First of all I apologize for the lenght of this question. I have some basic statistical mechanics facts that I am confused about, and in this subject it is probably better to be precise.

When foundational aspects of SM are discussed in textbooks, people appeal to some arguments in order to justify ensemble averages being equal to time averages. I pretty sure that specific details are not the same in different formulations so I will state broadly how I see this in order to be sure I am not misunderstanding something from the start:

The basic idea behind SM

People want to predict measurements of some variables made on a "big" system, for instance, the system is nitrogen gas in a container, and people want to predict what a measurement of the pressure of the gas will be. People realize that this measurement can't be instantaneous and that it will have to be made over some period of time, what one wants to predict then is the time average of a measurement of an observable $\langle P \rangle_{time}$. In order to predict this time average one puts into a bag all possible states the system can be in, and performs a weighted sum (average) over them, given some weight (probability distribution) that should be assigned somehow. People call this last average an average over an "ensemble" $\langle P \rangle_{ens}$. Afterwards, some foundational arguments are invoked in order to justify from a theoretical standpoint that ensemble and time averages are equal (which is experimentally confirmed) $\langle P \rangle_{time} = \langle P \rangle_{ens}$.

This holds true, for instance, for the "microcanonical ensemble", where states with a given energy are assumed to be equiprobable. If a system is determined to have a given "energy value" then we can use the microcanonical ensemble to predict time averages of observables. It also holds true for other ensambles but I don't want to get into that.

How I thought environmental interaction worked

Recently I realized that the mental argument I had been making in order to justify this in my head is different from the argument many people write down in books. I would have thought, in this case for instance, of a nitrogen gas system interacting extremely weakly with the walls of its container. The nitrogen gas starts of in a state $(q_0,p_0)$ (It is implied that there are many $q's$ and $p's$), and evolves a little while according to Hamilton's equations given a hamiltonian $H = T + V$ where $T$ is the kinetic energy of the gas and $V$ the potential energy of interaction between the molecules of the gas. After a short while the system interacts a bit with the gas in a random way, for instance, one molecule of the gas hits an atom on a wall of the container which was vibrating in a random manner, and gets an extra push or pull. I would say that during this time interval the system is evolving according to a hamiltonian $H' = T + V + \nu$, where $\nu$ is a term describing this random external interaction. When the system stops interacting with the environment then it is again evolving according to $H$, but it now has a different set of "initial" conditions $(q'_0,p'_0)$ as a result of the random interaction, which are hopefully not "very" correlated with the initial conditions $(q_0,p_0)$, the system then evolves a little bit more, then a second interaction kicks in, etc. All this, of course, is happening in extremely short time scales. Of course, one can always describe the system instead with a time-dependent hamiltonian $H = T + V + \nu(t)$, where the $\nu(t)$ term turns on and off randomly at different times and adopts different values (randomly as in, we don't know when that will happen but hope it will be sort of unpredictable due to the large number of environmental degrees of freedom).

Note that, because the states $(q'_0,p'_0)$ and $(q_0,p_0)$ are hopefully not very correlated the system eventually loses its knowledge of the initial conditions, and because the system evolves for a little while following the equations of motion before "jumping" to different initial conditions after some interaction takes place, the system ends up sampling the whole of phase space, hopefully quite quickly. That we have to assign equal probability density to the whole of phase space is not clear at all given this, and it is also not clear how small, how short, or how frequent these interactions should be, or how uncorrelated the different states can be, but at least I believed this to be plausible.

I adopted this way of looking at the matter after reading the first few chapters of Landau & Lifshitz book on SM and David Tong's lectures on SM, I am not sure if I misinterpreted those particular sources but that is what I took out of them.

How equilibrium could be achieved without this

I am starting to realize however that very many people would not think about it this way, but would instead think about an idealized system that is completely isolated from the environment, and whose constituent particles would never interact with the surroundings, they would then assign to that system a specific initial state $(q_0,p_0)$ and let it evolve according to Hamilton's equations forever. This is basically just a normal conservative classical system, without any kind of random or special element. These guys then hope to prove that this one trajectory, whatever it may be, samples every point of phase space, or at least comes very close to every point, or some related thing, in order to justify ensamble averages and time averages being equal.

I started to realize this after I read about Loschmidt's Paradox, the Poincare recurrence theorem, ergodic theory and the discussions between Poincare, Boltzmann, Loschmidt and Zermelo in different sources such as Sklar. At first I was extremely confused when I read about these things because most of the arguments and theorems discussed in this context assumed strictly time independent Hamiltonians for the system, which basically imply no interaction at all with the surroundings (from my point of view at least).

TL;DR: My question then is the following

Why is interaction with the environment usually not discussed when talking about the foundational aspects of SM? Is this intuitive idea of the environment slightly perturbing a system and helping it achieve equilibrium this way plain wrong, or a factor that is not crucial? Is there experimental or theoretical evidence pointing towards environmental interaction being needed, or not being needed in order to achieve equilibrium?

I am not asking this to confirm that my point of view is valid, I am asking because I believe I am missing some important piece of information or key ingredient from this picture.

As an addendum, I want to say I remember someone told me once that in molecular dynamics simulations without environmental interaction equilibration is achieved even when numerical errors (which would probably have a similar effect to random environmental interaction) are minimal. This was a comment in passing however and I am not sure (1) to what extent it is even true (2) what are the characteristic thermalization times for these kinds of systems and if they are comparable to the times observed in real life systems (3) if the range of validity of these results is wide or they are restricted to some small toy models (basically how relevant this is). If someone know about this it would also probably be useful in answering this question.


2 Answers 2


It seems there are two questions here.

1) Given isolated Hamiltonian system of particles with short-range interactions (no gravity), we believe it will evolve towards a state of internal thermodynamic equilibrium. Explain why this happens using mechanics and/or probability theory, using some plausible assumptions.

