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# Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle

by Tamás Sándor Biró * , Péter Ván, Gergely Gábor Barnaföldi and Károly Ürmössy
MTA Wigner FK RMI, Konkoly-Thege M. 29–33, Budapest H-1121, Hungary
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Entropy 2014, 16(12), 6497-6514; https://doi.org/10.3390/e16126497
Received: 3 November 2014 / Revised: 26 November 2014 / Accepted: 2 December 2014 / Published: 9 December 2014
(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)
Certain fluctuations in particle number, $$n$$, at fixed total energy, $$E$$, lead exactly to a cut-power law distribution in the one-particle energy, $$\omega$$, via the induced fluctuations in the phase-space volume ratio, $$\Omega_n(E-\omega)/\Omega_n(E)=(1-\omega/E)^n$$. The only parameters are $$1/T=\langle \beta \rangle=\langle n \rangle/E$$ and $$q=1-1/\langle n \rangle + \Delta n^2/\langle n \rangle^2$$. For the binomial distribution of $$n$$ one obtains $$q=1-1/k$$, for the negative binomial $$q=1+1/(k+1)$$. These results also represent an approximation for general particle number distributions in the reservoir up to second order in the canonical expansion $$\omega \ll E$$. For general systems the average phase-space volume ratio $$\langle e^{S(E-\omega)}/e^{S(E)}\rangle$$ to second order delivers $$q=1-1/C+\Delta \beta^2/\langle \beta \rangle^2$$ with $$\beta=S^{\prime}(E)$$ and $$C=dE/dT$$ heat capacity. However, $$q \ne 1$$ leads to non-additivity of the Boltzmann–Gibbs entropy, $$S$$. We demonstrate that a deformed entropy, $$K(S)$$, can be constructed and used for demanding additivity, i.e., $$q_K=1$$. This requirement leads to a second order differential equation for $$K(S)$$. Finally, the generalized $$q$$-entropy formula, $$K(S)=\sum p_i K(-\ln p_i)$$, contains the Tsallis, Rényi and Boltzmann–Gibbs–Shannon expressions as particular cases. For diverging variance, $$\Delta\beta^2$$ we obtain a novel entropy formula. View Full-Text
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Biró, T.S.; Ván, P.; Barnaföldi, G.G.; Ürmössy, K. Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle. Entropy 2014, 16, 6497-6514.