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Thermodynamics quantum thermodynamics is the study of the relations between thermodynamics and quantum mechanics. It investigates how thermodynamic concepts such as heat, work, entropy, irreversibility, and equilibrium emerge from quantum dynamics, especially in systems far from equilibrium and in individual quantum systems. This paper is often regarded as one of the starting points of quantum theory. In the following decades, quantum theory developed into an independent framework with its own mathematical foundations. Currently, quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics. It differs from quantum statistical mechanics in its stronger emphasis on dynamical processes out of equilibrium and on the behavior of individual quantum systems. The first university course titled "Quantum Thermodynamics" was offered at MIT in the spring of 1971 by George Hatsopoulos and Elias Gyftopoulos. There is an intimate connection between quantum thermodynamics and the theory of open quantum systems.

Dynamical view

There is an intimate connection between quantum thermodynamics and the theory of open quantum systems.^[1] In this framework, the entire world is regarded as a large closed system evolving unitarily under a global Hamiltonian. For a system coupled to a bath, the global Hamiltonian is written as $H = H_{S} + H_{B} + H_{SB}$ where $H_{S}$ is the system Hamiltonian, $H_{B}$ is the bath Hamiltonian, and $H_{SB}$ is the interaction Hamiltonian.

The state of the system is obtained from the combined state by tracing over the bath degrees of freedom: $ρ_{S} (t) = {Tr}_{B} (ρ_{SB} (t)) .$

Assuming Markovian dynamics, the reduced state of the system is commonly described by the Lindblad or GKLS equation:^[2]^[3] ${\dot{ρ}}_{S} = - \frac{i}{ℏ} [H_{S}, ρ_{S}] + L_{D} (ρ_{S}) .$

Here, the first term generates unitary evolution, while the dissipator $L_{D} (ρ_{S}) = \sum_{n} [V_{n} ρ_{S} V_{n}^{†} - \frac{1}{2} (ρ_{S} V_{n}^{†} V_{n} + V_{n}^{†} V_{n} ρ_{S})]$ describes the influence of the environment through the system operators $V_{n}$ . The Markov approximation assumes that the system and bath remain uncorrelated, $ρ_{SB} = ρ_{S} \otimes ρ_{B}$ , and leads to a steady-state solution satisfying ${\dot{ρ}}_{S} (t \to \infty) = 0$ .^[1]

In the Heisenberg picture, the evolution of an observable $O$ is given by $\frac{d O}{d t} = \frac{i}{ℏ} [H_{S}, O] + L_{D}^{*} (O) + \frac{\partial O}{\partial t} .$

First law

When $O = H_{S}$ , the dynamical form of the first law of thermodynamics emerges: $\frac{d E}{d t} = ⟨ \frac{\partial H_{S}}{\partial t} ⟩ + ⟨ L_{D}^{*} (H_{S}) ⟩ .$

The first term is interpreted as power, $P = ⟨ \frac{\partial H_{S}}{\partial t} ⟩,$ while the second is the heat current,^[4]^[5]^[6] $J = ⟨ L_{D}^{*} (H_{S}) ⟩ .$

To remain consistent with thermodynamics, the dissipator must satisfy additional constraints. In particular, the invariant state should become an equilibrium Gibbs state, and a unique consistent generator can be obtained in the weak system–bath coupling limit.^[1]^[7] This issue is especially important in periodically driven systems such as quantum heat engines and quantum refrigerators. Reexaminations of time-dependent heat currents and extensions beyond weak coupling have also been proposed.^[8]^[9]

Second law and entropy

The second law of thermodynamics expresses the irreversibility of dynamics and the breaking of time-reversal symmetry. In a static viewpoint for a closed quantum system, it can be understood as a consequence of unitary evolution applied to the whole system.^[10] Dynamically, the second law can be formulated through local entropy balances and entropy production in the baths.

In thermodynamics, entropy is related to the amount of energy that can be converted into work in a given process.^[11] In quantum theory, entropy can also be associated with measurement outcomes. If the observable $A$ has the spectral decomposition $A = \sum_{j} α_{j} P_{j},$ with outcome probabilities $p_{j} = Tr (ρ P_{j})$ , the entropy of that observable is the Shannon entropy $S_{A} = - \sum_{j} p_{j} \ln p_{j} .$

The most informative entropy measure is the von Neumann entropy, $S_{vn} = - Tr (ρ \ln ρ),$ introduced by John von Neumann. It is the minimum entropy over all possible observables and satisfies $S_{A} \geq S_{vn}$ . At thermal equilibrium, the energy entropy equals the von Neumann entropy.^[12]

A well-known illustration of the link between information and thermodynamics is Maxwell's demon, whose resolution connects entropy, information, and measurement.^[13]^[14]^[15]

A Clausius-type formulation for several coupled heat baths in steady state is $\sum_{n} \frac{J_{n}}{T_{n}} \geq 0.$ A dynamical version can be proven using Spohn's inequality: $Tr (L_{D} ρ [\ln ρ (\infty) - \ln ρ]) \geq 0,$ valid for any GKLS generator with stationary state $ρ (\infty)$ .^[4]

These thermodynamic constraints are also used to test quantum transport models. Some local master-equation models appeared to violate the second law,^[16] but later work showed that, when all energy and work contributions are accounted for, local master equations can be fully reconciled with thermodynamics.^[17]

