View source for Stochastic thermodynamics

{{Short description|Field of statistical mechanics}}

==Overview==
When a microscopic machine (e.g. a [[Engineering:Microelectromechanical systems|MEM]]) performs useful work it generates [[Physics:Heat|heat]] and [[Physics:Entropy|entropy]] as a byproduct of the process, however it is also predicted that this machine will operate in "reverse" or "backwards" over appreciable short periods. That is, heat energy from the surroundings will be converted into useful work. For larger engines, this would be described as a violation of the [[Physics:Second law of thermodynamics|second law of thermodynamics]], as entropy is consumed rather than generated. [[Physics:Loschmidt's paradox|Loschmidt's paradox]]{{sfn|Loschmidt|1876}} states that in a time reversible system, for every trajectory there exists a time-reversed anti-trajectory. As the entropy production of a trajectory and its equal anti-trajectory are of identical magnitude but opposite sign, then, so the argument goes, one cannot prove that entropy production is positive.{{sfn|Wang et al.|2002|p=050601-1}}

For a long time, exact results in thermodynamics were only possible in linear systems capable of reaching equilibrium, leaving other questions like the Loschmidt paradox unsolved. During the last few decades fresh approaches have revealed general laws applicable to non-equilibrium system which are described by nonlinear equations, pushing the range of exact thermodynamic statements beyond the realm of traditional linear solutions. These exact results are particularly relevant for small systems where appreciable (typically non-Gaussian) fluctuations occur. Thanks to stochastic thermodynamics it is now possible to accurately predict distribution functions of thermodynamic quantities relating to exchanged heat, applied work or entropy production for these systems.{{sfn|Seifert|2012|p=6}}

===Fluctuation theorem===
{{main|Physics:Fluctuation theorem}}
The mathematical resolution to Loschmidt's paradox is called the (steady state) fluctuation theorem (FT), which is a generalisation of the second law of thermodynamics. The FT shows that as a system gets larger or the trajectory duration becomes longer, entropy-consuming trajectories become more unlikely, and the expected second law behaviour is recovered.

The FT was first put forward by {{harvp|Evans et al.|1993}}{{sfnm|1a1=Gerstner|1y=2002|2a1=Wang et al.|2y=2002|2p=1|3a1=Seifert|3y=2008|3p=1|4a1=Seifert|4y=2012|4p=6|5a1=Jarzynski|5y=2011|5p=331|6a1=Campisi et al.|6y=2011|6p=3}} and much of the work done in developing and extending the theorem was accomplished by theoreticians and mathematicians interested in nonequilibrium statistical mechanics.{{efn|Evan's original numerical analysis was shown heuristically and related to a thermostatted shear-driven fluid in contact with a heat bath. It was later mathematically proven for a large class of systems using concepts from [[Chaos theory|chaotic dynamics]] by {{harvp|Gallavotti|Cohen|1995}}, for driven [[Physics:Langevin dynamics|Langevin dynamics]] by {{harvp|Kurchan|1998}} and for driven [[Physics:Diffusion|diffusive dynamic]]s by {{harvp|Lebowitz|Spohn|1999}}. A variant, a transient fluctuation theorem valid for relaxation towards the steady state was later found by {{harvp|Evans|Searles|1994}}.{{sfn|Seifert|2012|p=6}} }}{{sfn|Wang et al.|2002|p=050601-1}}

The first observation and experimental proof of Evan's fluctuation theorem (FT) was performed by {{harvp|Wang et al.|2002}}{{sfnm|2a1=Gerstner|2y=2002|1a1=Chalmers|1y=2002|3a1=Whitehouse|3y=2002}}

===Jarzynski equality===
{{main|Jarzynski equality}}
A recent review states that {{harvs|txt|last1=Jarzynski|year1=1997a|year2=1997b}} "proved a remarkable relation which allows to express the free energy difference between two equilibrium systems by a nonlinear average over the work required to drive the system in a non-equilibrium process from one state to the other. By comparing probability distributions for the work spent in the original process with the time-reversed one, Crooks found a “refinement” of the Jarzynski relation (JR),{{sfn|Crooks|1999}} now called the Crooks fluctuation theorem. Both, this relation and another refinement of the JR, the Hummer-Szabo relation became particularly useful for determining free energy differences and landscapes of biomolecules. These relations are the most prominent ones within a class of exact results (some of which found even earlier and then rediscovered) valid for non-equilibrium systems driven by time-dependent forces. A close analogy to the JR, which relates different equilibrium states, is the Hatano-Sasa relation that applies to transitions between two different non-equilibrium steady states".{{sfn|Seifert|2012|p=2}}

This is shown to be a special case of a more general relation.

===Stochastic energetics===

==History==
Classical thermodynamics, at its heart, deals with general laws governing the transformations of a system, in particular, those involving the exchange of heat, work and matter with an environment. As a central result, total entropy production is identified that in any such process can never decrease, leading, inter alia, to fundamental limits on the efficiency of heat engines and refrigerators.{{sfn|Seifert|2012|p=6}}

The thermodynamic characterisation of systems in equilibrium got its microscopic justification from equilibrium statistical mechanics which states that for a system in contact with a heat bath the probability to find it in any specific microstate is given by the Boltzmann factor. For small deviations from equilibrium, linear response theory allows to express transport properties caused by small external fields through equilibrium correlation functions. On a more phenomenological level, linear irreversible thermodynamics provides a relation between such transport coefficients and entropy production in terms of forces and fluxes. Beyond this linear response regime, for a long time, no universal exact results were available.{{sfn|Seifert|2012|p=6}}

During the last 20 years fresh approaches have revealed general laws applicable to non-equilibrium system thus pushing the range of validity of exact thermodynamic statements beyond the realm of linear response deep into the genuine non-equilibrium region. These exact results, which become particularly relevant for small systems with appreciable (typically non-Gaussian) fluctuations, generically refer to distribution functions of thermodynamic quantities like exchanged heat, applied work or entropy production.{{sfn|Seifert|2012|p=6}}

Stochastic thermodynamics combines the stochastic energetics introduced by {{harvp|Sekimoto|1998}}{{sfn|Seifert|2008|p=1}} with the idea that entropy can consistently be assigned to a single fluctuating trajectory.{{sfn|Seifert|2008|p=1}}

==Open research==
===Quantum stochastic thermodynamics===
{{main|Physics:Quantum thermodynamics}}
<!--[[File:Microreversibility for non autonomous quantum systems.png|thumb|right|286x286px|Using the mathematical definitions of non-equilibrium quantum mechanical work it is possible describe work fluctuation relations in certain quantum systems. The principle of micro-reversibility holds for quantum systems in much the same way for the classical case: but not only must the momenta be inverted (applying Θ - the quantum mechanical time reversal operator) but also the temporal sequence of Hamiltonian values.{{sfn|Campisi et al.|2011|p=10}}]]-->
Stochastic thermodynamics can be applied to driven (i.e. open) [[Physics:Open quantum system|quantum systems]] whenever the effects of quantum coherence can be ignored. The dynamics of an open quantum system is then equivalent to a classical stochastic one.  However, this is sometimes at the cost of requiring unrealistic measurements at the beginning and end of a process.{{efn|name=quantum|See {{harvp|Esposito et al.|2009}} and {{harvp|Campisi et al.|2011}} for academic review articles on non-equilibrium quantum fluctuations{{sfn|Seifert|2012|p=9}} }}{{sfn|Seifert|2012|p=9}}

Understanding non-equilibrium quantum thermodynamics more broadly is an important and active area of research. The efficiency of some computing and information theory tasks can be greatly enhanced when using [[Physics:Quantum correlation|quantum correlated]] states; quantum correlations can be used not only as a valuable resource in quantum computation, but also in the realm of quantum thermodynamics.{{sfn|Dillenschneider|Lutz|2009|p=6}} New types of quantum devices in non-equilibrium states function very differently to their classical counterparts.  For example, it has been theoretically shown that non-equilibrium quantum ratchet systems function far more efficiently then that predicted by classical thermodynamics.{{efn|See for example {{harvp|Yukawa et al.|1997}}, {{harvp|Reimann et al.|1997}}, {{harvp|Tatara et al.|1998}}{{sfn|Yukawa|2000|p=1}} }}{{sfn|Yukawa|2000|p=1}}  It has also been shown that quantum coherence can be used to enhance the efficiency of systems beyond the classical [[Physics:Carnot's theorem (thermodynamics)|Carnot limit]]. This is because it could be possible to extract work, in the form of photons, from a single heat bath. Quantum coherence can be used in effect to play the role of [[Physics:Maxwell's demon|Maxwell's demon]]{{sfn|Maruyama et al.|2009|p=20}} though the broader information theory based interpretation of the second law of thermodynamics is not violated.{{efn|See for example {{harvp|Scully|2001}},{{sfnm|1a1=Horodecki et al.|1y=2007|1p=80|2a1=Maruyama et al.|2y=2009|2p=20}}
{{harvp|Scully et al.|2003}},{{sfn|Maruyama et al.|2009|p=20}} {{harvp|Dillenschneider|Lutz|2009}},{{sfn|Modi et al.|2012|p=43}} {{harvp|Roßnagel et al.|2014}},{{Sfnm|1a1=Johannes Gutenberg Universitaet Mainz|1y=2014|2a1=Zyga|2y=2014}} and {{harvp|Roßnagel et al.|2016}}{{sfn|Cartlidge|2015}} }}{{sfn|Dillenschneider|Lutz|2009|pp=5-6}}

