

In a brief introduction to statistical mechanics, that is a part of a wider course on Solid State Physics I am taking, the teacher introduced the concept of microcanonical ensemble and the ergodic hypothesis, both in its general formulation as the equivalence between the average over time and the weighted average over the ensemble and in its application to the microcanonical ensemble as the fact that the distribution function is constant over all the accessible microstates corresponding to an equilibrium situation. In the lecture, he left somehow the idea that there is something not completely clear about this hypothesis, without pointing out what.

Studying Landau-Lifshitz "Course on Theoretical Physics, vol. 5", this hypothesis is somehow assumed to be wrong, even though it is used. Looking for a counter example to prove it wrong, on Wikipedia I found out that ferromagnetic materials in vacuum undergo a spontaneous symmetry breaking, exhibiting a magnetization and therefore a preferred set of microscopical configurations even though others, with the opposite direction of the magnetization vector, are possible, even in absence of any interaction.

Here comes my questions:

• the symmetry breaking in ferromagnetic materials below the Curie temperature could take place due to some microscopic interactions with something that is neglected in our description? The first thing that came to my mind is the vacuum fluctuations of the electromagnetic field, that maybe could locally break the symmetry and induce a macroscopic symmetry break. Equivalently, this question could be rephrased as: do microcanonical systems actually exist?

• the ergodic hypothesis does not say anything on how the system can pass from one microscopical configuration to another one. In my opinion, if between one configuration and the other there is an high enough barrier potential, the system will never change from one configuration to the other, while if we are neglecting (for non-microcanonical systems) interactions that can provide the energy needed for the system to cross this barrier, then the system could explore also this second configuration. Is it possible for the validity of the ergodic hypothesis to be related to the height of the energy barrier between allowed microscopic configurations? Again in the example of ferromagnetism, going from one direction of the magnetization to the other one will require an interaction between magnetic dipoles that is too high to be neglected, thus making the ergodic hypothesis not valid.

EDIT (12-dec-2017):

After a little further study, this problem seems to be related to what we have in the description of second order phase transitions given by Landau (and, in my case, applied to the example of magnetism considered and to superconductivity). Could a more complete description of phase transitions take into account the problem of spontaneous symmetry breaking? Is it something relevant to our limited knowledge of the system (if we knew every kind of interaction, we could predict it) or is it something much more deep and physically meaningful?

• Just a note, I remember L & L don't say it is wrong, they say it is unnecessary to do statistical mechanics, but I agree with you they say they don't need but they kinda use it almost immediately. – Run like hell Dec 11 '17 at 13:47
• With regards to your first question, what do you regard as the definition of a microcanonical system? In the example you give explicitly breaking the symmetry between different directions in a ferromagnet would not stop it from being a microcannonical system, simply change it to a different microcannonical system. – By Symmetry Dec 11 '17 at 14:06
• @BySymmetry In the definition that was given to me, a microcanonical system that cannot exchange neither particles nor energy with the surrounding environment. However, my question is how can such a system exist, if even in vacuum we have quanta corresponding to the zero point energy of fields. – JackI Dec 11 '17 at 14:40

## 5 Answers

First, I strongly advise you to read sections 24 and 25 of Tolman's excellent book on statistical mechanics. My answer will mainly go along the lines of what is in the book.

The ergodic hypothesis states that a system will eventually in some time interval $T$ visit all the states compatible with a given energy constraint. As you said, this implies an equivalence between the microcanonical ensemble average and the time average.

This hypothesis was introduced by Bolztmann and Maxwell in an attempt to give a physical (non-statistical) justification to statistical mechanics. The reasoning is that statistical mechanics would give a way to calculate averages over time of quantities which in turns links with averages over many repetitions of an experiment. Then, if the ergodic hypothesis could be justified with the laws of classical mechanics, statistical mechanics would not need to introduce any additional postulate (the postulate of equal probability or maximum entropy).

We now know that the ergodic hypothesis is flawed for two reasons :

1. Classical mechanics shows that systems do not explore the entire phase space corresponding to a given energy constraint, but only a subset of it. The trajectories can be very large but do not pass by each and every point. There is a quantum version of this statement.

However, if we let small perturbations from the environment affect a system, something like the ergodicity can becomes be true. This assumption is realistic because no system can ever be totally isolated. To go back to your example, a paramagnetic (or non-ferromagnetic) material would look like it is ergodic. It would explore most of the available states because of the small electromagnetic perturbations that affect it. On the other hand, a ferromagnetic material would never look like it is ergodic because small perturbations cannot make the magnet change its orientation. So you are right: systems in which there is a large energy barrier between states are definitely not ergodic.

1. Even in cases where something like the ergodicity holds, the time of recurrence $T$ can be very large, in fact be larger than the age of the universe.

Finally, some of your questions are more oriented on the concept of spontaneously broken symmetry. You may want to look at some other answers on this specific problem, for example this one.

EDIT : This article also gives good explainations, more specifically on the impossibility of strong ergodicity.

He [Boltzmann] put forth what he called the ergodic hypothesis, which postulated that the mechanical system, say for gas dynamics, starting from any point, under time evolution Pt, would eventually pass through every state on the energy surface. Maxwell and his followers in England called this concept the continuity of path (3). It is clear that under this assumption, time averages are equal to phase averages, but it is also equally clear to us today that a system could be ergodic in this sense only if phase space were one dimensional. Plancherel (14) and Rosenthal (15) published proofs of this, and earlier, Poincare (16) had expressed doubts about Boltzmann’s ergodic hypothesis.

• +1 Thank you for that answer, I learnt more than I really should have :) When you say Classical mechanics show that systems do not explore the entire phase space corresponding to a given energy constraint, but only a subset of it., do you mean absolutely, (that is, there is a proof), or in a time limited practical sense. A reference or link is fine, I can work away from there myself, thanks – user178231 Dec 14 '17 at 18:18
• Yes in the strong sense, ergodicity cannot be realised. Plancherel and Rosenthal have published proves of this as early as 1913. – Undead Dec 14 '17 at 19:38
• "Classical mechanics show that systems do not explore the entire phase space corresponding to a given energy constraint, but only a subset of it." How exactly does classical mechanics show this? And what about the Poincaré recurrence theorem? – valerio Dec 20 '17 at 17:20
• Take a look at my edited answer. I must say that I don't know the details of the proof. Poincaré recurrence theorem says that a system eventually comes back to a state very close to the initial state in a finite time, but it does not say anything about what happens in between, especially which proportion of the phase space is explored by the system during this time. – Undead Dec 21 '17 at 5:48
• True about the recurrence theorem, but I have never heard about this statement about the portion of phase space covered by a classical system during its motion... – valerio Dec 21 '17 at 13:51

The way I see it, your questions are closely related. Imagine a system at high temperature, rapidly exploring a lot of its phase space. You might say the ergodic hypothesis is correct in this situation. Then you start reducing the temperature, and the energy, and it may happen that for low energies there are two regions in phase space with the same energy, but far apart. Then the system, which is moving about chaotically, will be trapped, as the temperature falls below a certain limit, to one of these regions. This could be seen as a phase transition and the ergodic hypothesis would not be true in a naive sense (because not all microstates with given energy would be equally likely, only those in the connected component that was "chosen")

• I understand what you mean as a phase transition, but as far as I know if both of these regions of the phase space (in the final situation you have described) are allowed for the system, I think that applying the ergodic hypothesis both of them must be explored by the system. Therefore, the fact that the system is trapped somewhere to me seems to be a violation of the ergodic hypothesis. Am I right when I say that does not exist a microcanonical ensemble since we always have some interactions and, instead of an hypothesis, we are dealing with a condition, that can be verified depending... – JackI Dec 11 '17 at 14:44
• ... on the situation we are describing. – JackI Dec 11 '17 at 14:45
• The ergodic hypothesis is a statement not about the system, but about your knowledge of the system. When you say all states are equaly likely you are saying they are equaly likely to you, that is you cannot distinguish them. If you know that the system is trapped, then you should adjust your hypothesis to take this information into account. I don't think interactions are related to this point at all. – Marcel Dec 11 '17 at 14:49
• From the point of view of probability, I think understand your point. If I have knowledge about the fact that the system is in a region and not on the other, then I cannot neglect it. But, from a physical perspective, this seems strange to me. If I am describing such a system without any experimental knowledge, how is it possible to get the correct result? In the magnetization case, assuming the ergodic hypothesis I can obtain that the average magnetization is zero, that is different from what I get experimentally. How can I give a description not related to myself and my knowledge? – JackI Dec 11 '17 at 14:57
• I don't find it strange. The system is indeed trapped, and you know that it is trapped. Taking this into account is correct both in terms of probability and in terms of experimental facts. – Marcel Dec 11 '17 at 15:00

(Note: I have condensed matter background, therefore I would mostly use CMT terminologies -- fortunately, the FM example you gave belongs to CMT area.)

