Statistical Mechanical Equilibrium  

In General > s.a. ergodic theory; fluctuations; states; thermodynamics; wigner functions.
* Foundational problem: Physically, what we know or don't know about a system can't affect its evolution, but in a statistical mechanics interpretation of thermodynamics the information is crucial in explaining the evolution towards equilibrium.
* Calculations: All quantities of interest can be obtained from the distribution function; The easiest one to use is the canonical one.
@ Notions of equilibrium: Lavis SHPMP(05); Pitowsky SHPMP(06); Lavis PhSc(08)dec [degrees of equilibrium, incorporating the Boltzmann and Gibbs approaches]; Werndl & Frigg SHPMP(15)-a1510, a1510 [new definition of equilibrium, and results]; Werndl & Frigg a1606-in [general criteria for the existence of an equilibrium state], a1607-proc [with stochastic microdynamics]; Lazarovici a1809 [comment on Werndl & Frigg]; Limaa et al PRD-a1911 [in general relativity].
@ Notions of equilibrium, quantum systems: Bogdanov et al qp/06-conf [as an effect of quantum entanglement]; del Rio et al PRE(16)-a1401 [thermalization relative to a particular reference].

Approach to Equilibrium / Thermalization > s.a. history of physics.
* Approach to thermal equilibrium: The circumstances under which a system reaches thermal equilibrium, and how to derive this from basic dynamical laws, has been a major question from the very beginning of thermodynamics and statistical mechanics; Some results are known, such as the correlation between relaxation to equilibrium and chaos (Krylov showed that a sufficient condition is that the system be mixing), but it remains an open problem.
* Approaches: Statistical mechanics attempts to situate equilibrium at the macroscopic level in the Boltzmann approach and at the statistical level in the Gibbs approach; The issue has not really been settled.
* Remark: The distribution function in some phenomena, such as very unstable systems like K-flows, appears to be fundamental, and not just a way of encoding our ignorance.
@ History: Goldstein LNP-cm/01 [Boltzmann's analysis]; Leeds PhSc(03)jan [Albert's vs Boltzmann's approach].
@ Approach to equilibrium: Lebowitz RMP(99)mp/00; Flores-Hidalgo et al PRA(09)-a0903 [renormalized-coordinate approach]; Reimann & Evstigneev PRE(13)-a1311 [under experimentally realistic conditions]; Malabarba et al PRE(16)-a1604, Dong et al a1706 [classical vs quantum equilibration]; Ho & De Roeck a2011 [prethermalization, theory]; > s.a. arrow of time; diffusion; H-Theorem; Langevin Equation; thermodynamics [foundations].
@ And ergodicity and chaos: Srednicki PRE(94)cm, NYAS(95)cm/94, cm/94 [chaos]; Earman & Rédei BJPS(96) [and ergodic theory]; Gallavotti Chaos(98), cm/06 [rev; ensembles, ergodicity and chaoticity]; Vranas PhSc(98)dec [generalized, "epsilon" ergodicity]; Zaslavsky PT(99)aug [limitations of chaotic dynamics, relaxation to equilibrium and decay of fluctuations]; Srednicki JPA(99)cm [quantum chaotic system]; Castiglione et al 08 [and dynamical systems].
@ Microcanonical equilibrium: Bander cm/96; Hari Dass et al IJMPA(03)cm/01.
@ Related topics: Batterman PhSc(98)jun [Khinchin's program]; Wang et al AJP(07)may [equilibrium with few particles]; Akhmedov a2105 [in an expanding spacetime, equilibration]; > s.a. Maxwell-Boltzmann Distribution; Thermodynamic Limit.