Adiabatic conditions and quantum friction

Thermodynamic adiabatic processes involve no entropy change. In quantum mechanics, an externally driven isolated system evolves unitarily under a time-dependent Hamiltonian $H (t)$ , so $S_{vn}$ remains constant. A quantum adiabatic process is often defined by the constancy of the energy entropy $S_{E}$ , which implies no net change in the populations of the instantaneous energy levels.^[1]

When the adiabatic condition is not satisfied, extra work is required. In an isolated system, this work is, in principle, recoverable because the dynamics is unitary and reversible. In practice, interactions with a bath destroy the coherence stored in the off-diagonal terms of the density matrix, and the additional energy cost is lost as a quantum analog of friction.^[18]^[19] Such friction can be reduced by shortcuts to adiabaticity, which have been demonstrated experimentally in a unitary Fermi gas.^[20]

Third law

There are two common formulations of the third law of thermodynamics, both associated with Walther Nernst. The first is the Nernst heat theorem, which states that the entropy of a pure substance in thermodynamic equilibrium approaches zero as the temperature approaches zero. The second is the unattainability principle:^[21] no finite procedure can cool a system to absolute zero.

For a cooling process, the dynamics may be written as $J_{c} (T_{c} (t)) = - c_{V} (T_{c} (t)) \frac{d T_{c} (t)}{d t},$ where $c_{V}$ is the heat capacity of the cold bath. If the heat current scales as $J_{c} \propto T_{c}^{α + 1}$ , then the third law imposes restrictions on $α$ , ensuring that entropy production at the cold bath vanishes as $T_{c} \to 0$ .^[22]

Typicality and emergence

One explanation for the emergence of thermodynamic behavior in quantum mechanics is quantum typicality. The basic idea is that, in high-dimensional Hilbert spaces, the overwhelming majority of pure states with the same initial expectation value of a generic observable will display very similar future expectation values. As a result, the dynamics of a single pure state is often well described by an ensemble average.^[23]

The von Neumann quantum ergodic theorem gives a mathematically precise formulation of this idea, stating that for typical large systems, most wave functions in an energy shell evolve so that, for most times, they are macroscopically equivalent to the microcanonical state.^[24]

Resource theory

In modern formulations, the second law can be interpreted as constraining which state transformations are physically possible. Quantum thermodynamic resource theory studies these constraints for small systems interacting with a heat bath. Instead of a single macroscopic entropy inequality, microscopic systems are governed by a family of second laws, often expressed in terms of monotonicity of generalized free energies under thermal operations.^[25]^[26]

Noncommuting conserved charges

Thermodynamic systems typically conserve charges such as energy and particle number, and these charges are often assumed to commute. Quantum theory raises the question of what happens when conserved charges do not commute. This issue has become an active subject in quantum thermodynamics.^[27]

Noncommuting charges can alter the form of thermal states,^[28] increase entanglement,^[29] modify entropy production and transport,^[30] and even challenge the eigenstate thermalization hypothesis.^[31] Recent work suggests that noncommuting charges may either hinder or enhance thermalization depending on the setting.^[32]

Engineered reservoirs

At the nanoscale, quantum systems can be prepared in states with no classical analog, and the surrounding reservoirs can also be engineered. Such reservoirs may involve coherence, squeezing, or other nonequilibrium features that strongly modify thermodynamic behavior.^[33]^[34]^[35]

These reservoirs can generate effects such as efficiencies beyond the standard Otto bound, apparent violations of Clausius inequalities, or simultaneous extraction of heat and work from a reservoir.^[36]

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+{{Short description|Quantum Collection topic on Quantum Thermodynamics}}
+{{Quantum book backlink|Statistical mechanics and kinetic theory}}
+{{Quantum article nav|previous=Physics:Quantum Relaxation and thermalization|previous label=Relaxation and thermalization|next=Physics:Quantum Band structure|next label=Band structure}}
+<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">
-{{Quantum book backlink|Statistical mechanics and kinetic theory}}
+<div style="width:280px;">
+__TOC__
+</div>
-'''Quantum thermodynamics'''<ref>{{Cite book |last1=Deffner |first1=Sebastian |url=https://iopscience.iop.org/book/mono/978-1-64327-658-8 |title=Quantum Thermodynamics: An introduction to the thermodynamics of quantum information |last2=Campbell |first2=Steve |date=2019 |publisher=Morgan & Claypool Publishers |isbn=978-1-64327-658-8 |location=San Rafael, CA |doi=10.1088/2053-2571/ab21c6|bibcode=2019qtit.book.....D }}</ref><ref>{{Cite book |title=Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions |date=2018 |publisher=[[Company:Springer Publishing|Springer Publishing]] |isbn=978-3-319-99046-0 |editor-last=Adesso |editor-first=Gerardo |edition=1st |series=Fundamental Theories of Physics |location=Cham |editor-last2=Anders |editor-first2=Janet |editor-last3=Binder |editor-first3=Felix |editor-last4=Correa |editor-first4=Luis A. |editor-last5=Gogolin |editor-first5=Christian}}</ref> is the study of the relations between [[Physics:Thermodynamics|thermodynamics]] and [[Physics:Quantum mechanics|quantum mechanics]]. It investigates how thermodynamic concepts such as heat, work, entropy, irreversibility, and equilibrium emerge from quantum dynamics, especially in systems far from equilibrium and in individual quantum systems.
+<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
+'''Thermodynamics''' quantum thermodynamics is the study of the relations between thermodynamics and quantum mechanics. It investigates how thermodynamic concepts such as heat, work, entropy, irreversibility, and equilibrium emerge from quantum dynamics, especially in systems far from equilibrium and in individual quantum systems. This paper is often regarded as one of the starting points of quantum theory. In the following decades, quantum theory developed into an independent framework with its own mathematical foundations. Currently, quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics. It differs from quantum statistical mechanics in its stronger emphasis on dynamical processes out of equilibrium and on the behavior of individual quantum systems. The first university course titled "Quantum Thermodynamics" was offered at MIT in the spring of 1971 by George Hatsopoulos and Elias Gyftopoulos. There is an intimate connection between quantum thermodynamics and the theory of open quantum systems.
+</div>
-In 1905, [[Biography:Albert Einstein|Albert Einstein]] argued that consistency between [[Physics:Thermodynamics|thermodynamics]] and [[Electromagnetism|electromagnetism]]<ref>{{cite journal | last=Einstein | first=A. | title=Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt | journal=Annalen der Physik | volume=322 | issue=6 | year=1905 | issn=0003-3804 | doi=10.1002/andp.19053220607 | bibcode=1905AnP...322..132E | pages=132–148 | language=de|doi-access=free}}</ref> leads to the conclusion that light is quantized, obtaining the relation <math>E=h\nu</math>. This paper is often regarded as one of the starting points of [[Physics:Quantum|quantum]] theory. In the following decades, quantum theory developed into an independent framework with its own mathematical foundations.<ref>{{Cite book |last1=Neumann |first1=John von |url=https://books.google.com/books?id=JLyCo3RO4qUC |title=Mathematical Foundations of Quantum Mechanics |last2=Von Neumann |first2=John |date=1955 |publisher=Princeton University Press |isbn=978-0-691-02893-4 |series=Princeton landmarks in mathematics and physics |location=Princeton Chichester |language=en}}</ref>
+<div style="width:300px;">
+[[File:Quantum_heat_engine.png|thumb|280px|Quantum Thermodynamics.]]
+</div>
-Currently, quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics. It differs from [[Physics:Quantum statistical mechanics|quantum statistical mechanics]] in its stronger emphasis on dynamical processes out of equilibrium and on the behavior of individual quantum systems.<ref name="entropy1">{{cite journal | last=Kosloff | first=Ronnie | title=Quantum Thermodynamics: A Dynamical Viewpoint | journal=Entropy | volume=15 | issue=12 | date=2013-05-29 | issn=1099-4300 | doi=10.3390/e15062100 | arxiv=1305.2268 | bibcode=2013Entrp..15.2100K | pages=2100–2128|doi-access=free}}</ref> The first university course titled "Quantum Thermodynamics" was offered at [[Organization:Massachusetts Institute of Technology|MIT]] in the spring of 1971 by [[Biography:George N. Hatsopoulos|George Hatsopoulos]] and [[Biography:Elias Gyftopoulos|Elias Gyftopoulos]].<ref>MIT Bulletin 1970-71.</ref>
-<div style="float:right; border:1px solid #e0d890; background:#fff8dc; padding:6px; margin:0 0 1em 1em; width:400px;">
-[[File:Quantum_heat_engine.png|400px]]
 </div>
-{{Thermodynamics}}
-{{Quantum mechanics}}
 == Dynamical view ==
