States
in Statistical Mechanics| 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
given by the Boltzmann distribution =
exp{–
H}/Z,
with H the Hamiltonian, and Z = tr
exp{–H}
the partition function (Z stands for "Zustandsumme").
* 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)
[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].
@ Relationship with microcanonical: Lukkarinen JPA(99)cm/98;
Gurarie AJP(07)aug;
Touchette AJP(08)jan
[inequivalent]; > 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].
@ 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: Chazottes & Hochman a0907 [example with no zero-temperature
limit]; >
s.a. 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{–(E–
N)}/Z , Z:=
tr exp{–(E–N)}
, =
chemical potential .
@ 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].
@ 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 = 
(H–E)
/ tr[(H–E)],
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]; Kiessling JSP(09)-a0810 [Ruelle's
regularized microcanonical measure].
@ Quantum: Brody et al qp/05 [equilibrium],
PRS(07)qp/05 [finite-dimensional 
,
phase transition], JPCS(07)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 > 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)
[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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send feedback and suggestions to bombelli at olemiss.edu – modified 21
jul
2009