You have several questions, trying to relate the following issues: microcanonical ensemble, ergodicity, spontaneous symmetry breaking, and even Laudau phase transition theory. It's too complicated to clarify them all -- I'll try my best, and clearly it requires further discussion in the comment area to reach a final consensus...

First of all, some of your questions are not really well-defined, e.g. Laudau's theory is describing symmetry breaking phenomena, which could be a second order transition if the final symmetry group is a subgroup of the original group. I guess (not quite sure) what's in your mind is that you relate the phase transition to the system evolution in real time, which then seems to be related to ergodicity. However, Landau's transition theory has nothing to do with evolution, and is just a phenomenological description of the two sides of phase transition, characterizing the system by an order parameter introduced by hand (but measurable), which is a static point of view that is always describing the system after the equilibrium is reached. For a continuous second order transition, as long as the final symmetry group is containing or contained in the original group, Landau's theory always works. (Today, we know there are some exotic examples, e.g. deconfined quantum critical point, topological order, etc.)

Besides, go back to the FM example you mentioned, that's about the spontaneous symmetry breaking (SSB), which is much more tricky: to have a SSB does not require the system to choose a specific set of configuration, which is the classical picture in your mind; while quantum mechanically, even in a superposition state, you could still detect the off-diagonal long-range order of certain local order parameter to tell if there is a SSB or not. More precisely in the FM case, if you look at the original Heissenberg Hamiltonian, after block diagonalization, the effective hamiltonian in the ground state manifold must be an identity matrix times a constant energy value -- this verifies your thinking, that if the system was pinned into a certain direction due to some external perturbations or your measurement, say, of the $S_z$ component, then the time-evolution of this linear quantum system could never go to other directions since there is no off-diagonal term in hamiltonian at all -- remember, "due to something" means that theoretically you really could have a superposition state without "classical SSB (i.e. total magnetization choosing a direction)" if there is no perturbations at all. However, this superposition state is still not a "micorocanonical system", since it's a pure state and the linear evolution nature of QM forbids its path covering the whole ground state manifold, which I'm explaining in the following paragraph in details. But the basic idea of my above explanation is to clarify that SSB is not a proper example to help understand ergodicity.

Secondly, if there is a, in your word, "microcanonical system"? Well, in a statistic perspective, an ensemble should be made of infinite number of systems. There is nothing called "microcanonical system" at all! Only "microcanonical ensemble" is introduced in statistic physics. For example, for a given energy $E_0$ there are several degenerate states, then I could make several copies of the system forcing them satisfy the equal distribution assumption, then they together form an ensemble directly. So you don't need to worry about if there is an ensemble; rather, you should ask wether the ensemble approach for a single system is valid or not. More precisely, I think what you want to ask is: whether there is a single system (isolated), whose time average can be approximated by an ensemble average, i.e. ergodicity hypothesis holds there. Short answer (for an isolated system): classically possible, quantum mechanically not possible. Classically, you could find the famous example of free particles in a non-commensurate-length box. But for a quantum system, the evolution is linear: $|\Psi(t)\rangle = \sum_i\lambda_ie^{iE_it}|\psi_i\rangle$ where $E_i$ is the eigenenergy of the Hamiltonian -- this is clearly not ergodic. In the extreme case, you could simply just make an eigenstate at $t=0$, then it would never go to other degenerate states. Therefore, at least, ergodicity is impossible for an eigenstate of an isolated system. Hence you don't need to consider stuffs like "energy barrier" at all, if the system is really isolated.

But does that mean the traditional ensemble perspective is completely useless, even include canonical ensembles and grand canonical ensembles which is derived from microcanonical one? Not really. In a more modern point of view, there are, however, several tricky issues, e.g. eigenstate thermalization hypothesis (ETH), quantum chaos (probably... I'm not familiar with chaos at all, so I'll skip it, and you can ask someone else). I don't know much, but ETH provides a possibility that, for a subsystem of an isolated one, you could still use canonical ensemble for analysis (but this hypothesis would be violated in some cases, related to a very hot topic: many-body localization).

Two more things to mention about ergodicity and ensemble perspective:

(1) In the real world, there is nothing isolated in experiments -- here the energy barrier does matter, since external perturbations could bring the system to a "statistical equilibrium" only when the barrier is not too large.