-There is an intimate connection between quantum thermodynamics and the theory of [[Physics:Open quantum system|open quantum systems]].<ref name="entropy1" /> In this framework, the entire world is regarded as a large closed system evolving unitarily under a global [[Physics:Hamiltonian (quantum mechanics)|Hamiltonian]]. For a system coupled to a bath, the global Hamiltonian is written as
+There is an intimate connection between quantum thermodynamics and the theory of open quantum systems.<ref name="entropy1">{{cite journal | last=Kosloff | first=Ronnie | title=Quantum Thermodynamics: A Dynamical Viewpoint | journal=Entropy | volume=15 | issue=12 | date=2013-05-29 | issn=1099-4300 | doi=10.3390/e15062100 | arxiv=1305.2268 | bibcode=2013Entrp..15.2100K | pages=2100–2128|doi-access=free}}</ref> In this framework, the entire world is regarded as a large closed system evolving unitarily under a global Hamiltonian. For a system coupled to a bath, the global Hamiltonian is written as
 <math display="block">H=H_\text{S}+H_\text{B}+H_\text{SB}</math>
 where <math>H_\text{S}</math> is the system Hamiltonian, <math>H_\text{B}</math> is the bath Hamiltonian, and <math>H_\text{SB}</math> is the interaction Hamiltonian.
@@ Line 20: / Line 26: @@
 <math display="block">\rho_\text{S}(t)=\operatorname{Tr}_\text{B}(\rho_\text{SB}(t)).</math>
-Assuming [[Markov property|Markovian]] dynamics, the reduced state of the system is commonly described by the Lindblad or GKLS equation:<ref>{{cite journal | last=Lindblad | first=G. | s2cid=55220796 | title=On the generators of quantum dynamical semigroups | journal=Communications in Mathematical Physics | volume=48 | issue=2 | year=1976 | issn=0010-3616 | doi=10.1007/bf01608499 | bibcode=1976CMaPh..48..119L | pages=119–130| url=http://projecteuclid.org/euclid.cmp/1103899849 }}</ref><ref>{{cite journal | last=Gorini | first=Vittorio | title=Completely positive dynamical semigroups of N-level systems | journal=Journal of Mathematical Physics | volume=17 | issue=5 | year=1976 | issn=0022-2488 | doi=10.1063/1.522979 | bibcode=1976JMP....17..821G | pages=821–825}}</ref>
+Assuming Markovian dynamics, the reduced state of the system is commonly described by the Lindblad or GKLS equation:<ref>{{cite journal | last=Lindblad | first=G. | s2cid=55220796 | title=On the generators of quantum dynamical semigroups | journal=Communications in Mathematical Physics | volume=48 | issue=2 | year=1976 | issn=0010-3616 | doi=10.1007/bf01608499 | bibcode=1976CMaPh..48..119L | pages=119–130| url=http://projecteuclid.org/euclid.cmp/1103899849 }}</ref><ref>{{cite journal | last=Gorini | first=Vittorio | title=Completely positive dynamical semigroups of N-level systems | journal=Journal of Mathematical Physics | volume=17 | issue=5 | year=1976 | issn=0022-2488 | doi=10.1063/1.522979 | bibcode=1976JMP....17..821G | pages=821–825}}</ref>
 <math display="block">\dot\rho_\text{S}=-{i\over\hbar}[H_\text{S},\rho_\text{S}]+L_\text{D}(\rho_\text{S}).</math>
@@ Line 27: / Line 33: @@
 describes the influence of the environment through the system operators <math>V_n</math>. The Markov approximation assumes that the system and bath remain uncorrelated, <math>\rho_\text{SB}=\rho_\text{S}\otimes\rho_\text{B}</math>, and leads to a steady-state solution satisfying <math>\dot{\rho}_\text{S}(t\to\infty)=0</math>.<ref name="entropy1" />
-In the [[Physics:Heisenberg picture|Heisenberg picture]], the evolution of an observable <math>O</math> is given by
+In the Heisenberg picture, the evolution of an observable <math>O</math> is given by
 <math display="block">\frac{d O}{dt}=\frac{i}{\hbar}[H_\text{S},O]+L_\text{D}^*(O)+\frac{\partial O}{\partial t}.</math>
 == First law ==
-When <math>O=H_\text{S}</math>, the dynamical form of the [[Physics:First law of thermodynamics|first law of thermodynamics]] emerges:
+When <math>O=H_\text{S}</math>, the dynamical form of the first law of thermodynamics emerges:
 <math display="block">\frac{dE}{dt}=\left\langle\frac{\partial H_\text{S}}{\partial t}\right\rangle+\left\langle L_\text{D}^*(H_\text{S})\right\rangle.</math>
@@ Line 42: / Line 48: @@
 == Second law and entropy ==
-The [[Physics:Second law of thermodynamics|second law of thermodynamics]] expresses the irreversibility of dynamics and the breaking of time-reversal symmetry. In a static viewpoint for a closed quantum system, it can be understood as a consequence of unitary evolution applied to the whole system.<ref>{{cite journal | last1=Lieb | first1=Elliott H. | last2=Yngvason | first2=Jakob | s2cid=119620408 | title=The physics and mathematics of the second law of thermodynamics | journal=Physics Reports | volume=310 | issue=1 | year=1999 | issn=0370-1573 | doi=10.1016/s0370-1573(98)00082-9 | pages=1–96|arxiv=cond-mat/9708200| bibcode=1999PhR...310....1L }}</ref> Dynamically, the second law can be formulated through local entropy balances and entropy production in the baths.
+The second law of thermodynamics expresses the irreversibility of dynamics and the breaking of time-reversal symmetry. In a static viewpoint for a closed quantum system, it can be understood as a consequence of unitary evolution applied to the whole system.<ref>{{cite journal | last1=Lieb | first1=Elliott H. | last2=Yngvason | first2=Jakob | s2cid=119620408 | title=The physics and mathematics of the second law of thermodynamics | journal=Physics Reports | volume=310 | issue=1 | year=1999 | issn=0370-1573 | doi=10.1016/s0370-1573(98)00082-9 | pages=1–96|arxiv=cond-mat/9708200| bibcode=1999PhR...310....1L }}</ref> Dynamically, the second law can be formulated through local entropy balances and entropy production in the baths.