Quantum versions of stochastic thermodynamics have been studied for some time{{efn|See for example {{harvp|Yukawa|2000}}{{sfnm|1a1=Esposito et al.|1y=2009|1p=2|2a1=Jarzynski|2y=2011|2p=348|3a1=Campisi et al.|3y=2011|3p=8}}  and {{harvp|Mukamel|2003}}{{sfnm|1a1=Esposito et al.|1y=2009|1p=2, 8|2a1=Jarzynski|2y=2011|2p=348|3a1=Campisi et al.|3y=2011|3p=13}} }} and the past few years have seen a surge of interest in this topic.{{efn|name=quantum}} Quantum mechanics involves profound issues around the interpretation of reality (e.g. the [[Physics:Copenhagen interpretation|Copenhagen interpretation]], many-worlds, de Broglie-Bohm theory etc are all competing interpretations that try to explain the unintuitive results of quantum theory) . It is hoped that by trying to specify the quantum-mechanical definition of work, dealing with open quantum systems, analyzing exactly solvable models, or proposing and performing experiments to test non-equilibrium predictions,{{efn| See for example {{harvp|Huber et al.|2008}}{{sfnm|1a1=Campisi et al.|1y=2011|1p=16|2a1=Jarzynski|2y=2011|2p=348}} and {{harvp|An et al.|2014}}{{sfn|Roßnagel et al.|2016|p=1}} }} important insights into the interpretation of quantum mechanics and the true nature of reality will be gained.{{sfn|Jarzynski|2011|p=348}}

Applications of non-equilibrium work relations, like the Jarzynski equality, have recently been proposed for the purposes of detecting [[Quantum entanglement|quantum entanglement]] {{harv|Hide|Vedral|2010}} and to improving optimization problems (minimize or maximize a function of multivariables called the [[Loss function|cost function]]) via [[Quantum annealing|quantum annealing]] {{harv|Ohzeki|Nishimori|2011}}.{{sfn|Jarzynski|2011|p=348}}

===Active baths===
Until recently thermodynamics has only considered systems coupled to a thermal bath and, therefore, satisfying Boltzmann statistics. However, some systems do not satisfy these conditions and are far from equilibrium such as living matter, for which fluctuations are expected to be [[Gaussian function|non-Gaussian]].{{sfn|Bechinger et al.|2016|p=45}}

Active particle systems are able to take energy from their environment and drive themselves far from equilibrium. An important example of active matter is constituted by objects capable of self propulsion. Thanks to this property, they feature a series of novel behaviours that are not attainable by matter at thermal equilibrium, including, for example, swarming and the emergence of other collective properties.{{sfn|Bechinger et al.|2016|p=2}} A passive particle is considered in an active bath when it is in an environment where a wealth of active particles are present. These particles will exert nonthermal forces on the passive object so that it will experience non-thermal fluctuations and will behave widely different from a passive Brownian particle in a thermal bath. The presence of an active bath can significantly influence the microscopic thermodynamics of a particle. Experiments have suggested that the Jarzynski equality does not hold in some cases due to the presence of non-Boltzmann statistics in active baths.{{efn|See for example {{harvp|Argun et al.|2016}}{{sfn|Bechinger et al.|2016|p=12, 26, 45}} }} This observation points towards a new direction in the study of non-equilibrium statistical physics and stochastic thermodynamics, where also the environment itself is far from equilibrium.{{sfn|Bechinger et al.|2016|p=26}}

Active baths are a question of particular importance in biochemistry. For example, biomolecules within cells are coupled with an active bath due to the presence of molecular motors within the cytoplasm, which leads to striking and largely not yet understood phenomena such as the emergence of anomalous diffusion (Barkai et al., 2012). Also, protein folding might be facilitated by the presence of active fluctuations (Harder et al., 2014b) and active matter dynamics could play a central role in several biological functions (Mallory et al., 2015; Shin et al., 2015; Suzuki et al., 2015). It is an open question to what degree stochastic thermodynamics can be applied to systems coupled to active baths.{{sfn|Bechinger et al.|2016|p=45}}

== References ==
===Notes===
{{notelist|2}}

===Citations===
<!-- Inline citations added to your article will automatically display here. See https://en.wikipedia.org/wiki/WP:REFB for instructions on how to add citations. -->
{{reflist|4}}

