{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T22:16:06Z","timestamp":1766268966182,"version":"build-2065373602"},"reference-count":69,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2014,12,9]],"date-time":"2014-12-09T00:00:00Z","timestamp":1418083200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Hungarian National Science Fund OTKA","award":["K 104260","NK 106119"],"award-info":[{"award-number":["K 104260","NK 106119"]}]},{"name":"Hungarian Chinese Bilaterla Governmental Agreement","award":["NIH TET_12_CN-1-2012-0016"],"award-info":[{"award-number":["NIH TET_12_CN-1-2012-0016"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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\u2013Gibbs 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, R\u00e9nyi and Boltzmann\u2013Gibbs\u2013Shannon expressions as particular cases. For diverging variance, \\(\\Delta\\beta^2\\) we obtain a novel entropy formula.<\/jats:p>","DOI":"10.3390\/e16126497","type":"journal-article","created":{"date-parts":[[2014,12,9]],"date-time":"2014-12-09T10:18:35Z","timestamp":1418120315000},"page":"6497-6514","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":37,"title":["Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6188-8478","authenticated-orcid":false,"given":"Tam\u00e1s","family":"Bir\u00f3","sequence":"first","affiliation":[{"name":"MTA Wigner FK RMI, Konkoly-Thege M. 29\u201333, Budapest H-1121, Hungary"}]},{"given":"P\u00e9ter","family":"V\u00e1n","sequence":"additional","affiliation":[{"name":"MTA Wigner FK RMI, Konkoly-Thege M. 29\u201333, Budapest H-1121, Hungary"}]},{"given":"Gergely","family":"Barnaf\u00f6ldi","sequence":"additional","affiliation":[{"name":"MTA Wigner FK RMI, Konkoly-Thege M. 29\u201333, Budapest H-1121, Hungary"}]},{"given":"K\u00e1roly","family":"\u00dcrm\u00f6ssy","sequence":"additional","affiliation":[{"name":"MTA Wigner FK RMI, Konkoly-Thege M. 29\u201333, Budapest H-1121, Hungary"}]}],"member":"1968","published-online":{"date-parts":[[2014,12,9]]},"reference":[{"key":"ref_1","unstructured":"Newman, J. (July, January 20). On measures of entropy and information. Berkeley, CA, USA."},{"key":"ref_2","unstructured":"R\u00e9nyi, A. (1970). Probability Theory, North Holland."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1590\/S0103-97331999000100002","article-title":"Nonextensive statistics: Theoretical, experimental and computational evidences and connections","volume":"29","author":"Tsallis","year":"1999","journal-title":"Braz. J. Phys"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/S0378-4371(01)00644-6","article-title":"When is a quantity additive and when is it extensive?","volume":"305","author":"Touchette","year":"2002","journal-title":"Physica A"},{"key":"ref_6","unstructured":"Tsallis, C. (2009). Introduction to Nonextensive Statistical Mechanics, Springer."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"132302","DOI":"10.1103\/PhysRevLett.94.132302","article-title":"Power-law tails from multiplicative noise","volume":"94","year":"2005","journal-title":"Phys. Rev. Lett"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1603","DOI":"10.1016\/j.physa.2007.10.067","article-title":"Microscopic origin of non-Gaussian distributions of finacial returns","volume":"387","author":"Rosenfeld","year":"2008","journal-title":"Physica A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"162302","DOI":"10.1103\/PhysRevLett.95.162302","article-title":"Non-extensive Boltzmann equation and hadronization","volume":"95","author":"Purcsel","year":"2005","journal-title":"Phys. Rev. Lett"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1140\/epjb\/e2006-00112-3","article-title":"Two generalizations of the Boltzmann equation","volume":"50","author":"Kaniadakis","year":"2006","journal-title":"Eur. Phys. J. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1174","DOI":"10.1016\/j.physleta.2007.09.042","article-title":"Equilibration of two non-extensive subsystems in a parton cascade model","volume":"372","author":"Purcsel","year":"2008","journal-title":"Phys. Lett. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1016\/j.nuclphysa.2006.06.148","article-title":"Power-law tailed spectra from equilibrium","volume":"774","author":"Purcsel","year":"2006","journal-title":"Nucl. Phys. A"},{"key":"ref_13","first-page":"395","article-title":"Non-extensive equilibration in relativistic matter","volume":"7","author":"Purcsel","year":"2009","journal-title":"Cent. Eur. J. Phys"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1140\/epja\/i2009-10806-6","article-title":"Non-extensive approach to quark matter","volume":"40","author":"Purcsel","year":"2009","journal-title":"Eur. Phys. J. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/j.physletb.2005.10.064","article-title":"Limiting temperature from a parton gas with power-law tailed distribution","volume":"632","author":"Peshier","year":"2006","journal-title":"Phys. Lett. B"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"061147","DOI":"10.1103\/PhysRevE.83.061147","article-title":"Zeroth law compatibility of non-additive thermodynamics","volume":"83","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Bir\u00f3, T.S. (2011). Is There a Temperature? Conceptual Challenges at High Energy, Acceleration and Complexity, Springer.","DOI":"10.1007\/978-1-4419-8041-0"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"012002","DOI":"10.1088\/1742-6596\/394\/1\/012002","article-title":"Nonadditive thermostatistics and thermodynamics","volume":"394","year":"2012","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"56003","DOI":"10.1209\/0295-5075\/84\/56003","article-title":"Abstract composition rule for relativistic kinetic theory in the thermodynamical limit","volume":"84","year":"2008","journal-title":"Europhys. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3132","DOI":"10.1016\/j.physa.2013.03.028","article-title":"Ideal gas provides q-entropy","volume":"392","year":"2013","journal-title":"Physica A"},{"key":"ref_21","unstructured":"Bir\u00f3, T.S., Barnaf\u00f6ldi, G.G., V\u00e1n, P., and \u00dcrm\u00f6ssy, K. (2014). Statistical power-law spectra due to reservoir fluctuations."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Bir\u00f3, T.S., V\u00e1n, P., and Barnaf\u00f6ldi, G.G. (2013). Quark-gluon plasma connected to finite heat bath. Eur. Phys. J. A, 49.","DOI":"10.1140\/epja\/i2013-13110-0"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.physa.2014.07.086","article-title":"New entropy formula with fluctuating reservoir","volume":"417","year":"2015","journal-title":"Physica A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2047","DOI":"10.5506\/APhysPolB.43.2047","article-title":"Tsallis fits to pT spectra for pp collisions at the LHC","volume":"43","author":"Wong","year":"2012","journal-title":"Acta Phys. Pol. B"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"114007","DOI":"10.1103\/PhysRevD.87.114007","article-title":"Tsallis fits to pT spectra and multiple hard scattering in pp collisions at LHC","volume":"87","author":"Wong","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1140\/epja\/i2009-10803-9","article-title":"Power laws in elementary and heavy ion collisions: A story of fluctuations and non-extensivity?","volume":"40","author":"Wilk","year":"2009","journal-title":"Eur. Phys. J. A"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Wilk, G., and Wlodarczyk, Z. (2012). Consequences of temperature fluctuations in observables measured in high-energy collisions. Eur. Phys. J. A, 48.","DOI":"10.1140\/epja\/i2012-12161-y"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Wilk, G., and Wlodarczyk, Z. (2000). Interpretation of the nonextensivity parameter q in some applications of Tsallis statistics and Levy distribution. Phys. Rev. Lett, 84, http:\/\/dx.doi.org\/10.1103\/PhysRevLett.84.2770.","DOI":"10.1103\/PhysRevLett.84.2770"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.physletb.2010.04.037","article-title":"Cooper\u2013Frye Formula and non-extensive coalescence at RHIC energy","volume":"689","year":"2010","journal-title":"Phys. Lett. B"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.physletb.2012.10.025","article-title":"Microcanonical jet-fragmentation in proton-proton collisions at LHC energy","volume":"718","year":"2012","journal-title":"Phys. Lett. B"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.physletb.2011.03.073","article-title":"Generalised Tsallis statistics in electron-positron collisions","volume":"701","year":"2011","journal-title":"Phys. Lett. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1016\/S0378-4371(01)00353-3","article-title":"Generalized entropies from first principles","volume":"300","author":"Almeida","year":"2001","journal-title":"Physica A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"024904","DOI":"10.1103\/PhysRevC.78.024904","article-title":"Power-law in microcanonical ensemble with scaling volume fluctuations","volume":"78","author":"Begun","year":"2008","journal-title":"Phys. Rev. C"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"60004","DOI":"10.1209\/0295-5075\/99\/60004","article-title":"On the origin of power laws in equilibrium","volume":"99","author":"Campisi","year":"2012","journal-title":"Europhys. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"042126","DOI":"10.1103\/PhysRevE.88.042126","article-title":"Tsallis power-laws and finite baths with negative heat capacity","volume":"88","author":"Bagci","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1016\/j.physleta.2005.09.082","article-title":"Microcanonical ensemble extensive thermodynamics of Tsallis statistics","volume":"350","author":"Parvan","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"044902","DOI":"10.1103\/PhysRevC.78.044902","article-title":"Charged hadron multiplicity fluctuations in Au + Au and Cu + Cu collisions from sNN=22.5 to 200 GeV","volume":"78","author":"Adare","year":"2008","journal-title":"Phys. Rev. C"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"044910","DOI":"10.1103\/PhysRevC.88.044910","article-title":"Centrality dependence of \u03c0, K and p production in Pb-Pb collisions at sNN=2.76 TeV","volume":"88","author":"Abelev","year":"2013","journal-title":"Phys. Rev. C"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.physletb.2013.01.051","article-title":"Centrality dependence of charged particle production at large transverse momentum in Pb-Pb collisions at sNN=2.76 TeV","volume":"720","author":"Abelev","year":"2013","journal-title":"Phys. Lett. B"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"064903","DOI":"10.1103\/PhysRevC.80.064903","article-title":"Semi-inclusive observables in statistical models","volume":"80","author":"Begun","year":"2009","journal-title":"Phys. Rev. C"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1016\/S0375-9474(01)01228-3","article-title":"Fluctuations of rare particles as a measure of chemical equilibrium","volume":"697","author":"Jeon","year":"2002","journal-title":"Nucl. Phys. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"034901","DOI":"10.1103\/PhysRevC.70.034901","article-title":"Particle number fluctuations in a canonical ensemble","volume":"70","author":"Begun","year":"2004","journal-title":"Phys. Rev. C"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"024902","DOI":"10.1103\/PhysRevC.84.024902","article-title":"Identity method for particle number fluctuations and correlation","volume":"84","author":"Gorenstein","year":"2011","journal-title":"Phys. Rev. C"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"034903","DOI":"10.1103\/PhysRevC.89.034903","article-title":"Strongly intensive measures for the momentum and particle number fluctuations","volume":"89","author":"Gorenstein","year":"2014","journal-title":"Phys. Rev. C"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"054904","DOI":"10.1103\/PhysRevC.73.054904","article-title":"Particle number fluctuations in relativistic Bose and Fermi gases","volume":"73","author":"Begun","year":"2006","journal-title":"Phys. Rev. C"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/S0378-4371(03)00019-0","article-title":"Superstatistics","volume":"322","author":"Beck","year":"2003","journal-title":"Physica A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"031102","DOI":"10.1103\/PhysRevE.76.031102","article-title":"Superstatistics, thermodynamics and fluctuations","volume":"76","author":"Abe","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Ma, S.K. (1985). Statistical Mechanics, World Scientific.","DOI":"10.1142\/0073"},{"key":"ref_49","first-page":"2141","article-title":"Stohastic network view on hadron production","volume":"35","author":"Wilk","year":"2004","journal-title":"Acta Phys. Pol. B"},{"key":"ref_50","first-page":"568","article-title":"The imprints of superstatistics in multiparticle production processes","volume":"10","author":"Wilk","year":"2012","journal-title":"Cent. Eur. J. Phys"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"180601","DOI":"10.1103\/PhysRevLett.87.180601","article-title":"Dynamical foundations of nonextensive statistical mechanics","volume":"87","author":"Beck","year":"2001","journal-title":"Phys. Rev. Lett"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1209\/epl\/i2004-10506-9","article-title":"Dynamical correlations as origin of nonextensive entropy","volume":"70","author":"Kodama","year":"2005","journal-title":"Europhys. Lett"},{"key":"ref_53","unstructured":"Tsallis, C., and Haubold, H.J. (2014). Boltzmann-Gibbs entropy is sufficient but not necessary for the likelihood factorization required by Einstein. arXiv, 1407.6052."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"061105","DOI":"10.1103\/PhysRevE.63.061105","article-title":"General pseudoadditivity of composable entropy by the existence of equilibrium","volume":"63","author":"Abe","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1023\/A:1018811305766","article-title":"Thermodynamic uncertainty relations","volume":"29","author":"Uffink","year":"1999","journal-title":"Found. Phys"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1023\/A:1007836900527","article-title":"Comments on \u201cThermodynamic Uncertainty Relations\u201d by J. Uffink and J. van Lith","volume":"13","author":"Lavenda","year":"2000","journal-title":"Found. Phys. Lett"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1023\/A:1012385722039","article-title":"Thermodynamic uncertainty relations again: A reply to Lavenda","volume":"14","author":"Uffink","year":"2001","journal-title":"Found. Phys. Lett"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"3566","DOI":"10.1016\/j.physa.2011.05.002","article-title":"Generalized thermodynamic uncertainty relations","volume":"390","author":"Wilk","year":"2011","journal-title":"Physica A"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/S0375-9601(00)00337-6","article-title":"Axioms and uniqueness theorem for Tsallis entropy","volume":"271","author":"Abe","year":"2000","journal-title":"Phys. Lett. A"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1016\/S0375-9601(96)00832-8","article-title":"A note on the q-deformation theoretic aspect of the generalized entropies in nonextensive physics","volume":"224","author":"Abe","year":"1997","journal-title":"Phys. Lett. A"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/S0375-9601(00)00458-8","article-title":"Non-uniqueness of canonical ensemble theory entropy from microcanonical basis","volume":"272","author":"Abe","year":"2000","journal-title":"Phys. Lett. A"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"610","DOI":"10.1209\/epl\/i2000-00481-1","article-title":"Justification of power law canonical distributions based on generalized central limit theorem","volume":"52","author":"Abe","year":"2000","journal-title":"Europhys. Lett"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1209\/epl\/i2001-00373-4","article-title":"Macroscopic thermodynamics of equilibrium characterized by power law canonical distributions","volume":"55","author":"Abe","year":"2001","journal-title":"Europhys. Lett"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"016139","DOI":"10.1103\/PhysRevE.71.016139","article-title":"Necessity of q-expectation value in nonextensive statistical mechanics","volume":"71","author":"Abe","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1098\/rstl.1825.0026","article-title":"On the nature of the function expressing of the law of human mortality, and on a new mode of determining the value of life contingencies","volume":"115","author":"Gompertz","year":"1825","journal-title":"Philos. Trans. R. Soc"},{"key":"ref_66","first-page":"760","article-title":"The experimental alteration of malignacy with an homologous mammalian tumour material","volume":"21","author":"Casey","year":"1934","journal-title":"Am. J. Cancer"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.physleta.2005.10.082","article-title":"Euler\u2019s transform and a generalized Omori\u2019s law","volume":"351","author":"Apostol","year":"2005","journal-title":"Phys. Lett. A"},{"key":"ref_68","doi-asserted-by":"crossref","unstructured":"Hanel, R., and Thurner, S. (2011). A comprehensive classification of complex statsitical systems and an axiomatic derivation of their entropy and distribution functions. Europhys. Lett, 93.","DOI":"10.1209\/0295-5075\/93\/20006"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"260601","DOI":"10.1103\/PhysRevLett.105.260601","article-title":"Thermostatistics of overdamped motion of interacting particles","volume":"105","author":"Andrade","year":"2010","journal-title":"Phys. Rev. Lett"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/12\/6497\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:10:42Z","timestamp":1760217042000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/12\/6497"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,12,9]]},"references-count":69,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2014,12]]}},"alternative-id":["e16126497"],"URL":"https:\/\/doi.org\/10.3390\/e16126497","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2014,12,9]]}}}