2) Given Hamiltonian system of particles is interacting via short-range forces also with its environment which is in internal thermodynamic equilibrium (so it has temperature), we believe it will evolve towards a state where it is in thermodynamic equilibrium with the environment, in particular, it will have the same temperature. Again, explain using some theory and assumptions.

Now it is easy to see that if 1) is valid generally, we have 2) as corollary: just assume that the system in contact with environment is a small part of a supersystem (composed of environment and the system) that is isolated. If the supersystem evolves towards equilibrium, then the system must evolve towards equilibrium.

The reasoning cannot be done in the opposite direction. Even if 2) is generally true, this does not tell us anything about what happens to isolated systems. But from experience with systems that are "close" to isolated systems, we know that those will tend to equilibrium. We implicity assume the same is true for perfect isolated systems (which do not exist, but can be modelled by Hamiltonian mechanics). So we try to derive 1).

  • $\begingroup$ I would say that although I know almost closed systems achieve equilibrium, I am not sure that, for example, equilibration times don’t diverge as the interactions with the environment go to zero, or go to an impossibly high value, or that some derivatives of equilibration times with respect to the interaction don’t diverge. If this were the case then I would not try to derive (1) (except for fun, and I am not sure if it has been or not, honestly). $\endgroup$
    – Ignacio
    May 7, 2018 at 3:18
  • $\begingroup$ On the other hand, I wouldn’t rule out the “derivation” of (1) from (2) as a limiting case taking interactions to zero and sidestepping difficult questions of whether the system is ergodic and whatnot, since for any finite nonzero interaction the systems are slightly random and everything is fine. If there was a theoretical/experimental argument that pointed towards interactions not being relevant to real world equilibration on the other hand, I would see why that wouldn’t be a good idea. $\endgroup$
    – Ignacio
    May 7, 2018 at 3:18
  • $\begingroup$ Duration of equilibration to environment's temperature does diverge as interaction with the environment is weaned off. But internal equilibration does not, as this is due to internal interactions. Those internal interaction are necessary for internal equilibration. $\endgroup$ May 7, 2018 at 10:33
  • $\begingroup$ I guess a different way to describe what I am imagining in my question is a system negligibly coupled with an environment that has a finite temperature. Since the system-environment interaction is so small, the equilibration times with the environment will be extremely large, but the "internal equilibration" of the system will be shorter, and I hope we can say the system has achieved some sort of equilibrium after a while, even if it hasn't reached the environment's temperature. $\endgroup$
    – Ignacio
    May 10, 2018 at 0:13
  • 1
    $\begingroup$ I can imagine how the system can reach this internal equilibrium state this way, since interactions with the environment "help" it to sample phase space and forget its initial conditions, but I have trouble imagining how that could happen in absence of this negligible interaction, and I wonder if the internal equilibration times would not be severely affected by this, to the point of perhaps not even happening, or happening in $10^{something}$ years, so that the relevant physical process is the small environmental interaction and not interactions within the system. $\endgroup$
    – Ignacio
    May 10, 2018 at 0:15

I will address this question in the specific case for a quantum mechanical system.

The question of the conditions required for an interacting closed quantum system to come to equilibrium is actually a major topic of current research in many-body physics.

If one has interactions with a bath, which typically is assumed to have infinite degrees of freedom and an unbounded spectrum, then one can talk about equilibrium thermodynamics without too much trouble. So, indeed, often in statistical physics courses the assumption is made that the system is coupled weakly to a bath to allow it to reach thermodynamic equilibrium, or to thermalize. This is particularly important for studies of non-interacting ensembles, like the ideal Bose gas, which will not thermalize without such a bath. Careful authors will point this out, but not everyone is careful.

An interacting but isolated system, on the other hand, will never thermalize in the sense that the full many-body state will always reflect the initial conditions. This is a consequence of the unitarity of quantum mechanics, which means that there are no irreversible processes. An easy example is that the total energy of an isolated interacting ensemble is necessarily fixed, which limits the evolution of the system to states with the same total energy as the initial conditions.

However, there is still a reduced sense in which an interacting many-body system can thermalize. It might be the case that even though the global system remembers the initial conditions, if you look at any small part it still looks thermal. This idea is called the Eigenstate Thermalization Hypothesis (ETH)(1, 2), and it is the sensible generalization of thermalization to a closed quantum system.

The ETH is known to be true in a few model systems, and conjectured to apply to many others. However, there are also interacting quantum systems that never thermalize. These are called many-body localized (MBL) systems (3,4). The precise conditions required for a system to be many body localized or not, and therefore have either non-ergodic or ergodic behavior for few-body observables, is, as I mentioned above, quite non-trivial and not yet well understood.

I've focused on the theoretical side but I will mention in passing that there are some recent experiments showing evidence both for ETH thermalization (5) and MBL lack of thermalization (6,7) in highly isolated interacting quantum systems.

  • 2
    $\begingroup$ This is quite interesting, and seems like a whole field of study in itself. I implicitly assumed classical statistical mechanics in my question since didn't think quantum mechanics would play a big role in the thermalization times of the simple systems I had in mind, and I wanted to first have a clear idea how that happens, but I could be surprised. From what I read in your sources it seems like this process is quite different from anything that happens in classical mechancis; it is only the result that is analogous, however I will look more into it. $\endgroup$
    – Ignacio
    May 9, 2018 at 23:43
  • $\begingroup$ @Ignacio Indeed, it turns out, remarkably, that entanglement is fundamental to thermalization in quantum mechanical systems. I am not sure of the status of the foundations of statistical mechanics within a fully classical framework. $\endgroup$
    – Rococo
    May 10, 2018 at 14:39

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