Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles Page: 86
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where the U, V parametrization has been adopted. Alternatively, adopting the 3, V
parametrization the Tsallis escort ensemble would be:
[1 + (1 - q) (n - H(z; V))] 1-q
Pq(z;U,V)= q(.
f dz [1 + (1 - q)(2 - H(z; V))] 1-q
Applying Eqs. (174) and (175) gives the entropy. In the U, V representation, it reads:
(202) S['(U, V) = in V + in L,q(U) + In cn,
3
where
21
(U -Ul q) (1 -qq)n1--
(203) Ln>,q(U) =Uq dxx- 1 + 2(1 (1 -x)
In agreement with Eq. (196), the dependence of the entropy on U is of the type lIn U.
The integral L,2,q has been obtained by using the cut-off condition e = U 1 + - ,
and the change of variable x = e/U, where e denotes energy.
9.3.2. Gaussian Ensemble
Since the fundamental work of Khinchin [1] based on the application of the central
limit theorem, it is known that the distribution law for a large component of a large
Hamiltonian isolated systems of total energy a is well approximated by the following
Gaussian distribution:
2e(a-H) exp [ (A1-H)2
L 2a2
(204) p =
normalization
where the quantities Al and B2 being defined in terms of the Laplace transforms Zi (/3)
of the structure functions QS(x) of the system (i = 1) and the heat bath (i = 2):
d in Z1
Adf3=
d2 in Z2
B2
dp2
According to Khinchin the quantity A1 is a good approximation to the average energy
U of the system (u 1 O( 1)), where N is the number of particles in the system).86
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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/m1/96/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .