States in Statistical Mechanics  

In General > s.a. correlations; statistical mechanics and models.
* Idea: States in statistical mechanics are (possibly t-dependent) functions (q, p) on phase space, also called distribution functions; They can be interpreted in terms of the fraction of time a system spends near each point in phase space (Boltzmann), the probability that the system is found near that point, related to information theory (Jaynes, Katz), or an ensemble of macroscopically indistinguishable systems (Gibbs); The ensemble interpretation has greatly influenced the development and language of statistical mechanics, but is now mainly of historical interest.
@ Reviews: Hillery et al PRP(84).
@ General references: Challa & Hetherington PRL(88), PRA(88) [different ensembles]; Solomon et al qp/04-in [and combinatorics]; Turko a0711-in [statistical ensemble equivalence problem].

Canonical Ensemble
$ Def: The density matrix = exp{–H}/Z, with H the Hamiltonian, and Z = tr exp{–H} the partition function.
* Interpretation: It represents a system in thermal equilibrium with a thermal bath (microcanonical ensemble) at temperature T = 1/k.
* Properties: The energy fluctuation is E2 = kT 2CV.
* Applications: Very often used, even for systems not strictly in thermal equilibrium, because it makes calculations simple.
@ General references: Sorkin IJTP(79); Kalinin TMP(05) [ito two-particle partition function]; Parisio & de Aguiar PhyA(07) [semiclassical trace formula]; Reimann PRL(07), comment Brody PRL(08) [from more general state than microcanonical].
@ Relationship with microcanonical: Lukkarinen JPA(99)cm/98; Gurarie AJP(07); Touchette AJP(08) [inequivalent]; > s.a. Potts Model.
@ Generalized: Costeniuc et al JSP(05); Touchette et al PhyA(06).
@ Quantum: Brody & Hughston JMP(98)qp/97, JMP(99)qp/97; Tasaki PRL(98) [from quantum dynamics]; Albeverio et al m.PR/05/TMMS [estimates for quantum lattice systems]; Goldstein et al PRL(06) [from pure state of system + bath].
> Related topics: see Maxwell-Boltzmann Distribution; modified thermodynamics [relativistic].

Grand Canonical Ensemble
* Idea: It describes a system in thermal equilibrium at temperature T, with variable energy and particle number,

= exp{–(EN)}/Z ,   Z:= tr exp{–(EN)} ,    = chemical potential .

@ And canonical ensemble: Herzog & Olshanii PRA(97)at/96 [BEC]; Román et al AJP(99) [finite size effects]; Cancrini & Tremoulet JSP(04) [finite volume].
@ Quantum: Brody et al JPA(07)qp [ergodic theorem, unitary evolution of closed quantum systems leads to grand canonical ensemble].
@ Variations: in Beck PhyA(00) [non-extensive].

Microcanonical Ensemble > s.a. black hole entropy; fluctuations; lyapunov exponents; specific heat.
* Idea: The ensemble that correctly describes the equilibrium statistics of Hamiltonian systems; Classically, the manifold of all points in the N-body phase space with a given total energy.
$ Def: The density matrix = (HE) / tr[(HE)], for some fixed E.
* Advantages: Phase transitions can be defined for small systems (contrary to canonical ensemble situation).
* With long-range interactions: These systems in general are not additive, which can lead to an inequivalence of the microcanonical and canonical ensembles; The microcanonical ensemble may show richer behavior than the canonical one, including negative specific heats and other non-common behaviors.
* Quantum microcanonical state/postulate: A modification according to which for a system in microcanonical equilibrium all pure quantum states having the same energy expectation value are realised with equal probability.
@ General references: Rugh JPA(98) [geometric, dynamical approach]; Gross 01, PhyA(04); Bouchet & Barré JSP(05) [long-range interactions].
@ Quantum: Brody et al qp/05 [equilibrium], PRS(07)qp/05 [finite-dimensional , phase transition], qp/06-in [and grand microcanonical ensemble]; Bender et al JPA(05)qp; Naudts & Van der Straeten JSM(06)qp [alternative def]; Sugita cm/06-in [basis for use].

Other States
* Orthodes: (Boltzmann) Statistical ensembles that satisfy the heat theorem.
@ Stationary states: Cohen PhyA(06) [equilibrium and non-equilibrium].
@ Non-equilibrium steady states: Penrose & Coveney PRS(94), Evans & Coveney PRS(95) ["canonical" non-equilibrium ensemble]; Barré et al PRL(02) [as equilibrium of effective dynamics]; Dewar JPA(03) [properties, and information theory]; Piasecki & Soto PhyA(06) [example and approach]; Taniguchi & Cohen JSP(07) [Onsager-Machlup theory, fluctuation theorems]; Abou Salem mp/07 [fluctuations of macroscopic observables].
@ Other types: Hanel & Thurner PhyA(05) [power-law]; Volkov et al JSP(06) [bounded energy and particle number]; Toral PhyA(06); Campisi PhyA(07) [dual orthodes as generalized ensembles interpolating between canonical and microcanonical].


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