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; Eu 98;
Gaspard 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 & Jou 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].

@ __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 &
Estrada-Gil 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]; Brandăo 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 a1704 [Dirac structures]; > s.a. computational
physics; generalized thermodynamics [irreversible]; MaxEnt.

**Concepts and Phenomena** > s.a. arrow
of time; Detailed Balance;
ergodic theory; Heat
Flow; information; Master
Equation.

* __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].

@ __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]; > s.a. fokker-planck
equation; temperature.

> __Examples of
phenomena__: see dissipation; Nyquist
Theorem; Relaxation; Self-Organization;
superconductivity; Transport;
turbulence.

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send feedback and suggestions to bombelli at olemiss.edu – modified 14 apr
2017