===Academic references===
<!-- Please respect alphabetical order -->
{{refbegin|20em|indent=yes}}
* {{cite journal|last1=An|first1=Shuoming|last2=Zhang|first2=Jing-Ning|last3=Um|first3=Mark|last4=Lv |first4=Dingshun|last5=Lu|first5=Yao|last6=Zhang|first6=Junhua|last7=Yin|first7=Zhang-Qi|last8=Quan |first8=H. T. |last9=Kim|first9=Kihwan|title=Experimental test of the quantum Jarzynski equality with a trapped-ion system|journal=Nature Physics|volume=11|issue=2|year=2014|pages=193–199|issn=1745-2473 |doi=10.1038/nphys3197 |arxiv=1409.4485|ref={{harvid|An et al.|2014}} |bibcode=2015NatPh..11..193A|s2cid=118521687}}
* {{cite arXiv|last1=Argun|first1=Aykut|last2=Moradi|first2=Ali-Reza|last3=Pince|first3=Erçağ |last4=Bagci|first4=Gokhan Baris|last5=Volpe|first5=Giovanni|title=Experimental evidence of the failure of Jarzynski equality in active bath|date=2016|eprint=1601.01123|ref={{harvid|Argun et al.|2016}} |class=cond-mat.soft}}
* {{cite journal|last1=Bechinger|first1=Clemens|last2=Di Leonardo|first2=Roberto|last3=Löwen|first3=Hartmut|last4=Reichhardt|first4=Charles|last5=Volpe|first5=Giorgio|last6=Volpe |first6=Giovanni |title=Active Particles in Complex and Crowded Environments|journal=Reviews of Modern Physics|volume=88|issue=4|pages=045006|year=2016|issn=0034-6861|doi=10.1103/RevModPhys.88.045006 |arxiv=1602.00081v2 |bibcode=2016RvMP...88d5006B|ref={{harvid|Bechinger et al.|2016}} |hdl=11693/36533|s2cid=14940249}}
* {{cite journal|last1=Bertini|first1=Lorenzo|last2=De Sole|first2=Alberto|last3=Gabrielli|first3=Davide |last4=Jona-Lasinio|first4=Giovanni|last5=Landim|first5=Claudio|title=Macroscopic fluctuation theory |journal=Reviews of Modern Physics|volume=87|issue=2|year=2015|pages=593–636|issn=0034-6861 |doi=10.1103/RevModPhys.87.593|arxiv=1404.6466|bibcode=2015RvMP...87..593B|s2cid=119164130|ref={{harvid|Bertini et al.|2015}}}}
* {{cite journal|last1=Campisi|first1=Michele|last2=Hänggi|first2=Peter|last3=Talkner|first3=Peter |title=Colloquium: Quantum fluctuation relations: Foundations and applications|journal=Reviews of Modern Physics|volume=83 |issue=3|year=2011|pages=771–791|issn=0034-6861|doi=10.1103/RevModPhys.83.771 |arxiv=1012.2268|bibcode=2011RvMP...83..771C|ref={{harvid|Campisi et al.|2011}} |citeseerx=10.1.1.760.2265 |s2cid=119200058}}
* {{Cite journal|last=Crooks|first=Gavin E.|year=1999|title=Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences|journal=Physical Review E|volume=60|issue=3|pages=2721–2726 |doi=10.1103/PhysRevE.60.2721|pmid=11970075|arxiv=cond-mat/9901352 |bibcode=1999PhRvE..60.2721C |s2cid=1813818 |url=https://link.aps.org/doi/10.1103/PhysRevE.60.2721}}
* {{cite journal|last1=Dillenschneider|first1=R.|last2=Lutz|first2=E.|title=Energetics of quantum correlations|journal=EPL (Europhysics Letters)|volume=88|issue=5|year=2009|pages=50003|issn=0295-5075|doi=10.1209/0295-5075/88/50003|arxiv=0803.4067|bibcode=2009EL.....8850003D|s2cid=119262651}}
* {{cite journal|last1=Esposito|first1=Massimiliano|last2=Harbola|first2=Upendra|last3=Mukamel |first3=Shaul|title=Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems|journal=Reviews of Modern Physics|volume=81|issue=4|year=2009|pages=1665–1702|issn=0034-6861|doi=10.1103/RevModPhys.81.1665|arxiv=0811.3717|bibcode=2009RvMP...81.1665E|s2cid=56003679|ref={{harvid|Esposito et al.|2009}} |url=http://orbilu.uni.lu/handle/10993/12322}}
* {{cite journal|last1=Evans|first1=Denis J.|last2=Cohen|first2=E. G. D.|last3=Morriss|first3=G. P. |title=Probability of second law violations in shearing steady states|journal=Physical Review Letters |volume=71|issue=15 |year=1993|pages=2401–2404|issn=0031-9007|doi=10.1103/PhysRevLett.71.2401 |pmid=10054671|bibcode=1993PhRvL..71.2401E|ref={{harvid|Evans et al.|1993}} }}
