Non-Equilibrium Statistical Mechanics and Thermodynamics |

**In General**
> s.a. quantum statistical mechanics; statistical mechanics
[approach to equilibrium] / states and systems.

* __Idea__: The study of
properties of non-equilibrium states (find special states equivalent to
canonical ensembles for equilibrium statistical mechanics; Characterize
them in terms of order/chaos, at various scales and near/far from
equilibrium), and understand their dynamics (near-equilibrium transport
phenomena, the arrow of time, for which we need an irreversible,
non-unitary evolution for *ρ*), and estimate the fluctuations.

* __History__: XIX century,
Lord Kelvin; 1931, L Onsager proposed regression equations for
evolution of macroscopic variables, in terms of thermodynamic forces;
1953, Onsager & Machlup added white noise; More recently computer
simulations have been carried out, e.g., using cellular automata (G
Jona-Lasinio, C Laudin & M-E Vares).

* __Issue__: Many results
on non-equilibrium systems have been derived using arguments in which
microscopic fluctuations are not reliably treated, for lack of a good
statistical theory even in the steady-state case; For example, Fourier's
law that describes heat transfer in a normal wire fails at the nanoscale.

* __Features__: Far from
equilibrium a system can develop spontaneous ordered structures with
specific patterns (but there is no extremum principle to tell us which);
This led us not to believe anymore in the "thermal death" of the universe.

* __Tools__: Intensive
thermodynamic parameters can be associated to additive conserved
quantities (such as mass, volume, ...) using a statistical approach in
far-from-equilibrium steady-state systems, under few assumptions and
without a detailed balance requirement; In lattice systems dynamics can
be studied using numerical techniques such as matrix-product-state-based
methods, for continuum systems Hamiltonian truncation methods can be applied.

@ __Books__: de Groot & Mazur 62;
Balescu 75,
97;
Lavenda 85;
Keizer 87;
Brenig 89;
Gaspard 98;
Eu 98;
Zwanzig 01;
Chen 03 [without the assumption of molecular chaos];
Le Bellac et al 04;
Ebeling & Sokolov 05;
Öttinger 05;
Mazenko 07;
Evans & Morriss 07 [liquids];
Balakrishnan 08 [II/III];
Lebon et al 08;
Ódor 08; Streater 09 [stochastic approach];
Pottier 09
[and linear irreversible processes, r JSP(11)];
Krapivsky et al 10
[r JSP(11)];
Kamenev 11
[field-theoretical methods, r PT(12)nov];
Attard 12;
Wio et al 12;
Gallavotti 14-a1311 [and chaos, irreversibility];
Livi & Politi 17.

@ __Overviews__: Ruelle PhyA(99);
Gorban & Karlin cm/03 [geometrical];
Ruelle PT(04)may;
Pokrovski EJP(05);
Abou Salem mp/06 [quantum, and thermodyamics];
Gaspard PhyA(06);
Maes et al LNM(09)-mp/07;
Zia JSP(10) [and KLS model];
Jaksic et al JMP(14)#7;
Ribeiro et al AJP(16)dec-a1601 [small quantum systems, pedagogical introduction].

@ __Framework, tools__: Schlögl PRP(80) [stochastic measures];
Gaveau & Schulman PLA(97) [master equation];
Nieuwenhuizen cm/01-MG9;
Ghosh et al AJP(06)feb [dynamical framework];
Bertin et al PRL(06) [intensive parameters];
Astumian AJP(06)aug [use of equilibrium theory];
Qiao a0709/PhyA [based on subdynamics];
Bertini et al JSP(09)
[macroscopic description of driven diffusive systems];
Hernández-Lemus & EJTP(08)-a0908 [and theory of stochastic processes];
Sadhukhan & Bhattacharjee JPA(10);
Parmeggiani Phy(12) [new methods];
Deffner & Lutz PRE(13)-a1212 [far from equilibrium, Bures angle and thermodynamic length];
Bertini et al JSP(12) [work and thermodynamic transformations];
Kleeman JSP(15)-a1307 [path-integral formalism];
Etkin a1404 [from thermokinetics to thermostatics];
Brandão et al PRL(13) [resource theory];
Duong a1412;
Gay-Balmaz & Yoshimura a1510 [Lagrangian formalism];
Pavelka et al PhyD(16)-a1512 [Poisson brackets];
Rakovszky et al NPB(16)-a1607 [Hamiltonian truncation approach];
Gay-Balmaz & Yoshimura JMP(18)-a1704 [Dirac structures],
Ent(19)-a1904 [variational formulation, rev],
a1904 [variational to bracket formulations];
Aibara et al PTEP-a1807 [gravity analog model];
> s.a. computational physics;
Effective Field Theory;
generalized thermodynamics [irreversible];
MaxEnt.

**Concepts and Phenomena**
> s.a. arrow of time; Heat Flow;
information; Master Equation;
states and systems; Transport Phenomena.

* __Laws and constraints__:
There are constraints on the evolution of non-equilibrium systems, that
form a new family of second laws.

* __Phase
transitions__: Non-equilibrium phase transitions are situations
in which system properties related to non-equilibrium phenomena, such as
transport phenomena, undergo sudden changes with the system's parameters;
> s.a. critical phenomena;
quantum phase transitions [dynamical quasicondensation].

* __Entropy production__:
Prigogine suggested that there are two universal behaviors, (i) the entropy
production rate decreases when a system approaches a steady state, and (ii)
the entropy production rate reaches its minimal value at the steady state.

@ __Chaos__: Dorfman 99;
Klages 07 [transport and fractal techniques];
> s.a. quantum chaos.

@ __Entropy__:
Holian PRA(86);
Kandrup JMP(87);
Martyushev et al JPA(07),
Maes & Netocny JMP(07) [minimum entropy production];
Maes PS(12);
Lieb & Yngvason PRS(13)-a1305;
Wittkowski et al JPA(13) [microscopic approach to entropy production];
Beretta a1312-conf [steepest entropy ascent paths towards the MaxEnt distribution];
Kadanoff a1403 [kinetic entropy, etc];
Brunelli et al a1602 [entropy production in mesoscopic quantum systems];
Camati et al PRL(16) [control by Maxwell's demon];
Šafránek et al a1905 [dynamical coarse-grained entropy].

@ __Fluctuations__:
van Zon & Cohen PhyA(04);
Lucarini JSP(08)-a0710 [response to perturbations, causality];
Criado-Sancho et al PLA(09) [flux fluctuation theorem and non-equilibrium thermodynamic potential];
Chetrite & Gawedzki JSP(09)
[diffusion, Eulerian and Lagrangian pictures, and fluctuation-dissipation relations];
Boksenbojm et al PhyA(10) [work relations and the second law];
Altaner a1210 [Stochastic Thermodynamics approach];
Pagel et al NJP(13)-a1310 [fluctuation relations for harmonic oscillators];
Funo et al PRE(16)-a1609 [small systems, work fluctuation and total entropy production].

@ __Related topics__: Frieden et al PLA(02) [and Fisher information];
Merkli CMP(01)mp/04 [positive commutators, return to equilibrium];
Carati PhyA(05) [entropies from time averages];
Chiocchetta et al PRB(16)-a1606 [short-time universal exponents, functional renormalization-group approach];
Brandão et al PNAS-a1305 [second laws;
s.a. in Bernamonti et al a1803];
> s.a. Detailed Balance; ergodic theory;
fokker-planck equation; temperature.

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send feedback and suggestions to bombelli at olemiss.edu – modified 8 jun 2019