-In thermodynamics, [[Entropy|entropy]] is related to the amount of energy that can be converted into work in a given process.<ref name=EntropyDefinition>{{cite book|last1=Gyftopoulos|first1=E. P.|last2=Beretta|first2=G. P.|title=Thermodynamics: Foundations and Applications|year=2005|orig-date=1st ed., Macmillan, 1991|publisher=Dover Publications|place=Mineola (New York) |url=https://books.google.com/books?id=ISBN9780486439327}}</ref> In quantum theory, entropy can also be associated with measurement outcomes. If the observable <math>A</math> has the spectral decomposition
+In thermodynamics, entropy is related to the amount of energy that can be converted into work in a given process.<ref name=EntropyDefinition>{{cite book|last1=Gyftopoulos|first1=E. P.|last2=Beretta|first2=G. P.|title=Thermodynamics: Foundations and Applications|year=2005|orig-date=1st ed., Macmillan, 1991|publisher=Dover Publications|place=Mineola (New York) |url=https://books.google.com/books?id=ISBN9780486439327}}</ref> In quantum theory, entropy can also be associated with measurement outcomes. If the observable <math>A</math> has the spectral decomposition
 <math display="block">A=\sum_j \alpha_j P_j,</math>
 with outcome probabilities <math>p_j=\operatorname{Tr}(\rho P_j)</math>, the entropy of that observable is the Shannon entropy
 <math display="block">S_A=-\sum_j p_j \ln p_j.</math>
-The most informative entropy measure is the [[Physics:Von Neumann entropy|von Neumann entropy]],
+The most informative entropy measure is the von Neumann entropy,
 <math display="block">S_\text{vn}=-\operatorname{Tr}(\rho\ln\rho),</math>
-introduced by [[Biography:John von Neumann|John von Neumann]]. It is the minimum entropy over all possible observables and satisfies <math>S_A\ge S_\text{vn}</math>. At thermal equilibrium, the energy entropy equals the von Neumann entropy.<ref>{{cite journal | last=Polkovnikov | first=Anatoli | s2cid=118412733 | title=Microscopic diagonal entropy and its connection to basic thermodynamic relations | journal=Annals of Physics | volume=326 | issue=2 | year=2011 | issn=0003-4916 | doi=10.1016/j.aop.2010.08.004 | pages=486–499|arxiv=0806.2862| bibcode=2011AnPhy.326..486P }}</ref>
+introduced by John von Neumann. It is the minimum entropy over all possible observables and satisfies <math>S_A\ge S_\text{vn}</math>. At thermal equilibrium, the energy entropy equals the von Neumann entropy.<ref>{{cite journal | last=Polkovnikov | first=Anatoli | s2cid=118412733 | title=Microscopic diagonal entropy and its connection to basic thermodynamic relations | journal=Annals of Physics | volume=326 | issue=2 | year=2011 | issn=0003-4916 | doi=10.1016/j.aop.2010.08.004 | pages=486–499|arxiv=0806.2862| bibcode=2011AnPhy.326..486P }}</ref>
-A well-known illustration of the link between information and thermodynamics is [[Physics:Maxwell's demon|Maxwell's demon]], whose resolution connects entropy, information, and measurement.<ref>{{cite journal | last=Szilard | first=L. | s2cid=122038206 | title=Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen |trans-title=On the minimization of entropy in a thermodynamic system with interferences of intelligent beings| journal=Zeitschrift für Physik | volume=53 | issue=11–12 | year=1929 | issn=1434-6001 | doi=10.1007/bf01341281 | bibcode=1929ZPhy...53..840S | pages=840–856 | language=de}}</ref><ref>{{Cite book |last=Brillouin |first=Léon |url=https://archive.org/details/scienceinformati0000bril |title=Science and information theory |date=1956 |publisher=[[Company:Academic Press|Academic Press]] |location=New York |page=107 |url-access=registration}}</ref><ref>{{cite journal | last1=Maruyama | first1=Koji | last2=Nori | first2=Franco | last3=Vedral | first3=Vlatko | s2cid=18436180 | title=Colloquium: The physics of Maxwell's demon and information | journal=Reviews of Modern Physics | volume=81 | issue=1 | date=2009-01-06 | issn=0034-6861 | doi=10.1103/revmodphys.81.1 | pages=1–23|arxiv=0707.3400| bibcode=2009RvMP...81....1M }}</ref>
+A well-known illustration of the link between information and thermodynamics is Maxwell's demon, whose resolution connects entropy, information, and measurement.<ref>{{cite journal | last=Szilard | first=L. | s2cid=122038206 | title=Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen |trans-title=On the minimization of entropy in a thermodynamic system with interferences of intelligent beings| journal=Zeitschrift für Physik | volume=53 | issue=11–12 | year=1929 | issn=1434-6001 | doi=10.1007/bf01341281 | bibcode=1929ZPhy...53..840S | pages=840–856 | language=de}}</ref><ref>{{Cite book |last=Brillouin |first=Léon |url=https://archive.org/details/scienceinformati0000bril |title=Science and information theory |date=1956 |publisher=Academic Press |location=New York |page=107 |url-access=registration}}</ref><ref>{{cite journal | last1=Maruyama | first1=Koji | last2=Nori | first2=Franco | last3=Vedral | first3=Vlatko | s2cid=18436180 | title=Colloquium: The physics of Maxwell's demon and information | journal=Reviews of Modern Physics | volume=81 | issue=1 | date=2009-01-06 | issn=0034-6861 | doi=10.1103/revmodphys.81.1 | pages=1–23|arxiv=0707.3400| bibcode=2009RvMP...81....1M }}</ref>
 A Clausius-type formulation for several coupled heat baths in steady state is