* {{cite journal|last1=Evans|first1=Denis J.|last2=Searles|first2=Debra J.|title=Equilibrium microstates which generate second law violating steady states|journal=Physical Review E|volume=50|issue=2|year=1994 |pages=1645–1648|issn=1063-651X|doi=10.1103/PhysRevE.50.1645|pmid=9962139|bibcode=1994PhRvE..50.1645E |url=https://espace.library.uq.edu.au/data/UQ_282872/UQ282872_OA.pdf?Expires=1485950218&Signature=eUBZHEJA3prgm9GwaQku-HKTPc69--dGpJIykI8jhYJZyY8G~4qIkyRNw5fSEK0-kc1v5zW3EuWl3V1wrPenYFpux-xmQNQ6hAu5d9F5D3OgCVo9GHTomPn~1QWVgUbcY9DP7pZ91uq1OofnoBArWKsb7217CXJN0ElMkqX6-lfZQwi467u1R8Acd2O82ZgX8r8WCVN5Aoxun0mmFV1cihayaz7Wg~JT~CQkGzlwXYMHaR-dy92kLY28EXvgju7F2d2XdkVm3EX~pisQB5vPmLsc6SUiUgQOepNWmsmaj8idywP76JEQCm0BNZ0Zt-sdwf0Ww}}
* {{cite journal|last1=Gallavotti|first1=G.|last2=Cohen|first2=E. G. D.|date=2 April 1995 |title=Dynamical Ensembles in Nonequilibrium Statistical Mechanics|journal=Physical Review Letters |volume=74|issue=14|pages=2694–2697 |issn=0031-9007|doi=10.1103/PhysRevLett.74.2694|pmid=10057994 |arxiv=chao-dyn/9410007|bibcode=1995PhRvL..74.2694G|s2cid=259762}}
* {{cite journal|last1=Hide|first1=Jenny|last2=Vedral|first2=Vlatko|title=Detecting entanglement with Jarzynski's equality|journal=Physical Review A|volume=81|issue=6|pages=062303|year=2010|issn=1050-2947 |doi=10.1103/PhysRevA.81.062303|arxiv=0907.0179|bibcode=2010PhRvA..81f2303H|s2cid=119109512}}
* {{cite journal|last1=Horodecki|first1=Ryszard|last2=Horodecki|first2=Paweł|last3=Horodecki|first3=Michał |last4=Horodecki|first4=Karol|date=20 Apr 2007|title=Quantum entanglement|journal=Reviews of Modern Physics |volume=81 |issue=2 |pages=865–942 |doi=10.1103/RevModPhys.81.865 |arxiv=quant-ph/0702225v2|bibcode=2009RvMP...81..865H |s2cid=59577352 |ref={{harvid|Horodecki et al.|2007}}}}<!--This PDF version has page numbers-->
** {{cite journal|last1=Horodecki|first1=Ryszard|last2=Horodecki|first2=Paweł|last3=Horodecki |first3=Michał|last4=Horodecki|first4=Karol|author1-mask=1|author2-mask=1|author3-mask=1|author4-mask=1 |date=17 June 2009 |title=Quantum entanglement|journal=Reviews of Modern Physics |volume=81|issue=2 |pages=865–942|doi=10.1103/revmodphys.81.865|arxiv=quant-ph/0702225 |bibcode=2009RvMP...81..865H |s2cid=59577352<!--Blacklisted site: |url=https://sci-hub.se/10.1103/RevModPhys.81.865 |access-date=2023-03-07-->|ref=none}}<!--The journal article is only 78 pages-->
* {{cite journal|last1=Huber|first1=Gerhard|last2=Schmidt-Kaler|first2=Ferdinand|last3=Deffner |first3=Sebastian|last4=Lutz|first4=Eric|title=Employing Trapped Cold Ions to Verify the Quantum Jarzynski Equality |journal=Physical Review Letters|volume=101|issue=7|pages=070403|year=2008|issn=0031-9007|doi=10.1103/PhysRevLett.101.070403|pmid=18764513|arxiv=0808.0334|bibcode=2008PhRvL.101g0403H |s2cid=15750989|ref={{harvid|Huber et al.|2008}} }}
* {{cite journal|last1=Jarzynski|first1=C.|title=Nonequilibrium Equality for Free Energy Differences |journal=Physical Review Letters|volume=78|issue=14|year=1997a|pages=2690–2693|issn=0031-9007 |doi=10.1103/PhysRevLett.78.2690|arxiv=cond-mat/9610209 |bibcode=1997PhRvL..78.2690J|s2cid=16112025}}
* {{cite journal|last1=Jarzynski|first1=C.|title=Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach|journal=Physical Review E|volume=56|issue=5|year=1997b |pages=5018–5035 |issn=1063-651X|doi=10.1103/PhysRevE.56.5018|arxiv=cond-mat/9610209 |bibcode=1997PhRvE..56.5018J|s2cid=119101580}}
* {{cite journal|last1=Jarzynski|first1=Christopher|title=Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale|journal=Annual Review of Condensed Matter Physics |volume=2 |issue=1|year=2011|pages=329–351|issn=1947-5454|doi=10.1146/annurev-conmatphys-062910-140506 |url=http://newton.kias.re.kr/~brane/wc2012/download/Jarzynski_annurev-conmatphys-062910-140506.pdf |bibcode=2011ARCMP...2..329J}}
* {{cite journal|last1=Kurchan|first1=Jorge|title=Fluctuation theorem for stochastic dynamics |journal=Journal of Physics A: Mathematical and General|volume=31|issue=16|year=1998|pages=3719–3729 |issn=0305-4470 |doi=10.1088/0305-4470/31/16/003|arxiv=cond-mat/9709304|bibcode=1998JPhA...31.3719K |citeseerx=10.1.1.305.2208|s2cid=11226039}}
