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Huang 7.6-7.8

  1. Mar 29, 2013 #1
    I am specifically referring to textbook "Statistical Mechanics" by Kerson Huang. It is not a general physics question, I am just trying to understand what is written in this specific book. So you can only really help if are familiar with it and have it at hand.

    I'm having trouble following what is going on in these sections and would appreciate if anyone could help me out. Prior to 7.6 he shows that when ∂P/∂v < 0 then the distribution of N in the grand canonical ensemble is extremely narrow and the grand canonical ensemble reduces to the canonical ensemble. From what I gather, he then wants to show that even in the case ∂P/∂v=0 the two ensembles are still equivalent. Then I'm lost.

    I do not see how 7.62, 7.63 represents this mathematically. I can see that if 7.62 is true, then the grand canonical sum is dominated by a single term (a single N) and so may 'reduce to a canonical ensemble' in the sense that if I fix N and use the canonical ensemble I should get the same results.

    But then I'm not sure what the second condition is for. I thought that it may be something to do with the single dominant term in 7.62 not necessarily referring to the canonical ensemble that refers to the actual physical system, it's just a term in the sum. The canonical ensemble corresponding to the physical system might not have 'N_{max}' particles but some other N=M. Perhaps this condition guarantees we can find the 'right' N for a given fugacity. But then surely the fugacity is fixed physically as well.

    I'm also confused about the reference to canonical and grand canonical and what physical systems we are referring to. I was visualizing a large system in equilibrium with a thermal reservoir, and with a fixed N,V. Inside there is a subsystem with N varying but fixed V which we are treating in the grand canonical ensemble. He says "....it is a basic experimental fact that we can obtain the same thermodynamic information whether we look at the whole system or a subvolume." which suggests he is thinking along the same lines. However, some of the math does not seem consistent with this.

    If someone could help me get my head round 7.62/7.63 I think the rest will follow.
     
  2. jcsd
  3. Mar 29, 2013 #2

    Bill_K

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    Well I'll give it a shot, however I seem to have a different edition of Huang, and although the text seems to follow along Ok, the equations are numbered differently. I think you're referring to my eqs 8.52 and 8.53, and in particular question (b): "Does there always exist a value of z for which N has any given positive value?"

    What he's trying to show is that, given the grand partition function (z, V, T), one can recover the same description as came from the canonical ensemble. Looking back at the last equation in the previous section, he derived a pair of relations involving log . The first one gives the Helmholtz free energy (but still in terms of z) and the second one gives N. He then says to eliminate z between the two to get the real A(N, V, T).

    Here also he leaves the same question b hanging: can you find a value of z that corresponds to any desired value of N? The point, I think, is that this is necessary for consistency. If you can't do this, then you won't have a legitimate function A(N, V, T) after all. A must be defined for all N.
     
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