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I have lately been reading Gibbs' book Elementary Principles in Statistical Mechanics, and I'm surprised how much in that book seems to have been ignored by later textbook writers. In particular, Gibbs looks carefully into the question of whether in the microcanonical ensemble we can sensibly define entropy and temperature, and his answer, to paraphrase, is "meh, kind of".

Gibbs pays close attention to exactly deriving properties of systems with any number of degrees of freedom (not just macroscopic systems), and shows that the canonical and grand canonical ensembles give perfect explanations for thermodynamic equations, even extending them to small systems. This is not so for the microcanonical ensemble:

  • There are two valid definitions of microcanonical entropy (phase space "surface" entropy (Shannon/Boltzmann entropy) and phase space "volume" entropy, to use the modern terminology). These are both valid since they become equivalent in the thermodynamic limit.
  • Neither definition, when differentiated against energy, leads to a satisfactory value of temperature, because two microcanonical systems, when they are combined and allowed to exchange energy, do not maintain the same temperature that they had before. (Except in the thermodynamic limit.)
  • For a system with 1 or 2 quadratic degrees of freedom (e.g. particle in a 1D or 2D box), these temperatures behave super strangely. As in, being always negative or always infinite, and the system becoming colder with increasing energy. Again, the thermodynamic limit escapes these problems.

It seems like the only value of the microcanonical ensemble is in the thermodynamic limit where these two expressions of entropy and temperature become equivalent, and start to behave properly. On the other hand, in a lot of textbooks the microcanonical ensemble is taken to be somehow the "fundamental" ensemble, even though the ensembles become equivalent in the thermodynamic limit anyway!

So, putting the question in a slightly different way, are there any modern textbooks on statistical mechanics which are as careful as Gibbs, or are learners simply expected to read Gibbs for the "real thing"? I would guess that proper treatment of small systems is a big deal for, e.g. nanotechnology.

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+1 This is a very good question. I am trying to recall something related to what you're looking for but I'm coming up blank. There was lots of work in relating ensembles and small systems in the context of dynamical systems/chaos but apart from that I can't think of any text book dealing with equilibrium statistical physics in that sense. People work mostly in out of equilibrium problems these days where everything is of course completely different. –  Bubble Sep 3 '13 at 22:06
    
I just found a link to these lectures here which you may find interesting: pirsa.org/11050009 Source: physics.stackexchange.com/questions/27402/… –  Bubble Sep 3 '13 at 22:13
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Hi @udiboy: Concerning retagging: If you haven't done this already please read the wiki tags for the ref.req. and books tags. I believe the books tag is appropriate here, while the ref.req. tag is not. –  Qmechanic Sep 3 '13 at 22:27
    
One of the classic textbooks is Landau & Lifshitz vol 5. –  Pulsar Sep 4 '13 at 9:36
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up vote 4 down vote accepted

Your second point, which is the most important one I think, is right but is not so problematic I think. You make a point about temperature but the same thing could be said of the density. You can consider a gas (ideal gas to make it simple) in either microcanonical or canonical ensembles and find that if you partition the box into two halves, the 1-particle density on each side is not necessarily the same and becomes exactly the same in the thermodynamic limit only.

Note also that, although the density need not be uniform, the most likely macrostate characterised by the number of particles in one of the halves corresponds to the case where the density is the same in the two parts of the box.

What you describe is exactly the same thing but with the energy instead of the number of particles for a gas.

Now, some people have tried to understand deeply what statistical mechanics is about after Gibbs and have come with some original and important ideas, among them you will find:

  • Khinchin on a mathematical formulation of statistical mechanics

  • Jaynes on a statistical inference interpretation of statistical mechanics

  • Fermi with the Pasta-Ula-Fermi problem

  • Kolmogorov-Arnold and Moser for the KAM theorem

  • Oliver Penrose (brother of Roger Penrose) has devised his own theory to give a rational to statistical mechanics

  • Roger Balian for continuing the work of Jaynes and extending it to quantum systems

  • Vulpiani on the relation between deterministic chaos and statistical mechanics

  • Lawrence Sklar on the philosophical issues of the fundations of statistical mechanics

This is not an exhaustive list but these are the authors that really made me change my mind on many misconceptions I had on statistical mechanics.

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Thanks, this is really good. As to the example you gave about combining canonical ensembles, I would understand that as the problem of describing the chemical potential of a canonical ensemble. Analogous to how the canonical ensemble introduces temperature in a consistent and satisfying way, I would say that the grand canonical ensemble introduces chemical potential in a consistent and satisfying way. –  Nanite Sep 4 '13 at 19:28
    
I've read some of Jaynes before but I'm not totally convinced, and his combative tone puts me off. I'm quite curious though to hear the same from someone else so I'll try to find a copy of Balian's. –  Nanite Sep 4 '13 at 19:35
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