Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles Page: 76
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8.5.1. Canonical MEP
Let q -- 1 in Eq. (157). As is well known [66], Rinyi entropy of order 1 is nothing
but Shannon's entropy, thus Eq. (157) becomes:
(165) F[p] - Jpln p - p A (Jdzp -1) (JdzHp - U)
whose maximization (see for example [18]) is the canonical ensemble.
8.5.2. Microcanonical MEP
It is more interesting, though, to consider the limit q -- -oc. In this limit the
integrand p1q/ tends to 1. We have to be careful with the integration domain at
this point, because if the integration is carried over the whole space R6n, the integral
would diverge. Instead, if one considers that the domain of integration is bounded
by the condition that the system's energy does not exceed the total energy of system
plus bath, which is fixed, we obtain the following cut-off condition:
T
(166) H(z) < U + T
1 q
where U is the system's average energy and T/(1 -q) = CvT is the bath's average
energy. Now if q goes to 1, the right hand side of Eq. (167) goes to oc and the
integration domain becomes IR6n for the canonical case. Instead, for q tending to -oc
the cut-off becomes
(167) H(z) < U.
Thus the microcanonical distribution maximizes the following:
(168) F[p] l = In dz A (Jdzp - - (JdzHp -U)
JfH<U J
(the explicit maximization of the functional in (168) is carried in appendix B). Thus
we obtain the pcMEP in the form of constrained maximization of the volume entropy
of Eq. (8) with the normalization constraint and the average energy constraint. This
is in contrast with the traditional derivation of the microcanonical ensemble from a
maximum entropy principle. According to the commonly accepted derivation, in order76
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Campisi, Michele. Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles, dissertation, May 2008; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc6128/m1/86/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .