Non-Extensive Statistical Mechanics  

In General > s.a. generalized thermodynamics [relativistic].
* Idea: (Tsallis) non-extensive thermo-statistics is based on a natural generalization of entropy for systems with long-range interactions, for which extensive thermo-statistics fails, such as gravity and electromagnetism.
* Status: 2005, There are growing theoretical indications of the need for this generalization for large cosmological structures, where the observed pseudo-temperature is generally different from the true thermodynamic one.
@ General references: Tsallis JSP(88); Czachor & Naudts PRE(99)qp/98 [foundation]; Naudts RVMP(00)mp/99; Tsallis PhyA(04); Plastino PhyA(04); Ferri et al PhyA(05); García-Morales & Pellicer PhyA(06) [microcanonical foundation and fractal phase space]; Parvan PLA(06) [microcanonical foundation], PLA(06) [extensive thermodynamic limit]; Campisi PLA(07) [limiting cases]; de Almeida PhyA(08)-a0708 [formal equivalence with extended Boltzmann-Gibbs statistics]; Ohara PLA(07) [geometric aspects]; Feng & Liu PhyA(09) [correlations and energy fluctuations]; Naudts 11; Pressé et al PRL(13) [consistency issue with the Shore and Johnson axioms]; Plastino & Rocca PhyA-a1503 [two types of Tsallis probability distributions].
@ Entropy: Sakhnovich a1103 [and energy]; Anghel & Parvan a1803 [physical interpretation, mesoscopic systems]; > s.a. entropy [extensivity].
@ Non-extensive thermodynamics: Abe & Rajagopal PRL(03) [quantum, second law]; Ou & Chen PhyA(06) [energy additivity and 0th law]; Carrete et al PhyA(08) [microcanonical equations]; Scarfone PLA(10) [intensive variables]; > s.a. specific heat; temperature.
@ Generalizations: Beck & Cohen PhyA(03), Beck PhyA(04) [superstatistics].
@ Related topics: Carati PhyA(08) [and fractal dimension of orbits]; Liu & Du PhyA(08) [ensemble equivalence]; Takahashi PhyA(09) [free energy]; Du ChPB(10)-a0905 [and Fokker-Planck equation dynamics]; Ubriaco PLA(09) [entropies based on fractional calculus]; Boon & Lutsko PLA(10)-a1003 [and continuous Hamiltonian systems]; Babacan PLA(11) [density of states calculation]; Barreira RVMP(10) [results in almost-additive thermodynamic formalism]; Guo & Du PhyA(12) [energy distribution and fluctuations]; Wilk & Włodarczyk PhyA(14) [Tsallis distribution with complex non-extensivity parameter q].
> Related topics: see Coarse-Graining; entropy; statistical-mechanical states [grand canonical]; temperature.

Tsallis Non-Extensive Entropy > s.a. entropy [relative and conditional entropy].
* Idea: Applicable when microscopic interactions and memory are long-ranged; The Tsallis definition is

Sq = k (1–∑i piq) / (q–1),     with     q > 1;

The Boltzmann–Gibbs expression is recovered in the q → 1 limit.
@ General references: Tsallis JSP(88); Suyari JPA(02); Furuichi et al JMP(04), Furuichi JMP(06) [properties]; Sattin PS(05) [interpretation in terms of incomplete knowledge]; Piasecki PhyA(06) [quasi-additivity]; Dukkipati et al PhyA(07) [measure-theoretic aspects]; Furuichi JMP(09) [maximum-entropy principle]; Tsallis EPJA(09)-a0812 [rev]; Du BASI-a1001 [properties]; Sadeghi et al PRA(12) [in phase-space quantum mechanics]; Creaco & Kalogeropoulos JPCS(13)-a1209; Rufeil Fiori & Plastino PhyA(13) [and Shannon entropy]; Kalogeropoulos AIP(13)-a1308 [hyperbolicity, and consequences], EPJB(14)-a1403 [extensive limit]; Tsallis CP(14) [introduction, and thermostatistical approach to inanimate and living matter]; Petz & Virosztek MIA(15)-a1403 [inequalities]; Kalogeropoulos a1704 [Legendre transforms]; > s.a. quantum statistical mechanics.
@ Interpretation: Wilc & Włodarczyk PhyA(08); Nelson & Umarov PhyA(10) [in terms of non-linear coupling of statistical states]; Jizba et al PRE(17)-a1610 [as a statistical physics of random chains].
@ Vs Rényi entropy: Jizba & Arimitsu PhyA(04)cm/03; Masi PLA(05) [common framework]; Figueiredo et al PhyA(06) [statistical]; Campisi & Bagci PLA(07) [and Tsallis ensemble]; Campos PhyA(10) [and escort probability distribution].
@ Related topics: Oikonomou PhyA(11) [multinomial coefficients method]; Trindade & Vianna PhyA(12) [and quantum groups]; Kalogeropoulos a1206 [escort distributions]; Plastino & Rocca PhyA(13) [q-Laplace transform]; Kalogeropoulos IJGMP(14)-a1401 [almost additive, properties], a1502-conf [and generalized Ricci curvature]; Bhattacharyya et al a1608 [example and thermodynamic calculations]; Korbel a1705 [rescaling the non-additivity parameter].
@ Generalizations: Santos et al PLA(11) [generalized quantum entropies within the Tsallis and Kaniadakis frameworks]; Weberszpil & Helayël-Neto PhyA(16)-a1511 [Tsallis and Kaniadakis frameworks, q-deformed algebras and fractional-derivative operators].

Applications > s.a. chaos; critical phenomena; gas [ideal gas]; networks [Kaniadakis statistics]; open systems; turbulence [in astrophysics].
@ General references: Chamati et al PhyA(06) [black-body radiation]; Ochiai & Nacher PhyA(09) [for complex networks]; Saguia & Sarandy PLA(10) [in disordered quantum spin-S chains]; Lukes-Gerakopoulos et al PhyA(08), Kalogeropoulos QSC(12)-a1104 [and weak chaos]; Kalogeropoulos QSC(13)-a1203 [and systems with vanishing largest Lyapunov exponent].
@ In gravity and cosmology: Du ASS(06)nl/04 [self-gravitating systems]; Mavromatos & Sarkar PRD(09)-a0812 [stringy spacetime foam models]; Frigerio et al MNRAS(15)-a1409 [on galactic scales, rotation curves]; Kalogeropoulos a1601-proc; Majhi a1703 [black-hole entropy].
@ In particle physics: Beck PhyA(00) [particle spectra], PhyA(02) [turbulence, and cosmology], PhyA(04) [cosmic rays]; Kohyama & Niegawa PTP(06)ht [quantum field theory, quarks and gluons].
@ Other applications: Sattin JPA(03) [granular gas]; Chakrabarti et al PhyA(08) [diatomic molecule, specific heat]; Saguia & Sarandy PLA(10)-a0912 [disordered antiferromagnetic quantum spin chains]; Hasegawa PhyA(10) [interpolation approximation, and some important systems]; Keshavarzi et al PhyA(10) [oscillators, statistics and thermodynamics]; Hasegawa PhyA(11) [Hubbard dimers, thermal entanglement]; > s.a. Scaling.
> Related topics: see Central-Limit Theorem; distributions; numbers [deformation of the reals]; stochastic processes; uncertainty relations.
> Gravity-related applications: see cosmic microwave background; cosmic rays; early-universe cosmology and nucleosynthesis; galaxy distribution.

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