The entropy in finite $N$-unit nonextensive systems: the ordinary average and $q$-average

Abstract

We have discussed the Tsallis entropy in finite $N$-unit nonextensive systems, by using the multivariate $q$-Gaussian probability distribution functions (PDFs) derived by the maximum entropy methods with the normal average and the $q$-average ($q$: the entropic index). The Tsallis entropy obtained by the $q$-average has an exponential $N$ dependence: $S_q^{(N)}/N\simeq e^{(1-q)N{S}_1^{(1)}}$ for large $N$ ($\gg \frac{1}{(1-q)}>0$). In contrast, the Tsallis entropy obtained by the normal average is given by $S_q^{(N)}/N\simeq [1/(q-1)N]$ for large $N$ ($\gg \frac{1}{(q-1)}>0)$. $N$ dependences of the Tsallis entropy obtained by the $q$- and normal averages are generally quite different, although the both results are in fairly good agreement for $|q-1|\ll 1.0$. The validity of the factorization approximation to PDFs which has been commonly adopted in the literature, has been examined. We have calculated correlations defined by $C_m=⟨(\delta x_i\delta x_j{)}^m⟩-⟨(\delta x_i{)}^m⟩⟨(\delta x_j{)}^m⟩$ for $i\ne j$ where $\delta x_i=x_i-⟨x_i⟩$, and the bracket $⟨\cdot ⟩$ stands for the normal and $q$-averages. The first-order correlation ($m=1$) expresses the intrinsic correlation and higher-order correlations with $m\ge 2$ include nonextensivity-induced correlation, whose physical origin is elucidated in the superstatistics.

23 pages, 5 figures: the final version accepted in J. Math. Phys

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Cited by in corpus (4)

• Classical small systems coupled to finite baths
• Specific heat and entropy of $N$-body nonextensive systems
• Thermal entanglement of Hubbard dimers in the nonextensive statistics
• Scaled Bregman divergences in a Tsallis scenario