Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles Page: 58
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generates the heat differential both in the U, V and /3, V parameterizations. Let us
treat the U, V case first. Remember that within this parametrization 0 is a function
of U, V. Therefore:Jdz[1 (1
-Nq U (H(1 q)(H - U)]1
q)4(H - U)] 0-0(H
8UU) + ]
U),, + N,
NN9
INq/ T q%Jdz[1
dz[1 -(1
Nf (K(1 - q)/(H - U)]1
q)O(H - U)]I [- 0(U
V (aH]
H) -H) - Nq/ <Vj
P
where according to the general state definition of Eq. (96) Pq = _S and U
Then, combining Eqs. (112), (115) and (116) together, we obtain:(117)
dNq dUq + PdV
dSq - Tqwhich proves orthodicity in the U, V parametrization.
Similarly, within the /3, V parametrization one has U0q
0/3a dz[1
80U(, V), and thus:
(1 q)(H - U)]1-
(1 -q)3(H
)] , [
ou
(H- U) + -N, (H - U) + N U
00dV
Sdz[1 (1(1 - q)/3 (H- U)] 1-q
OU OH
BV B58
OUq
(115)
and
Aq
DV(116)
Uq.
I dz[1
(118)
and
aq U
Aq
dV
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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/m1/68/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .