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.
* Tsallis non-extensive entropy: 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.
* 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]; Ou & Chen PhyA(06) [energy additivity and 0th law]; 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].
@ Non-extensive thermodynamics: Abe & Rajagopal PRL(03) [quantum, second law]; Carrete et al PhyA(08) [microcanonical equations]; > s.a. entropy [extensivity]; specific heat; temperature.
@ Tsallis non-extensive entropy: 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]; Lukes-Gerakopoulos et al PhyA(08) [and weak chaos]; Wilc & Wlodarczyk PhyA(08) [interpretation]; Furuichi JMP(09) [maximum-entropy principle]; Tsallis EPJA(09)-a0812 [rev]; Ochiai & Nacher PhyA(09) [for complex networks]; > s.a. quantum statistical mechanics.
@ Tsallis 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].
@ 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]; Jiulin a0905 [and Fokker-Planck equation dynamics]; Ubriaco PLA(09) [entropies based on fractional calculus].
> Related topics: see Coarse-Graining; entropy; statistical-mechanical states [grand canonical]; temperature.

Applications > s.a. chaos; turbulence [in astrophysics].
@ In particle physics and gravity: 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]; Mavromatos & Sarkar PRD(09)-a0812 [stringy spacetime foam models].
@ Other applications: Sattin JPA(03) [granular gas]; Chamati et al PhyA(06) [black-body radiation]; Chakrabarti et al PhyA(08) [diatomic molecule, specific heat]; > s.a. Scaling.
> Related topics: see Central Limit Theorem; critical phenomena; stochastic processes; uncertainty relations.
> Gravity-related applications: see cosmic microwave background, cosmic rays; early-universe cosmology; galaxies.

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