# Does Decoherence explain why constituents of composite systems in textbook Statistical Mechanics are assumed to occupy energy eigenstates?

This question is based off of the first chapter of Pathria's text on Statistical Mechanics.

Consider the system of an Ideal Gas. Consider the system to be in some macrostate specified by $$(N, V, E)$$. Pathria then constructs an argument to count the number of microstates for this specified macrostate. In particular, Pathria assumes that the constituent gas molecules which make up the composite Ideal Gas system can only occupy energy eigenstates. This reduces the problem to counting the number of ways $$N$$ particles each occupying some energy eigenstate $$\lvert \epsilon_n \rangle$$ can collectively have a total energy equal to $$E.$$

Does the theory of Decoherence justify this assumption? In particular, are we to assume (or perhaps we can work out) that each gas molecule interacts with its environment (the other $$N-1$$ gas molecules) such that the selected class of pointer states of the system are self-energy eigenstates?

• Do you know the information theoretic treatment of statistical mechanics? See e.g. the work of Jaynes. Sep 12, 2023 at 6:21
• I do not ! @TobiasFünke Sep 12, 2023 at 7:37

In particular, are we to assume (or perhaps we can work out) that each gas molecule interacts with its environment (the other $$N−1$$ gas molecules) such that the selected class of pointer states of the system are self-energy eigenstates?

This is not a valid assumption. We know that the gas molecule will continually interact again with the other gas molecules. Let us entertain the situation that a gas molecule in such a system, at some instant, gets close to being decohered. It will, within a few multiples of the mean free path time scale, interact again and become entangled again. This is always true when you have such strong overlaps, in this case due to confinement. So decoherence cannot be the correct explanation for this behaviour.

In particular, Pathria assumes that the constituent gas molecules which make up the composite Ideal Gas system can only occupy energy eigenstates. This reduces the problem to counting the number of ways $$N$$ particles each occupying some energy eigenstate $$\left|\varepsilon_n\right>$$ can collectively have a total energy equal to $$E$$.

I think this is the essence of your question. You think this is suspect, and I think it is important to dispel this suspicion, regardless of decoherence. Here are some important facts relevant to this discussion:

1. We are already assuming constant total energy $$E$$; in fact, it is this assumption that is utterly unphysical and quickly needs to be abandoned in favour of the grand canonical viewpoint. But if you accept that $$E=$$ const, then how else but summing individual energy eigenstates are you going to achieve this when we have turned off interactions?
2. Again, taking the unphysical but tolerable assumption that individual energy eigenstates $$\left|\varepsilon_n\right>$$ are worth talking about, then from the postulates of QM, really about Hilbert spaces, we know that energy eigenstates span the Hilbert space, and thus we know that any possible quantum state of interest can be studied in the energy eigenbasis. There is thus not much of an assumption right here.

Here I take a detour and talk about why the assumption is tolerable despite being somewhat unphysical. We know from experience trying to simulate / handle / experiment with a bunch of similar/identical atoms/molecules that when many things of the same type comes together, each single energy level, upon tiny interactions treated as perturbations, turn into a bunch of closely spaced sublevels, forming energy bands in the limit of infinitely many particles coming together. This is the basis of band theory that is so fundamental to the modern physics understanding of crystalline solids, and absolutely foundational to the understanding of how semiconductors work. Textbooks give the impression that the splitting between the sublevels is symmetric, i.e. they look, locally in the energy spectrum, like QHO energy levels, but this is not necessary for this discussion—turning off the interaction and having all of the energy eigenstates be on the same energy level is fine for the discussion too. The energy sublevel splitting might involve superpositions of the individual energy eigenstates, but the mean value of all the eigenstates being filled up is always going to be near the mean value of unperturbed, non-interacting energy level. i.e. it literally just mucks up the details of the discussion without providing any useful insight, and so pedagogically it is better to ignore them for the exposition to be smoother. 3) The work to be done is in counting the number of possible states. It does not at all matter that, in reality, the system is definitely much more likely to be in superpositions of the various energy eigenstates in a non-trivial way. What is important is that the counting using complicated superpositions is going to be necessarily equal to the simplistic counting of the number of individual energy eigenstates, because this count is just the size of the Hilbert subspace of total energy $$E$$, and it does not matter how you choose to count it. This size is proved to be unique as long as it is the finite positive integer that it is. Does the theory of Decoherence justify this assumption?

Amusingly, the answer is a mix of yes and no. As above, there is a huge "no" component, but decoherence will explain why we should expect, on human time and length scales, to expect that the gas will be observed to be in some almost-constant $$E$$ state. i.e. while each gas molecule cannot be decohered from the rest of the gas molecules, the gas chamber itself can approximately be decohered to behave as a classical ignorance style statistical probabilistic mixture of different energy $$E$$ eigenstates. This is the part that decoherence can help, though, again, most of the heavy lifting is nothing to do with decoherence.

Note, also, that if we abandon the unphysical assumption that $$E=$$ const and go towards the grand canonical treatment, then decoherence would not even be needed at all.

By postponing the discussion of decoherence, even though I like decoherence, I hope you can understand that the treatment by Pathria would not be too meaningfully informed by decoherence, and that plenty of reasonable physicists with favoured interpretations differing from decoherence ought to still be able to tolerate that treatment.