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I just started studying statistical mechanics and I'm doing a multiple choice quiz but I'm confused about one (well, two) questions in particular. The main one is the following:

"A microcanonical ensemble of systems corresponds to a collection of systems:

Select one or more: (a) All having a different macrostate. (b) All with the same energy. (c) In every different microstate. (d) All having the same microstate."

I know that by definition a microcanonical ensemble has a constant total energy, which leads me to think that b is an answer, but then the answer would refer to the 'collection of systems all having the same energy'. If by collection it means the individual microstates then I don't think this is true as I'm pretty sure the microstates don't all have the same energy. Also, I think that c is correct too since the microstates aren't identical but I don't really know what it means by 'in every different microstate'. Any help will be greatly appreciated!

Edit: Sorry, I forgot to include the other one I was a bit unsure of as well. It is to do with a system in contact with a heat bath, i.e. the canonical ensemble. The question says:

"A system in contact with a heat bath has: (a) The same temperature as the heat bath. (b) A constant mean internal energy. (c) The same energy as the heat bath. (d) A fixed, constant internal energy."

I know that (a) is definitely one of the answers, and I am sure that (c) can't be an answer because on Wikipedia it says that 'The system can exchange energy with the heat bath, so that the states of the system will differ in total energy.' which means the system surely can't have the same energy as the heat bath. So that leaves either (b) or (d). My notes say that in the case of a canonical ensemble 'we should now talk about an average energy $\bar E$ rather than $E$' so according to that it should be (b) but I'm struggling to distinguish between b and d really.

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The problem with multiple choice questions is that they do not give you the opportunity to demonstrate your knowledge by saying things like "well, number 4 is true if by blah we understand blah ...".

The best way to define microcanonical is to discuss the concept of an ensemble in this area of physics, and contrast it with canonical. Such a discussion would probably include that (b) is ok but not the whole truth, and (c) is ok with some strong conditions on the usage of the term "every". It cannot really mean "every", including all those microstates where the system has more internal energy than the entire observable universe, for example. It might be "every microstate with the given total energy, volume and particle number".

Now let's consider canonical. Just to rule out (c) for sure, take the case where the system is a cup of water and the bath is the pacific ocean. The one has an energy of about a hundred kilojoules (compared with its energy at absolute zero) while the pacific ocean has an energy some billions of times that. So not, I think, equal!

(b) and (d) are concerned with fluctuations. When in thermal contact with a reservoir, the internal energy of the system fluctuates a bit, so it is not constant over time, but its average value is constant (if we take it that the volume, temperature and particle number are not changing). To be precise about this one makes the hypothesis that the time-average for one such system is equal to the average, at any given time, over an ensemble of such systems.

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  • $\begingroup$ Thank you, and I agree with your first point. I realised I forgot to include the other question that I was semi-unsure of so I just edited my post now to include that (it's about the canonical ensemble) $\endgroup$
    – user328183
    Feb 15 at 12:45
  • $\begingroup$ Perfect explanation, thank you so much :) $\endgroup$
    – user328183
    Feb 15 at 14:39
  • $\begingroup$ @Andrew steane great explantion, but if may I ask, you say in 2nd paragraph "system has more internal energy than the entire observable universe," how is it even possible? the system is part from the observable universe. I am not sure why we need to consider this scenario $\endgroup$
    – Sagigever
    Feb 15 at 20:22
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    $\begingroup$ @Sagigever that was an example of hyperbola, that it, exaggeration for the sake of effect. You are quite right of course. So to be more careful I should not take such an extreme example, but the point is the word "every" is a bit vague on its own; one should say something more like "every available state subject to the constraints ..." and then specify what the constraints are. $\endgroup$ Feb 15 at 23:00

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