Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles Page: 82
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Note also that, from Eqs. (172) and (175), G g in this specific case, therefore
the generalized equipartition theorem (180) gives T p = . In this way the canonical
equipartition theorem (106) is recovered too.
9.1.1.2. Microcanonical. The microcanonical ensemble is recovered with the choice
f(x) W 6(x)
F(x) 0(x)
From the properties of the Dirac delta the distribution in Eq. (171) is:
(185) p(z; U, V) f6 ((U - H)) (U - H)
f dz6((U - H)) f dz3(U - H)
As with the canonical case (184), the last term in (185), does not depend explicitly
on 3, hence the microcanonical case is also a case of "hidden dual statistics". The
microcanonical equipartition theorem (15) is also recovered. From (180) one gets:
1 g fdzO(U - H) _P
(186) T - ______
(186) TP $3G f dzS(U- H) -Q'
where we have used the relations 0(ax) = 0(x) (for a > 0) and (ax) = a-16(x).
9.1.1.3. Tsallis (escort). The Tsallis escort case is recovered with the choice
(187) f (x) = [1 +(1 q)x] 1-q
F(x) = [1 + (1 -q)x] 1-q
(188)
In this case one finds g ANq and G = Nq in Eqs. (172) and (175), so from Eq. (180)
the Tsallis equipartition theorem (112), is recovered:
10 1N
(189) T. 'i lN
Canonical and microcanonical cases are both included in the family of Tsallis dis-
tributions as special cases corresponding to the values q = 1 and q = -oc, as from
Chap. 8.82
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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/m1/92/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .