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
Article Menu

Export Article

Open AccessArticle
Entropy 2014, 16(12), 6497-6514; doi:10.3390/e16126497

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.
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)
View Full-Text   |   Download PDF [224 KB, uploaded 24 February 2015]   |  

Abstract

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
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

Scifeed alert for new publications

Never miss any articles matching your research from any publisher
  • Get alerts for new papers matching your research
  • Find out the new papers from selected authors
  • Updated daily for 49'000+ journals and 6000+ publishers
  • Define your Scifeed now

SciFeed Share & Cite This Article

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

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top