States in Statistical Mechanics |

**In General** > s.a. correlations; Macrostates and
Microstates; statistical mechanics and models.

* __Idea__: States in classical statistical
mechanics are (possibly *t*-dependent) functions *ρ*(*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-conf [and combinatorics];
Turko a0711-conf [statistical ensemble equivalence problem];
Werndl a1310 [typicality measures in Boltzmannian statistical mechanics].

**Canonical Ensemble or Gibbs State** > s.a. statistical mechanics
and quantum statistical mechanics [approach to equilibrium, thermalization].

$ __Def__: The distribution function
given by the Boltzmann distribution *ρ*(*q*, *p*) =
exp{–*βH*}/*Z*, were *H*(*q*, *p*) is the Hamiltonian,
and *Z* the canonical partition function.

$ __Partition function__: The function
*Z* = ∑_{states α}
exp{–*βE*_{α}}
(*Z* stands for "Zustandsumme"; *Q* is another common symbol),
where for states labeled by continuous variables (e.g., *q* and *p*) the
sum over states is really a suitably normalized integral over phase space.

$ __Boltzmann factor__: The factor
exp{–*βH*(*q*, *p*} that determines the relative probabilities
of different states for a system in a thermal state.

* __Interpretation__: A canonical ensemble
represents a system in thermal equilibrium with a thermal bath (microcanonical ensemble)
at temperature *T* = 1/*kβ*.

* __Properties__: The energy fluctuation
is *σ*_{E}^{2}
= *kT*^{ 2}*C*_{V};
This also shows that the heat capacity is always positive.

* __Applications__: It is very often
used, even for systems that are not strictly in thermal equilibrium, because it makes
calculations simple.

@ __General references__: Sorkin IJTP(79);
Kalinin TMP(05) [in terms of two-particle partition function];
Parisio & de Aguiar PhyA(07) [semiclassical trace formula];
Reimann PRL(07),
comment Brody PRL(08),
Plastino & Daffertshofer EPL(08) [from more general state than microcanonical];
Velazquez & Curilef JPA(09) [energy fluctuation relation];
Müller EJP(14) [Boltzmann factor, simplified derivation]; Dai et al PRA(16)-a1508 [joint system-bath state].

@ __Relationship with microcanonical__: Lukkarinen JPA(99)cm/98;
Gurarie AJP(07)aug;
Touchette AJP(08)jan
[inequivalent]; Zhang et al PRE(10)-a1007 [in coupled-spin systems]; Brandão & Cramer a1502 [equivalence for non-critical quantum systems]; Tasaki a1609 [local equivalence, for quantum spin systems]; Cancrini & Olla JSP(17)-a1701 [equivalence, and fluctuations]; > s.a. non-extensive
statistical mechanics [in Tsallis statistics]; Potts Model.

@ __Generalized__: Costeniuc et al JSP(05);
Touchette et al PhyA(06);
Kakorin AJP(09)jan [finite particle-number correction];
Malyshev RMS(01)gq [on random discrete sets];
Ambruş & Winstanley PLB(14)-a1406 [rotating];
Halpern et al nComm(16)-a1512 [non-Abelian thermal state];
Pachón et al a1401 [deviations from quantum uncertainty relations].

@ __Related topics__: Chazottes & Hochman CMP(10)-a0907 [example
with no zero-temperature limit]; Plastino et al PhyA(14) [partition function and entropy from thermodynamics]; Navrotskaya a1406 [multi-particle inverse problem]; Tatsuta & Shimizu PRA(18)-a1703 [conversion into a superposition of macroscopically distinct states]; > s.a. Maxwell-Boltzmann
Distribution; modified thermodynamics [relativistic].

**Grand Canonical Ensemble** > s.a. states in quantum statistical mechanics.

* __Idea__: It describes a system in thermal equilibrium at temperature *T*,
with variable energy and particle number,

*ρ* = exp{–*β*(*E*–*μN*)}/*Z* , *Z*:=
tr exp{–*β*(*E*–*μN*)}
, *μ* = chemical potential .

@ __General references__: Smii a1001-wd [graphical representation].

@ __And canonical ensemble__: Herzog & Olshanii PRA(97)at/96 [BEC];
Román et al AJP(99)dec
[finite size effects]; Cancrini & Tremoulet JSP(04)
[finite volume].

@ __Variations__: in Beck PhyA(00)
[non-extensive]; Klein & Yang MPAG(12)-a1009 [general relativistic, in curved spacetimes].

**Microcanonical Ensemble** > s.a. Envariance; fluctuations; lyapunov
exponents; specific
heat; states in quantum statistical mechanics.

* __Idea__: The ensemble that correctly describes the equilibrium statistics
of Hamiltonian systems; The manifold of all points in phase space with a given total energy.

$ __Def__: The distribution function *ρ* = δ(*H*–*E*)
/ tr[δ(*H*–*E*)],
for some fixed value *E* of the energy, where the "tr" stands for integrating over the energy shell.

* __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.

@ __General references__: Rugh JPA(98) [geometric, dynamical approach];
Gross 01, PhyA(04);
Bouchet & Barré JSP(05) [long-range interactions];
Kiessling JSP(09)-a0810 [Ruelle's regularized microcanonical measure];
Fine & Hantschel PS(12)-a1010 [alternative]; Chiribella & Scandolo a1608 [purity as a resource, beyond the thermodynamic limit].

@ __Examples__: Baeten & Naudts Ent(11)-a1009 [monatomic ideal gas, finite *N*]; Miranda & Bertoldi EJP(13) [small systems]; > s.a. black-hole
entropy; Einstein Model.

**Other States** > s.a. non-equilibrium states.

* __Orthodes__: (Boltzmann)
Statistical ensembles that satisfy the heat theorem.

@ __Stationary states__: Cohen PhyA(06) [equilibrium and non-equilibrium].

@ __Other types__: Hanel & Thurner PhyA(05)
[derivation of power-law distributions]; 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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