Quantum
Statistical Mechanical States |

**In General** \ s.a. quantum statistical mechanics.

* __Quantum
microcanonical postulate__:
A modification of the microcanonical postulate according to which for a
system in microcanonical equilibrium
all pure quantum states having the same energy expectation value are
realised with equal probability.

$ __Microcanonical state__:
The density matrix *ρ* = δ(*H*–*E*) / tr[δ(*H*–*E*)],
for
some fixed value *E* of the energy.

* __Schrödinger-Park
paradox__:
A fundamental
difficulty undermining the concept of individual "state" in the present
formulations of quantum statistical mechanics (and in its quantum
information theory
interpretation as well), which is an unavoidable consequence of an
observation
by Schrödinger and Park; To resolve it, we must either reject as unsound
the concept of state,
or reformulate quantum theory and the role of statistics in it.

@ __Equilibrium states__: Linden et al NJP(10)-a0907
[speed
of fluctuations];
Goldstein et al PRL(15)-a1506
[pure states, macroscopic vs microscopic thermal equilibrium]; Goldstein et al a1610 [macroscopic vs microscopic thermal equilibrium]; > s.a. equilibrium [thermalization].

@ __Distribution functions__: Lee PRP(95);
Lukkarinen
JPA(00)
[non-canonical]; Pandya & Tumulka JSP(14)-a1306
[canonical].

@ __Microcanonical state__: Brody et al qp/05
[equilibrium], PRS(07)qp/05
[finite-dimensional Hilbert space,
phase transition], JPCS(07)qp/06
[and grand microcanonical ensemble];
Bender et al JPA(05)qp;
Naudts & Van der Straeten JSM(06)qp
[alternative
definition]; Sugita NPCS(07)cm/06
[basis for use]; > s.a. foundations of
quantum theory.

@ __Grand canonical ensemble__: Brody et al JPA(07)qp [ergodic theorem, unitary evolution
of closed quantum systems leads to grand canonical ensemble]; Zannetti EPL(15)-a1507 [grand canonical catastrophe and condensation of fluctuations].

__Related topics__:
see mixed
states; phase-space
formulation
and wigner
functions; states
in statistical mechanics.

**Canonical Ensemble, Thermal State**

$ __Canonical partition
function__: The function *Z* = tr exp{–*βH*} (*Z* stands for "Zustandsumme"; *Q* is another common symbol); Its
value is a measure of the number of states effectively available to the
system at temperature *β* and for a given set of values for the
parameters in the Hamiltonian *H*.

@ __General references__: Brody & Hughston JMP(98)qp/97,
JMP(99)qp/97;
Tasaki PRL(98)
[from
quantum dynamics]; Albeverio et al TMMS-m.PR/05
[estimates
for quantum lattice systems]; Goldstein et al PRL(06)
[from
pure state of system + bath]; Gu PS(10)
[decomposition]; Seglar & Pérez EJP(13)
[classical limit]; Magnus & Brosens a1505
[projection operator approach for the partition function]; Perarnau-Llobet
et al a1512
[generalised Gibbs ensembles, work and entropy production].

@ __Thermal pure quantum states__: Sugiura & Shimizu PRL(12)-a1112,
PRL(13)-a1302;
Sugiura & Shimizu a1312-proc;
Hyuga et al PRB(14)-a1405
[infinite-dimensional Hilbert spaces]; Kaufman et al Sci(16)aug-a1603
[local thermalization of a globally pure state]; Dymarsky & Liu a1702 [universality of (approximately) canonical states].

@ __Eigenstate thermalization hypothesis__: Chandran et al PRA(16)-a1607 [constrained Hilbert spaces]; Shiraishi & Mori PRL(17)-a1702, comment Mondaini et al a1711 [counterexamples]; Musumbu et al a1703 [discrete time quantum walk simulations on lattices]; Lan & Powell a1706 [in quantum dimer models].

@ __Schrödinger-Park paradox__: Beretta MPLA(06)qp
[and
the Hatsopoulos-Gyftopoulos proposal]; > s.a. Paradoxes.

@ __Related topics__: Cunden et al JPA(13)-a1304
[polarized ensembles of random pure states]; Alonso et al PRE-a1403
[ensemble leading to non-extensive thermodynamic functions].

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