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

MTA Wigner FK RMI, Konkoly-Thege M. 29–33, Budapest H-1121, Hungary
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Entropy 2014, 16(12), 6497-6514; https://0-doi-org.brum.beds.ac.uk/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. https://0-doi-org.brum.beds.ac.uk/10.3390/e16126497

AMA Style

Biró TS, Ván P, Barnaföldi GG, Ürmössy K. Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle. Entropy. 2014; 16(12):6497-6514. https://0-doi-org.brum.beds.ac.uk/10.3390/e16126497

Chicago/Turabian Style

Biró, Tamás S., Péter Ván, Gergely G. Barnaföldi, and Károly Ürmössy. 2014. "Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle" Entropy 16, no. 12: 6497-6514. https://0-doi-org.brum.beds.ac.uk/10.3390/e16126497

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