 <math display="block">\sum_n \frac{J_n}{T_n}\ge 0.</math>
-A dynamical version can be proven using [[Biography:Herbert Spohn|Spohn]]'s inequality:
+A dynamical version can be proven using Spohn's inequality:
 <math display="block">\operatorname{Tr}\left(L_\text{D}\rho\left[\ln\rho(\infty)-\ln\rho\right]\right)\ge 0,</math>
 valid for any GKLS generator with stationary state <math>\rho(\infty)</math>.<ref name="spohn lebowitz 1978" />
@@ Line 64: / Line 70: @@
 == Adiabatic conditions and quantum friction ==
-Thermodynamic [[Physics:Adiabatic process|adiabatic process]]es involve no entropy change. In quantum mechanics, an externally driven isolated system evolves unitarily under a time-dependent Hamiltonian <math>H(t)</math>, so <math>S_\text{vn}</math> remains constant. A quantum adiabatic process is often defined by the constancy of the energy entropy <math>S_E</math>, which implies no net change in the populations of the instantaneous energy levels.<ref name="entropy1" />
+Thermodynamic adiabatic processes involve no entropy change. In quantum mechanics, an externally driven isolated system evolves unitarily under a time-dependent Hamiltonian <math>H(t)</math>, so <math>S_\text{vn}</math> remains constant. A quantum adiabatic process is often defined by the constancy of the energy entropy <math>S_E</math>, which implies no net change in the populations of the instantaneous energy levels.<ref name="entropy1" />
-When the adiabatic condition is not satisfied, extra work is required. In an isolated system, this work is, in principle, recoverable because the dynamics is unitary and reversible. In practice, interactions with a bath destroy the coherence stored in the off-diagonal terms of the density matrix, and the additional energy cost is lost as a quantum analog of friction.<ref>{{cite journal | last1=Kosloff | first1=Ronnie | last2=Feldmann | first2=Tova | s2cid=9292108 | title=Discrete four-stroke quantum heat engine exploring the origin of friction | journal=Physical Review E | volume=65 | issue=5 | date=2002-05-16 | issn=1063-651X | doi=10.1103/physreve.65.055102 | pmid=12059626 | page=055102(R)|arxiv=physics/0111098| bibcode=2002PhRvE..65e5102K }}</ref><ref>{{cite journal | last1=Plastina | first1=F. | last2=Alecce | first2=A. | last3=Apollaro | first3=T. J. G. | last4=Falcone | first4=G. | last5=Francica | first5=G. | last6=Galve | first6=F. | last7=Lo Gullo | first7=N. | last8=Zambrini | first8=R. | s2cid=9353450 |display-authors=5| title=Irreversible Work and Inner Friction in Quantum Thermodynamic Processes | journal=Physical Review Letters | volume=113 | issue=26 | date=2014-12-31 | issn=0031-9007 | doi=10.1103/physrevlett.113.260601 | pmid=25615295 | article-number=260601|arxiv=1407.3441| bibcode=2014PhRvL.113z0601P }}</ref> Such friction can be reduced by [[Physics:Shortcuts to adiabaticity|shortcuts to adiabaticity]], which have been demonstrated experimentally in a unitary Fermi gas.<ref>{{cite journal |last1=Deng|first1= S.|last2= Chenu|first2=A.|last3=Diao|first3= P.|last4=Li|first4=F.|last5= Yu|first5=S.|last6=Coulamy|first6=I.|last7=del Campo|first7=A|last8=Wu|first8=H.|date=2018| title= Superadiabatic quantum friction suppression in finite-time thermodynamics | journal= Science Advances | volume=4 |issue= 4|article-number=eaar5909|doi=10.1126/sciadv.aar5909|pmid= 29719865|pmc= 5922798|arxiv=1711.00650| bibcode=2018SciA....4.5909D }}</ref>
+When the adiabatic condition is not satisfied, extra work is required. In an isolated system, this work is, in principle, recoverable because the dynamics is unitary and reversible. In practice, interactions with a bath destroy the coherence stored in the off-diagonal terms of the density matrix, and the additional energy cost is lost as a quantum analog of friction.<ref>{{cite journal | last1=Kosloff | first1=Ronnie | last2=Feldmann | first2=Tova | s2cid=9292108 | title=Discrete four-stroke quantum heat engine exploring the origin of friction | journal=Physical Review E | volume=65 | issue=5 | date=2002-05-16 | issn=1063-651X | doi=10.1103/physreve.65.055102 | pmid=12059626 | page=055102(R)|arxiv=physics/0111098| bibcode=2002PhRvE..65e5102K }}</ref><ref>{{cite journal | last1=Plastina | first1=F. | last2=Alecce | first2=A. | last3=Apollaro | first3=T. J. G. | last4=Falcone | first4=G. | last5=Francica | first5=G. | last6=Galve | first6=F. | last7=Lo Gullo | first7=N. | last8=Zambrini | first8=R. | s2cid=9353450 |display-authors=5| title=Irreversible Work and Inner Friction in Quantum Thermodynamic Processes | journal=Physical Review Letters | volume=113 | issue=26 | date=2014-12-31 | issn=0031-9007 | doi=10.1103/physrevlett.113.260601 | pmid=25615295 | article-number=260601|arxiv=1407.3441| bibcode=2014PhRvL.113z0601P }}</ref> Such friction can be reduced by shortcuts to adiabaticity, which have been demonstrated experimentally in a unitary Fermi gas.<ref>{{cite journal |last1=Deng|first1= S.|last2= Chenu|first2=A.|last3=Diao|first3= P.|last4=Li|first4=F.|last5= Yu|first5=S.|last6=Coulamy|first6=I.|last7=del Campo|first7=A|last8=Wu|first8=H.|date=2018| title= Superadiabatic quantum friction suppression in finite-time thermodynamics | journal= Science Advances | volume=4 |issue= 4|article-number=eaar5909|doi=10.1126/sciadv.aar5909|pmid= 29719865|pmc= 5922798|arxiv=1711.00650| bibcode=2018SciA....4.5909D }}</ref>
 == Third law ==
