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Average Energy of Grand Canonical Ensemble

  1. Apr 20, 2012 #1
    Hello,

    The entropy of the Grand Canonical Ensemble (GCE) is:

    S = KB ln ZG + ([itex]\bar{E}[/itex]/T) - μo[itex]\bar{N}[/itex]/T

    Helmholtz function is:

    F = [itex]\bar{E}[/itex] - TS = [itex]\bar{E}[/itex] - TKB ln ZG - [itex]\bar{E}[/itex] + μo[itex]\bar{N}[/itex]
    = -TKB ln ZG + μo[itex]\bar{N}[/itex]

    But

    [itex]\partial[/itex]F/[itex]\partial[/itex]T = -S (From thermodynamics).

    Then,

    -TKB [itex]\partial[/itex]ln ZG/[itex]\partial[/itex]T - KBln ZG = -kB ln ZG - [itex]\bar{E}[/itex]/T + μo[itex]\bar{N}[/itex]/T

    This gives:

    [itex]\bar{E}[/itex] = kBT2 [itex]\partial[/itex]ln ZG/[itex]\partial[/itex]T + μo[itex]\bar{N}[/itex]

    This is not the correct answer. The correct answer does not have the μo[itex]\bar{N}[/itex] term, what's wrong ???
     
  2. jcsd
  3. Apr 27, 2012 #2
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