I think that when particles and energy are allowed to freely leave the grand ensemble like that, there'll always be fluctuation in temperature despite the heat bath. But I only see particle and energy fluctuations, and temperature seems to always be considered at constant. So how do I find the T fluctuation, and is it necessary? Thanks.

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    $\begingroup$ The temperature doesn't fluctuate in the grand canonical ensemble. Temperature is more like a parameter you put in to specify which ensemble you're working with. $\endgroup$ – knzhou Oct 17 '16 at 6:09
  • $\begingroup$ But I find in here: link that T is not fixed in a canonical ensemble? Is it different in the GCE? $\endgroup$ – Kim Dong Oct 17 '16 at 9:26

Temperature is fixed in the grand canonical ensemble. In fact, for the three main ensembles we have:

  • Microcanonical ensemble: fixed $N,V,E$
  • Canonical ensemble: fixed $N,V,T$
  • Grand canonical ensemble: fixed $\mu,V,T$

where $\mu$ is the chemical potential.

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