* {{cite journal|last1=Lebowitz|first1=Joel L.|last2=Spohn|first2=Herbert|title=A Gallavotti–Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics|journal=Journal of Statistical Physics |volume=95 |issue=1/2|year=1999|pages=333–365|issn=0022-4715|doi=10.1023/A:1004589714161 |arxiv=cond-mat/9811220|bibcode=1999JSP....95..333L|s2cid=16878926}}
* {{cite journal|last1=Loschmidt|first1=Joseph|title=Über den Zustand des Wärmegleichgewichtes eines Systems von Körpern mit Rücksicht auf die Schwerkraft|journal=Sitzungsber. Der Kais. Akad. D. W. Math. Naturw. II |date=1876|volume=73|pages=128|language=de|trans-title=About the state of thermal equilibrium of a system of bodies with regard to gravity}}
* {{cite journal|last1=Maruyama|first1=Koji|last2=Nori|first2=Franco|last3=Vedral|first3=Vlatko |title=Colloquium: The physics of Maxwell's demon and information|journal=Reviews of Modern Physics |volume=81|issue=1|year=2009 |pages=1–23|issn=0034-6861|doi=10.1103/RevModPhys.81.1|arxiv=0707.3400 |bibcode=2009RvMP...81....1M|s2cid=18436180|ref={{harvid|Maruyama et al.|2009}} }}
* {{cite journal|last1=Modi|first1=Kavan|last2=Brodutch|first2=Aharon|last3=Cable|first3=Hugo |last4=Paterek|first4=Tomasz|last5=Vedral|first5=Vlatko|title=The classical-quantum boundary for correlations: Discord and related measures|journal=Reviews of Modern Physics|volume=84|issue=4|year=2012 |pages=1655–1707|issn=0034-6861|doi=10.1103/RevModPhys.84.1655|arxiv=1112.6238 |bibcode=2012RvMP...84.1655M|s2cid=119698121|ref={{harvid|Modi et al.|2012}}}}
* {{cite journal|last1=Mukamel|first1=Shaul|title=Quantum Extension of the Jarzynski Relation: Analogy with Stochastic Dephasing|journal=Physical Review Letters|volume=90|issue=17|pages=170604|year=2003 |issn=0031-9007 |doi=10.1103/PhysRevLett.90.170604|pmid=12786064|bibcode=2003PhRvL..90q0604M |s2cid=13392358|url=http://mukamel.ps.uci.edu/publications/pdfs/457.pdf}}
* {{cite journal|last1=Pal|first1=P. S.|last2=Deffner|first2=Sebastian|title=Stochastic thermodynamics of relativistic Brownian motion|journal=New Journal of Physics|volume=22|year=2020|issue=7|pages=073054 |doi=10.1088/1367-2630/ab9ce6|arxiv=2003.02136|bibcode=2020NJPh...22g3054P|doi-access=free}}
* {{Cite book|last1=Peliti|first1=Luca|last2=Pigolotti|first2=Simone|year=2021|title=Stochastic Thermodynamics: An Introduction|location=Princeton|publisher=Princeton University Press|isbn=978-0-521-85103-9}}
* {{cite journal|last1=Reimann|first1=Peter|last2=Grifoni|first2=Milena|last3=Hänggi|first3=Peter |title=Quantum Ratchets|journal=Physical Review Letters|volume=79|issue=1|year=1997|pages=10–13 |issn=0031-9007 |doi=10.1103/PhysRevLett.79.10|bibcode=1997PhRvL..79...10R|s2cid=14640168 |url=https://pdfs.semanticscholar.org/eb04/b17564b443e5bcb7d9b8f579bbfb501db507.pdf|archive-url=https://web.archive.org/web/20170211080214/https://pdfs.semanticscholar.org/eb04/b17564b443e5bcb7d9b8f579bbfb501db507.pdf|url-status=dead|archive-date=2017-02-11|ref={{harvid|Reimann et al.|1997}} }}
* {{cite journal|last1=Roßnagel|first1=J.|last2=Abah|first2=O.|last3=Schmidt-Kaler|first3=F. |last4=Singer|first4=K.|last5=Lutz|first5=E.|title=Nanoscale Heat Engine Beyond the Carnot Limit |journal=Physical Review Letters |volume=112|issue=3|pages=030602|year=2014|issn=0031-9007 |doi=10.1103/PhysRevLett.112.030602|pmid=24484127|arxiv=1308.5935|bibcode=2014PhRvL.112c0602R |s2cid=1826585|ref={{harvid|Roßnagel et al.|2014}} }}
* {{cite journal|last1=Roßnagell|first1=J.|last2=Dawkins|first2=S. T.|last3=Tolazzi|first3=K. N. |last4=Abah|first4=O.|last5=Lutz|first5=E.|last6=Schmidt-Kaler|first6=F.|last7=Singer|first7=K.|title=A single-atom heat engine |journal=Science|volume=352|issue=6283|year=2016|pages=325–329|issn=0036-8075 |doi=10.1126/science.aad6320|pmid=27081067|arxiv=1510.03681|bibcode=2016Sci...352..325R|s2cid=44229532 |ref={{harvid|Roßnagel et al.|2016}} }}
* {{cite journal|last1=Scully|first1=Marlan O.|title=Extracting Work from a Single Thermal Bath via Quantum Negentropy|journal=Physical Review Letters|volume=87|issue=22|pages=220601|year=2001|issn=0031-9007 |doi=10.1103/PhysRevLett.87.220601|pmid=11736390|bibcode=2001PhRvL..87v0601S}}