-There are two common formulations of the [[Physics:Third law of thermodynamics|third law of thermodynamics]], both associated with [[Biography:Walther Nernst|Walther Nernst]]. The first is the [[Physics:Nernst heat theorem|Nernst heat theorem]], which states that the entropy of a pure substance in thermodynamic equilibrium approaches zero as the temperature approaches zero. The second is the unattainability principle:<ref>{{cite journal | last=Landsberg | first=P. T. | title=Foundations of Thermodynamics | journal=Reviews of Modern Physics | volume=28 | issue=4 | date=1956-10-01 | issn=0034-6861 | doi=10.1103/revmodphys.28.363 | bibcode=1956RvMP...28..363L | pages=363–392}}</ref> no finite procedure can cool a system to [[Physics:Absolute zero|absolute zero]].
+There are two common formulations of the third law of thermodynamics, both associated with Walther Nernst. The first is the Nernst heat theorem, which states that the entropy of a pure substance in thermodynamic equilibrium approaches zero as the temperature approaches zero. The second is the unattainability principle:<ref>{{cite journal | last=Landsberg | first=P. T. | title=Foundations of Thermodynamics | journal=Reviews of Modern Physics | volume=28 | issue=4 | date=1956-10-01 | issn=0034-6861 | doi=10.1103/revmodphys.28.363 | bibcode=1956RvMP...28..363L | pages=363–392}}</ref> no finite procedure can cool a system to absolute zero.
 For a cooling process, the dynamics may be written as
@@ Line 78: / Line 84: @@
 One explanation for the emergence of thermodynamic behavior in quantum mechanics is quantum typicality. The basic idea is that, in high-dimensional Hilbert spaces, the overwhelming majority of pure states with the same initial expectation value of a generic observable will display very similar future expectation values. As a result, the dynamics of a single pure state is often well described by an ensemble average.<ref>{{cite journal | last1=Bartsch | first1=Christian | last2=Gemmer | first2=Jochen | s2cid=34603425 | title=Dynamical Typicality of Quantum Expectation Values | journal=Physical Review Letters | volume=102 | issue=11 | date=2009-03-19 | issn=0031-9007 | doi=10.1103/physrevlett.102.110403 | pmid=19392176 | article-number=110403|arxiv=0902.0927| bibcode=2009PhRvL.102k0403B }}</ref>
-The [[Biography:John von Neumann|von Neumann]] quantum ergodic theorem gives a mathematically precise formulation of this idea, stating that for typical large systems, most wave functions in an energy shell evolve so that, for most times, they are macroscopically equivalent to the microcanonical state.<ref>{{cite journal | last1=Goldstein | first1=Sheldon | last2=Lebowitz | first2=Joel L. | last3=Mastrodonato | first3=Christian | last4=Tumulka | first4=Roderich | last5=Zanghì | first5=Nino | s2cid=816619 | title=Normal typicality and von Neumann's quantum ergodic theorem | journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | volume=466 | issue=2123 | date=2010-05-20 | issn=1364-5021 | doi=10.1098/rspa.2009.0635 | arxiv=0907.0108 | bibcode=2010RSPSA.466.3203G | pages=3203–3224}}</ref>
+The von Neumann quantum ergodic theorem gives a mathematically precise formulation of this idea, stating that for typical large systems, most wave functions in an energy shell evolve so that, for most times, they are macroscopically equivalent to the microcanonical state.<ref>{{cite journal | last1=Goldstein | first1=Sheldon | last2=Lebowitz | first2=Joel L. | last3=Mastrodonato | first3=Christian | last4=Tumulka | first4=Roderich | last5=Zanghì | first5=Nino | s2cid=816619 | title=Normal typicality and von Neumann's quantum ergodic theorem | journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | volume=466 | issue=2123 | date=2010-05-20 | issn=1364-5021 | doi=10.1098/rspa.2009.0635 | arxiv=0907.0108 | bibcode=2010RSPSA.466.3203G | pages=3203–3224}}</ref>
 == Resource theory ==
-In modern formulations, the [[Physics:Second law of thermodynamics|second law]] can be interpreted as constraining which state transformations are physically possible. Quantum thermodynamic resource theory studies these constraints for small systems interacting with a heat bath. Instead of a single macroscopic entropy inequality, microscopic systems are governed by a family of second laws, often expressed in terms of monotonicity of generalized free energies under thermal operations.<ref>{{cite journal | last1=Brandão | first1=Fernando | last2=Horodecki | first2=Michał | last3=Ng | first3=Nelly | last4=Oppenheim | first4=Jonathan | last5=Wehner | first5=Stephanie | title=The second laws of quantum thermodynamics | journal=Proceedings of the National Academy of Sciences | volume=112 | issue=11 | date=2015-02-09 | issn=0027-8424 | doi=10.1073/pnas.1411728112 | pmid=25675476 | pmc=4372001 | arxiv=1305.5278 | bibcode=2015PNAS..112.3275B | pages=3275–3279|doi-access=free}}</ref><ref>{{cite journal | last1=Goold | first1=John | last2=Huber | first2=Marcus | last3=Riera | first3=Arnau | last4=Rio | first4=Lídia del | last5=Skrzypczyk | first5=Paul | title=The role of quantum information in thermodynamics—a topical review | journal=Journal of Physics A: Mathematical and Theoretical | volume=49 | issue=14 | date=2016-02-23 | issn=1751-8113 | doi=10.1088/1751-8113/49/14/143001 | article-number=143001|doi-access=free|arxiv=1505.07835| bibcode=2016JPhA...49n3001G }}</ref>