Thermalization of a Quantum System > s.a. quantum statistical mechanics; states in quantum statistical mechanics.
* Idea: The first approach was von Neumann's "quantum ergodic hypothesis"; 2020, a common approach is based on understanding the spreading of entanglement in out-of-equilibrium quantum many-body systems; Other approaches are based on the eigenstate thermalization hypothesis and on entropy production; One can get mixed states from pure states by coarse-graining, or some self-thermalization.
@ General references: Albert BJPS(94) [and the collapse of a quantum wave function]; Hari Dass et al IJMPA(03)cm/01 [self-thermalization]; Scarani EPJST(07)-a0707 [entanglement and irreversibility]; Reimann PRL(08), comment Gong & Duan a1109 [realistic quantum system]; Linden et al PRE(09)-a0812; Lychkovskiy PRE(10)-a0903; Yuan et al JPSJ(09)-a0904 [decoherence and thermalization]; Cho & Kim PRL(10)-a0911 [from pure quantum states]; Yuan et al JPSJ(09)-a0904 [decoherence and thermalization]; Cho & Kim PRL(10)-a0911 [from pure quantum states]; Tasaki a1003 [proof and examples]; Ponomarev et al PRL(11)-a1004; Dizadji-Bahmani PhSc(11) [Aharonov approach]; Riera et al PRL(12)-a1102 [in nature and on a quantum computer]; Ponomarev et al EPL(12)-a1107 [between two finite systems, from equipartition]; Cui et al a1110; Cramer NJP(12)-a1112 [randomized local Hamiltonians]; Larson JPB(13)-a1304 [and non-integrability]; Yang et al PRE(14)-a1311 [canonical vs non-canonical equilibration dynamics]; Xiong et al SRep(15)-a1311 [and non-Markovian dynamics]; Khlebnikov & Kruczenski a1312 [isolated system]; Gogolin & Eisert RPP(16)-a1503 [rev]; Ithier & Benaych-Georges a1510 [from first principles], PRA(17)-a1706 [and random interactions]; Zhdanov et al PRL(17)-a1706 [not without correlations]; Grimmer et al a1805; Dymarsky a1806 [isolated system]; Parker et al PRX(19); Koukoulekidis et al a1912 [emergence of the Gibbs state from passive states]; Jacob et al a2012 [from quantum scattering].
@ Eigenstate Thermalization Hypothesis: Rigol & Srednicki PRL(12) [alternatives]; De Palma et al PRL(15)-a1506; Hosur & Qi PRE(16)-a1507; D'Alessio et al AiP(16)-a1509 [and consequences]; Deutsch RPP(18) [rev]; Richter et al PRE(19)-a1805 [and the route to equilibrium]; Foini & Kurchan PRE(19)-a1809 [and out-of-time-order correlators]; Inozemcev & Volovich a1811 [modified formulation]; Campos Venuti & Liu a1904 [and quantum ergodicity]; Wilming et al PRL(19); Inozemcev & Volovich a2002 [and thermalization]; Sugimoto et al a2005 [test].
@ Entropy production: Esposito et al NJP(10); Ptaszyński & Esposito PRL(19) [open system].
@ Time scales: Brandão et al PRE(12)-a1108 [under a random Hamiltonian]; Short & Farrelly NJP(12)-a1110 [bound]; Goldstein et al PRL(13)-a1307, NJP(15)-a1402; García-Pintos et al PRX(17)-a1509; Farrelly NJP(16)-a1512.
@ Macroscopic, many-body systems: Goldstein et al PRE(10)-a0911; Reimann NJP(10); Torres-Herrera et al PS(15)-a1403 [many-body systems, numerical and analytical studies]; Tasaki JSP(16)-a1507 [isolated, and typicality]; Ostilli & Presilla PRA(17)-a1611 [phenomenological theory]; Swingle & Yao Phy(17) scrambling of information in spin systems]; Anza a1808-PhD [and quantum gravity]; > s.a. Superpositions.
@ Other types of systems: Belton CMP(10) [quantum random walks]; Kota et al JSM(11)-a1102 [two-body random ensemble]; Veniaminov a1112 [interacting particles in a random medium]; Blanc & Lewin JMP(12)-a1201 [disordered quantum Coulomb systems]; Kay a1209 [system and bath of comparable sizes, and black holes]; Sedlmayr et al PRL(13)-a1212 [1D fermionic chain]; Zhuang & Wu PRE(13)-a1308 [chaotic system]; Campbell et al SRep(16)-a1507 [bosonic atoms in a double-well potential]; Cherian et al EPL(19)-1604 [two-level quantum system]; Tang et al PRX(18) [dipolar quantum Newton's cradle]; Jaschke et al a1805 [1D Ising chain]; Gluza et al a1809 [non-interacting lattice fermions]; van Enk a1810 [two identical bosons]; Wang et al a1905 [small quantum systems].

Examples of Systems
@ General references: Rothstein AJP(57)nov [irreversibility and information in nuclear spin echo]; Nauenberg AJP(04)mar [radiation].
@ Breakdown of thermalization: news pw(06)apr [gas that does not approach equilibrium]; Rigol PRL(09) [and integrability]; Wang et al EPJD(13)-a1207 [non-Markovian dynamics in a spin star system].

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