* {{cite journal|last1=Scully|first1=Marlan O.|last2=Zubairy|first2=M. Suhail|last3=Agarwal |first3=Girish S.|last4=Walther|first4=Herbert.|title=Extracting Work from a Single Heat Bath via Vanishing Quantum Coherence |journal=Science|volume=299|issue=5608|year=2003|pages=862–864|issn=0036-8075|doi=10.1126/science.1078955|bibcode=2003Sci...299..862S|pmid=12511655|s2cid=120884236|ref={{harvid|Scully et al.|2003}} |doi-access=free}}
* {{cite journal|last1=Seifert|first1=U.|title=Stochastic thermodynamics: principles and perspectives |journal=The European Physical Journal B|volume=64|issue=3–4|year=2008|pages=423–431|issn=1434-6028 |doi=10.1140/epjb/e2008-00001-9|arxiv=0710.1187|bibcode=2008EPJB...64..423S|s2cid=18607099}}
* {{cite journal|last1=Ohzeki|first1=Masayuki|last2=Nishimori|first2=Hidetoshi|title=Quantum annealing with Jarzynski equality|journal=Computer Physics Communications|volume=182|issue=1|year=2011|pages=257–259|issn=0010-4655|doi=10.1016/j.cpc.2010.07.008|pmid=20867896|bibcode=2011CoPhC.182..257O|s2cid=274174 |arxiv=1007.1277}}
* {{cite journal|last1=Seifert|first1=Udo|title=Stochastic thermodynamics, fluctuation theorems and molecular machines|journal=Reports on Progress in Physics|volume=75|issue=12|year=2012|pages=126001 |issn=0034-4885 |doi=10.1088/0034-4885/75/12/126001|pmid=23168354|arxiv=1205.4176|bibcode=2012RPPh...75l6001S|s2cid=782930}}
* {{cite journal|last1=Sekimoto|first1=Ken|title=Langevin Equation and Thermodynamics|journal=Progress of Theoretical Physics Supplement|volume=130|year=1998|pages=17–27|issn=0375-9687|doi-access=free |doi=10.1143/PTPS.130.17 |bibcode=1998PThPS.130...17S|url=http://ci.nii.ac.jp/els/110001208700.pdf?id=ART0001546607&type=pdf&lang=en&host=cinii&order_no=&ppv_type=0&lang_sw=&no=1485783327&cp=}}
* {{cite journal|last1=Tatara|first1=Gen|last2=Kikuchi|first2=Makoto|last3=Yukawa|first3=Satoshi |last4=Matsukawa|first4=Hiroshi|title=Dissipation Enhanced Asymmetric Transport in Quantum Ratchets |journal=Journal of the Physical Society of Japan|volume=67|issue=4|year=1998|pages=1090–1093|issn=0031-9015|doi=10.1143/JPSJ.67.1090|arxiv=cond-mat/9711045|bibcode=1998JPSJ...67.1090T|s2cid=11253455|ref={{harvid|Tatara et al.|1998}}}}
* {{cite journal|last1=Wang|first1=G. M.|last2=Sevick|first2=E. M.|last3=Mittag|first3=Emil |last4=Searles|first4=Debra J.|last5=Evans|first5=Denis J.|title=Experimental Demonstration of Violations of the Second Law of Thermodynamics for Small Systems and Short Time Scales|journal=Physical Review Letters|volume=89|issue=5|pages=050601|year=2002|issn=0031-9007|doi=10.1103/PhysRevLett.89.050601 |url=https://espace.library.uq.edu.au/data/UQ_282878/UQ282878_OA.pdf?Expires=1485941550&Signature=GMF-QnYunEhzAIL7LoBi5JXUu6w6yy5nTAmd1p6~pi~3hWHN5ZpsVyoF~mRY92XtSTK6zyOYVxRnICr1QrD7RXx1Fyga19kMXa0qGS6bIVka~I0alobclJBOtI81XUcrUZcKYIHRXc5NJMA7RHNw2xwUVwu4WhGOqxUwN3K-83rzry83FWyG3Oxzaszh2fY1VOHH-vFUNM-CiJM63PIPG-rHCeZIhYcLgxU7O2TUqY3D1kuhpDxOQJL21lyQL74LmgsCIJ9ZOb2zwNpZ9SlXFxJFST6aQiq26OCeU~Ddou6nqXGYBRCyzSjUfMrNtRSoIBi9Qh4~N0jDQd6fLV-KPQ__&Key-Pair-Id=APKAJKNBJ4MJBJNC6NLQ |pmid=12144431|bibcode=2002PhRvL..89e0601W |hdl=10440/854 |hdl-access=free|ref={{harvid|Wang et al.|2002}}}}
* {{cite journal|last1=Yukawa|first1=Satoshi|last2=Kikuchi|first2=Macoto|last3=Tatara|first3=Gen |last4=Matsukawa|first4=Hiroshi|title=Quantum Ratchets|journal=Journal of the Physical Society of Japan |volume=66|issue=10|year=1997|pages=2953–2956|issn=0031-9015|doi=10.1143/JPSJ.66.2953|arxiv=cond-mat/9706222|bibcode=1997JPSJ...66.2953Y|s2cid=16578514|ref={{harvid|Yukawa et al.|1997}} }}
* {{cite journal|last1=Yukawa|first1=Satoshi|title=A Quantum Analogue of the Jarzynski Equality |journal=Journal of the Physical Society of Japan|volume=69|issue=8|year=2000|pages=2367–2370|issn=0031-9015 |doi=10.1143/JPSJ.69.2367|arxiv=cond-mat/0007456|bibcode=2000JPSJ...69.2367Y|s2cid=119097589}}
{{refend}}