+In modern formulations, the second law can be interpreted as constraining which state transformations are physically possible. Quantum thermodynamic resource theory studies these constraints for small systems interacting with a heat bath. Instead of a single macroscopic entropy inequality, microscopic systems are governed by a family of second laws, often expressed in terms of monotonicity of generalized free energies under thermal operations.<ref>{{cite journal | last1=Brandão | first1=Fernando | last2=Horodecki | first2=Michał | last3=Ng | first3=Nelly | last4=Oppenheim | first4=Jonathan | last5=Wehner | first5=Stephanie | title=The second laws of quantum thermodynamics | journal=Proceedings of the National Academy of Sciences | volume=112 | issue=11 | date=2015-02-09 | issn=0027-8424 | doi=10.1073/pnas.1411728112 | pmid=25675476 | pmc=4372001 | arxiv=1305.5278 | bibcode=2015PNAS..112.3275B | pages=3275–3279|doi-access=free}}</ref><ref>{{cite journal | last1=Goold | first1=John | last2=Huber | first2=Marcus | last3=Riera | first3=Arnau | last4=Rio | first4=Lídia del | last5=Skrzypczyk | first5=Paul | title=The role of quantum information in thermodynamics—a topical review | journal=Journal of Physics A: Mathematical and Theoretical | volume=49 | issue=14 | date=2016-02-23 | issn=1751-8113 | doi=10.1088/1751-8113/49/14/143001 | article-number=143001|doi-access=free|arxiv=1505.07835| bibcode=2016JPhA...49n3001G }}</ref>
 == Noncommuting conserved charges ==
@@ Line 91: / Line 97: @@
 At the nanoscale, quantum systems can be prepared in states with no classical analog, and the surrounding reservoirs can also be engineered. Such reservoirs may involve coherence, squeezing, or other nonequilibrium features that strongly modify thermodynamic behavior.<ref>{{Cite journal |last1=Scully |first1=Marlan O. |last2=Zubairy |first2=M. Suhail |last3=Agarwal |first3=Girish S. |last4=Walther |first4=Herbert |date=2003-02-07 |title=Extracting Work from a Single Heat Bath via Vanishing Quantum Coherence |url=https://www.science.org/doi/10.1126/science.1078955 |journal=Science |language=en |volume=299 |issue=5608 |pages=862–864 |doi=10.1126/science.1078955 |pmid=12511655 |bibcode=2003Sci...299..862S |issn=0036-8075|url-access=subscription }}</ref><ref>{{Cite journal |last1=Manzano |first1=Gonzalo |last2=Galve |first2=Fernando |last3=Zambrini |first3=Roberta |last4=Parrondo |first4=Juan M. R. |date=2016-05-10 |title=Entropy production and thermodynamic power of the squeezed thermal reservoir |url=https://link.aps.org/doi/10.1103/PhysRevE.93.052120 |journal=Physical Review E |language=en |volume=93 |issue=5 |article-number=052120 |doi=10.1103/PhysRevE.93.052120 |pmid=27300843 |issn=2470-0045|arxiv=1512.07881 |bibcode=2016PhRvE..93e2120M |hdl=10261/134146 }}</ref><ref>{{Cite journal |last1=de Assis |first1=Rogério J. |last2=de Mendonça |first2=Taysa M. |last3=Villas-Boas |first3=Celso J. |last4=de Souza |first4=Alexandre M. |last5=Sarthour |first5=Roberto S. |last6=Oliveira |first6=Ivan S. |last7=de Almeida |first7=Norton G. |date=2019-06-19 |title=Efficiency of a Quantum Otto Heat Engine Operating under a Reservoir at Effective Negative Temperatures |url=https://link.aps.org/doi/10.1103/PhysRevLett.122.240602 |journal=Physical Review Letters |language=en |volume=122 |issue=24 |article-number=240602 |doi=10.1103/PhysRevLett.122.240602 |pmid=31322364 |arxiv=1811.02917 |bibcode=2019PhRvL.122x0602D |issn=0031-9007}}</ref>
-These reservoirs can generate effects such as efficiencies beyond the standard Otto bound, apparent violations of Clausius inequalities, or simultaneous extraction of heat and work from a reservoir.<ref>{{Cite journal |last1=Sánchez |first1=Rafael |last2=Splettstoesser |first2=Janine |last3=Whitney |first3=Robert S. |date=2019 |title=Nonequilibrium System as a Demon |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |language=en |volume=123 |issue=21 |article-number=216801 |doi=10.1103/PhysRevLett.123.216801 |pmid=31809128 |arxiv=1811.02453 |bibcode=2019PhRvL.123u6801S |issn=0031-9007}}</ref>
+These reservoirs can generate effects such as efficiencies beyond the standard Otto bound, apparent violations of Clausius inequalities, or simultaneous extraction of heat and work from a reservoir.<ref>{{Cite journal |last1=Sánchez |first1=Rafael |last2=Splettstoesser |first2=Janine |last3=Whitney |first3=Robert S. |date=2019 |title=Nonequilibrium System as a Demon |journal=Physical Review Letters |language=en |volume=123 |issue=21 |article-number=216801 |doi=10.1103/PhysRevLett.123.216801 |pmid=31809128 |arxiv=1811.02453 |bibcode=2019PhRvL.123u6801S |issn=0031-9007}}</ref>
 == See also ==
 {{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
-* [[Physics:Quantum statistical mechanics|Quantum statistical mechanics]]
-* [[Physics:Thermal quantum field theory|Thermal quantum field theory]]
-== References ==
+= References =
 {{reflist|3}}
 {{Author|Harold Foppele}}
 {{Sourceattribution|Physics:Quantum Thermodynamics|1}}

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Physics:Quantum Thermodynamics: Difference between revisions

Latest revision as of 00:31, 24 May 2026

Dynamical view

First law

Second law and entropy

Adiabatic conditions and quantum friction

Third law

Typicality and emergence

Resource theory

Noncommuting conserved charges

Engineered reservoirs

See also

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