(2) Even you idealize something isolated, the way that you theoretically model it is always not complete enough: there are always some internal interaction mechanism neglected by us. Hence, to examine the theoretical prediction of an experiment, traditional ensemble perspective can still be applied in a lot of cases.

As everything else in physics, a microcanonical ensemble is an idealization, useful to get started and to build some intuition.

Classical physics, where ergodicity may be invoked for simple enough systems, is also an idealization. In quantum mechanics, which is the more accurate theory, the notion of ergodicity has not even a place.

Thus you are right - there are no microcanonical systems. At equilibrium, the typical systems are grand canonical.

The symmetry breaking in ferromagnetic materials below the Curie temperature could take place due to some microscopic interactions with something that is neglected in our description?

No, not really. You have spontaneous symmetry breaking in the Ising model in $d>1$ without the need for any additional interaction term in the Hamiltonian.

Equivalently, this question could be rephrased as: do microcanonical systems actually exist?

What do you mean exactly with "microcanonical system"?

If you mean a system with fixed energy and number of particles, than of course, like everything in physics, this is an idealization, because no system can ever be perfectly isolated in practice.

However, the adjective "microcanonical" is more often referred to the microcanonical ensemble. An ensemble is an ideal infinite collection of copies of a system that differ for their microstate at a given instant in time. A microcanonical ensemble is an ensemble of system with the same energy $E$, number of particles $N$ and volume $V$.

With this meaning, the concept of "microcanonical ensemble" is of course an idealization, because we will never have access to an infinite number of copies of the same system. The usefulness of the concept of "ensemble" resides precisely in the ergodic hypothesis, that tells you that averages taken on this ideal ensemble (which does not exist in practice) is equivalent to taking time averages, if the time interval over which you take the average is "long enough". So the ergodic hypothesis is, if we use the "ensemble" approach, the core hypothesis over which all of statistical mechanics is built (1).

There are several issues in the ergodic hypothesis, the first one being that "long enough": sometimes, the relaxation time of a system can be extremely long, longer than any timescale a human being can possibly measure. For such systems, we cannot even tell if ergodicity is really broken or if the time that we must wait for the system to thermalize is simply too large for us to measure.

Is it possible for the validity of the ergodic hypothesis to be related to the height of the energy barrier between allowed microscopic configurations?

Absolutely. This is exactly what happens is a glass: if a liquid is cooled rapidly enough, it won't be able to crystallize and get trapped in a metastable state that is not its true ground state at that temperature (which is the crystal). The time that the system must wait to escape from this metastable state is approximately proportional to $\exp(\beta \Delta F)$, where $\Delta F$ is the height of the free energy barrier that the system must overcome. So the relaxation time increases exponentially with the height of the barrier; when this time is larger than any time we can possibly measure, we say that ergodicity is broken.

I would like to stress, however, that what we are actually saying is that ergodicity is broken... for us: if we wait a "large enough" time, the system will eventually escape from this metastable state and reach its real ground state. There is actually a theorem, called the Poincaré recurrence theorem, that says that if a system has bounded energy and volume, than almost every initial state will be visited an infinite amount of times during its dynamical evolution ("almost every" means that there will be a discrete set of initial states that may violate the theorem). However, the recurrence time increases exponentially with the size of the system, and since a typical real life system contains something like $10^{23}$ particles we get recurrence times that are completely out of the domain of physics (the age of the universe being a "mere" $10^{17}$ seconds).

Let's take as an example your ferromagnet, and more specifically the Ising model description of it. There are two possible fundamental states: the all-spins-up state and the all-spins-down state. To switch from one ground state to the other, you need to flip half of the spins plus one, and the other half will immediately follow; the energy cost associated with this operation is therefore proportional to $N/2+1$, where $N$ is the size of your system (the number of spins). Therefore, the time $\tau$ you will have to wait for a fluctuation to make your system jump from one ground state to the other scales approximately as $\tau \propto \exp(\beta \epsilon N)$, where $\epsilon$ is some energy. This means that if $N$ is "large enough" $\tau$ will be longer than the age of the universe and therefore the system will have, for any practical purpose, broken ergodicity.

(1) It is actually possible to lay the foundation of statistical mechanics starting from weaker statements, but this is another matter. Take for example a look at the first chapter of Landau's book (Statistical Physics) or at this question and answers.