Next Article in Journal
Geometric Thermodynamics: Black Holes and the Meaning of the Scalar Curvature
Next Article in Special Issue
Tsallis Distribution Decorated with Log-Periodic Oscillation
Previous Article in Journal
An Evolutionary Algorithm for the Texture Analysis of Cubic System Materials Derived by the Maximum Entropy Principle
Open AccessArticle

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
Author to whom correspondence should be addressed.
Entropy 2014, 16(12), 6497-6514;
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
Keywords: generalized q-entropy; fluctuations; hadronization generalized q-entropy; fluctuations; hadronization
MDPI and ACS Style

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.

Show more citation formats Show less citations formats

Article Access Map by Country/Region

Only visits after 24 November 2015 are recorded.
Search more from Scilit
Back to TopTop