===Press===
{{refbegin|20em|indent=yes}}
* {{cite web|last1=Cartlidge|first1=Edwin|title=Scientists build heat engine from a single atom |website=Science Magazine|url=https://www.science.org/content/article/scientists-build-heat-engine-single-atom |archiveurl=https://archive.today/20170201155641/http://www.sciencemag.org/news/2015/10/scientists-build-heat-engine-single-atom|archivedate=1 February 2017|date=21 October 2015}}
* {{cite web|last1=Chalmers|first1=Matthew|title=Second law of thermodynamics "broken"|website=New Scientist |url=https://www.newscientist.com/article/dn2572-second-law-of-thermodynamics-broken/ |archiveurl=https://archive.today/20170131095800/https://www.newscientist.com/article/dn2572-second-law-of-thermodynamics-broken/|archivedate=31 January 2017|date=19 July 2002}}
* {{cite journal|last1=Gerstner|first1=Ed|title=Second law broken: Small-scale energy fluctuations could limit minaturization|journal=Nature |url=http://www.nature.com/news/1998/020722/full/news020722-2.html |archiveurl=https://archive.today/20130129135021/http://www.nature.com/news/1998/020722/full/news020722-2.html|archivedate=29 January 2013|date=23 July 2002|doi=10.1038/news020722-2}}
* {{cite web|author1=Johannes Gutenberg Universitaet Mainz|title=Prototype of single ion heat engine created|url=https://www.sciencedaily.com/releases/2014/02/140203101031.htm|website=ScienceDaily|date=3 February 2014}}
* {{cite web|last1=Whitehouse|first1=David|date=18 July 2002|title=Beads of doubt|website=BBC News |url=http://news.bbc.co.uk/1/hi/sci/tech/2135779.stm |archivedate=2 September 2013 |archiveurl=https://archive.today/20130902141858/http://news.bbc.co.uk/2/hi/science/nature/2135779.stm}}
* {{cite web|last1=Zyga|first1=Lisa|date=27 January 2014|title=Nanoscale heat engine exceeds standard efficiency limit|url=https://phys.org/news/2014-01-nanoscale-standard-efficiency-limit.html |website=Phys.org |archiveurl=https://archive.today/20150404110141/http://phys.org/news/2014-01-nanoscale-standard-efficiency-limit.html|archivedate=4 April 2015}}
{{refend}}

{{Physics-footer}}
{{Statistical mechanics topics}}
{{Stochastic processes}}

[[Category:Statistical mechanics]]
[[Category:Thermodynamics]]
[[Category:Non-equilibrium thermodynamics]]
[[Category:Branches of thermodynamics]]
[[Category:Stochastic models]]
[[Category:Stochastic processes| ]]

{{Sourceattribution|Stochastic thermodynamics}}