Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles Page: 88
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55
50
+ 6=2
45 x o= 10
40 0o = 50
35
S30-
C
< 25-
20-
15-
10
5
0 20 40 60 80 100
n
FIGURE 9.1. Solution of Eq. (209), for different values of a and n
In agreement with Eq. (196), the dependence of the entropy on U is of the type
SIn U. The integral Y, has been obtained by using the change of variable x = e/U,
where e denotes energy.
The Gaussian ensemble of Eq. (206) interpolates between canonical and micro-
canonical ensembles as does the Tsallis escort ensemble [79]. In fact, on one hand
f (x) - ex as u-a +oo, and, on the other f(x) - /2w86(x) as 6- 0 (the vanishing
term 27o is not a problem because it will be cancelled with the normalization).
Furthermore, based on the fact that the Gaussian ensemble of index u well de-
scribes the statistics of a large component of a large isolated system, we deduce that
it must well approximate the Tsallis statistics of index cq = u in the case n, cq 1.
This fact is illustrated in Figure 9.2, where we have plotted the unnormalized Tsallis88
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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/m1/98/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .