Quantum thermodynamics

There is an intimate connection of quantum thermodynamics with the theory of open quantum systems.{{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}} Quantum mechanics inserts dynamics into thermodynamics, giving a sound foundation to finite-time-thermodynamics. The main assumption is that the entire world is a large closed system, and therefore, time evolution is governed by a unitary transformation generated by a global Hamiltonian. For the combined system

bath scenario, the global Hamiltonian can be decomposed into:

$$H=H\_\backslash text\{S\}+H\_\backslash text\{B\}+H\_\backslash text\{SB\}$$

where $H\_\backslash text\{S\}$ is the system Hamiltonian, $H\_\backslash text\{B\}$ is the bath Hamiltonian and $H\_\backslash text\{SB\}$ is the system-bath interaction.

The state of the system is obtained from a partial trace over the combined system and bath:

$\backslash rho\_\backslash text\{S\}\; (t)\; =\; \backslash operatorname\{Tr\}\_\backslash text\{B\}\; (\backslash rho\_\backslash text\{SB\}\; (t))\; .$

Reduced dynamics is an equivalent description of the system dynamics utilizing only system operators.

Assuming Markov property for the dynamics the basic equation of motion for an open quantum system is the Lindblad equation (GKLS):{{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 }}{{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}}

$$\backslash dot\backslash rho\_\backslash text\{S\}=-\{i\backslash over\backslash hbar\}[H\_\backslash text\{S\},\backslash rho\_\backslash text\{S\}]+L\_\backslash text\{D\}(\backslash rho\_\backslash text\{S\})$$

$H\_\backslash text\{S\}$ is a (Hermitian) Hamiltonian part and $L\_\backslash text\{D\}$:

$$L\_\backslash text\{D\}(\backslash rho\_\backslash text\{S\})\; =\; \backslash sum\_n\; \backslash left[V\_n\; \backslash rho\_\backslash text\{S\}\; V\_n^\backslash dagger\; -\; \backslash tfrac\{1\}\{2\}\; \backslash left(\backslash rho\_\backslash text\{S\}\; V\_n^\backslash dagger\; V\_n\; +\; V\_n^\backslash dagger\; V\_n\; \backslash rho\_\backslash text\{S\}\backslash right)\backslash right]$$

is the dissipative part describing implicitly through system operators $V\_n$ the influence of the bath on the system.

The Markov property imposes that the system and bath are uncorrelated at all times $\backslash rho\_\backslash text\{SB\}=\backslash rho\_s\; \backslash otimes\; \backslash rho\_\backslash text\{B\}$. The L-GKS equation is unidirectional and leads any initial state $\backslash rho\_\backslash text\{S\}$ to a steady state solution which is an invariant of the equation of motion $\backslash dot\{\backslash rho\}\_\backslash text\{S\}(t\; \backslash to\; \backslash infty\; )\; =\; 0$.

The Heisenberg picture supplies a direct link to quantum thermodynamic observables. The dynamics of a system observable represented by the operator, $O$, has the form:

$$\backslash frac\{d\; O\}\{dt\}\; =\backslash frac\{i\}\{\backslash hbar\}\; [H\_\backslash text\{S\},\; O\; ]\; +L\_\backslash text\{D\}^*(O)$$

+\frac{\partial O}{\partial t}

where the possibility that the operator, $O$ is explicitly time-dependent, is included.

When $O=\; H\_\backslash text\{S\}$ the first law of thermodynamics emerges:

\frac{d E}{dt} = \left\langle \frac{\partial H_\text{S}}{\partial t }\right\rangle + \langle L_\text{D}^* (H_\text{S}) \rangle

where power is interpreted as

$$P\; =\; \backslash left\backslash langle\; \backslash frac\{\backslash partial\; H\_\backslash text\{S\}\}\{\backslash partial\; t\; \}\backslash right\backslash rangle$$

and the heat current{{Cite book |last1=Spohn |first1=Herbert |url=https://cmsr.rutgers.edu/images/people/lebowitz_joel/publications/1978spohn_leb.pdf |title=Irreversible Thermodynamics for Quantum Systems Weakly Coupled to Thermal Reservoirs |last2=Lebowitz |first2=Joel L. |date=1978 |publisher=Wiley |isbn=978-0-471-03883-2 |editor-last=Rice |editor-first=Stuart A. |volume=38 |pages=109–142 |language=en |doi=10.1002/9780470142578.ch2}}{{cite journal |last=Alicki |first=R |year=1979 |title=The quantum open system as a model of the heat engine |journal=Journal of Physics A: Mathematical and General |volume=12 |issue=5 |pages=L103–L107 |bibcode=1979JPhA...12L.103A |doi=10.1088/0305-4470/12/5/007 |issn=0305-4470}}{{cite journal |last=Kosloff |first=Ronnie |date=1984-02-15 |title=A quantum mechanical open system as a model of a heat engine |journal=The Journal of Chemical Physics |volume=80 |issue=4 |pages=1625–1631 |bibcode=1984JChPh..80.1625K |doi=10.1063/1.446862 |issn=0021-9606}}

$$J\; =\; \backslash left\backslash langle\; L\_\backslash text\{D\}^*\; (H\_\backslash text\{S\})\; \backslash right\backslash rangle.$$

Additional conditions have to be imposed on the dissipator $L\_\backslash text\{D\}$ to be consistent with thermodynamics.

First the invariant $\backslash rho\_\backslash text\{S\}(\backslash infty)$ should become an equilibrium Gibbs state. This implies that the dissipator $L\_\backslash text\{D\}$ should commute with the unitary part generated by $H\_\backslash text\{S\}$. In addition an equilibrium state is stationary and stable. This assumption is used to derive the Kubo-Martin-Schwinger stability criterion for thermal equilibrium i.e. KMS state.

A unique and consistent approach is obtained by deriving the generator, $L\_\backslash text\{D\}$, in the weak system bath

coupling limit.{{cite journal | last=Davies | first=E. B. | s2cid=122552267 | title=Markovian master equations | journal=Communications in Mathematical Physics | volume=39 | issue=2 | year=1974 | issn=0010-3616 | doi=10.1007/bf01608389 | bibcode=1974CMaPh..39...91D | pages=91–110| url=http://projecteuclid.org/euclid.cmp/1103860160 }} In this limit, the interaction energy can be neglected. This approach represents a thermodynamic idealization: it allows energy transfer, while keeping a tensor product separation

between the system and bath, i.e., a quantum version of an isothermal partition.

Markovian behavior involves a rather complicated cooperation between system and bath dynamics. This means that in

phenomenological treatments, one cannot combine arbitrary system Hamiltonians, $H\_\backslash text\{S\}$, with a given L-GKS generator. This observation is particularly important in the context of quantum thermodynamics, where it is tempting to study Markovian dynamics with an arbitrary control Hamiltonian. Erroneous derivations of the quantum master equation can easily lead to a violation of the laws of thermodynamics.

An external perturbation modifying the Hamiltonian of the system will also modify the heat flow. As a result, the L-GKS generator has to be renormalized. For a slow change, one can adopt the adiabatic approach and use the instantaneous system’s Hamiltonian to derive $L\_\backslash text\{D\}$. An important class of problems in quantum thermodynamics is periodically driven systems. Periodic quantum heat engines and power-driven refrigerators fall into this class.

A reexamination of the time-dependent heat current expression using quantum transport techniques has been proposed.{{cite journal | last1=Ludovico | first1=María Florencia | last2=Lim | first2=Jong Soo | last3=Moskalets | first3=Michael | last4=Arrachea | first4=Liliana | last5=Sánchez | first5=David | s2cid=119265583 | title=Dynamical energy transfer in ac-driven quantum systems | journal=Physical Review B | volume=89 | issue=16 | date=2014-04-21 | issn=1098-0121 | doi=10.1103/physrevb.89.161306 | page=161306(R)|arxiv=1311.4945| bibcode=2014PhRvB..89p1306L }}

A derivation of consistent dynamics beyond the weak coupling limit has been suggested.{{cite journal | last1=Esposito | first1=Massimiliano | last2=Ochoa | first2=Maicol A. | last3=Galperin | first3=Michael | s2cid=11498686 | title=Quantum Thermodynamics: A Nonequilibrium Green's Function Approach | journal=Physical Review Letters | volume=114 | issue=8 | date=2015-02-25 | issn=0031-9007 | doi=10.1103/physrevlett.114.080602 | pmid=25768745 | page=080602|arxiv=1411.1800| bibcode=2015PhRvL.114h0602E }}

Phenomenological formulations of irreversible quantum dynamics consistent with the second law and implementing the geometric idea of "steepest entropy ascent" or "gradient flow" have been suggested to model relaxation and strong coupling.{{cite journal | last1=Tabakin | first1=Frank | title=Model dynamics for quantum computing | journal=Annals of Physics | volume=383 | date=2017-06-03 | doi=10.1016/j.aop.2017.04.013 | page=33 | arxiv=1611.00664 | bibcode=2017AnPhy.383...33T | s2cid=119718818 }}{{cite journal | last1=Beretta | first1=Gian Paolo | title=The fourth law of thermodynamics: steepest entropy ascent | journal=Philosophical Transactions of the Royal Society A | volume=378 | issue=2170 | date=2020-05-01 | issn=1471-2962 | doi=10.1098/rsta.2019.0168 | page=20190168| pmid=32223406 |arxiv=1908.05768 | bibcode=2020RSPTA.37890168B | s2cid=201058607 }}

The second law of thermodynamics is a statement on the irreversibility of dynamics or, the breakup of time reversal symmetry (T-symmetry). This should be consistent with the empirical direct definition: heat will flow spontaneously from a hot source to a cold sink.

From a static viewpoint, for a closed quantum system, the 2nd law of thermodynamics is a consequence of the unitary evolution.{{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 }} In this approach, one accounts for the entropy change before and after a change in the entire system. A dynamical viewpoint is based on local accounting for the entropy changes in the subsystems and the entropy generated in the baths.

In thermodynamics, entropy is related to the amount of energy of a system that can be converted into mechanical work in a concrete process.{{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}} In quantum mechanics, this translates to the ability to measure and manipulate the system based on the information gathered by measurement. An example is the case of Maxwell’s demon, which has been resolved by Leó Szilárd.{{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}}{{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 |pages=107 |url-access=registration}}{{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 }}

The entropy of an observable is associated with the complete projective measurement of an observable,$\backslash langle\; A\; \backslash rangle$, where the operator $A$ has a spectral decomposition:

$$A\; =\; \backslash sum\_j\; \backslash alpha\_j\; P\_j,$$ where $P\_j$ are the projection operators of the eigenvalue $\backslash alpha\_j.$

The probability of outcome $j$ is $p\_j\; =\; \backslash operatorname\{Tr\}(\backslash rho\; P\_j).$ The entropy associated with the observable $\backslash langle\; A\; \backslash rangle$ is the Shannon entropy with respect to the possible outcomes:

$$S\_A\; =-\backslash sum\_j\; p\_j\; \backslash ln\; p\_j$$

The most significant observable in thermodynamics is the energy represented by the Hamiltonian operator $H,$ and its associated energy entropy, $S\_E.${{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 }}

John von Neumann suggested to single out the most informative observable to characterize the entropy of the system. This invariant is obtained by minimizing the entropy with respect to all possible observables. The most informative observable operator commutes with the state of the system. The

entropy of this observable is termed the Von Neumann entropy and is equal to

$$S\_\backslash text\{vn\}\; =\; -\; \backslash operatorname\{Tr\}(\; \backslash rho\; \backslash ln\; \backslash rho).$$

As a consequence, $S\_A\; \backslash ge\; S\_\backslash text\{vn\}$ for all observables. At thermal equilibrium the energy entropy is equal to the von Neumann entropy: $S\_E\; =S\_\backslash text\{vn\}.$

$S\_\backslash text\{vn\}$ is invariant to a unitary transformation changing the state. The Von Neumann entropy $S\_\backslash text\{vn\}$ is additive only for a system state that is composed of a tensor product of its subsystems:

$$\backslash rho\; =\; \backslash Pi\_j\; \backslash otimes\; \backslash rho\_j$$

No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature.

This statement for N-coupled heat baths in steady state becomes

$$\backslash sum\_n\; \backslash frac\{J\_n\}\{T\_n\}\; \backslash ge\; 0$$

A dynamical version of the II-law can be proven, based on Spohn's inequality:

$$\backslash operatorname\{Tr\}\; \backslash left(\; L\_\backslash text\{D\}\; \backslash rho\; \backslash left[\backslash ln\; \backslash rho(\backslash infty)\; -\; \backslash ln\; \backslash rho\; \backslash right]\; \backslash right)\; \backslash ge\; 0,$$

which is valid for any L-GKS generator, with a stationary state, $\backslash rho(\backslash infty)$.

Consistency with thermodynamics can be employed to verify quantum dynamical models of transport. For example, local models for networks where local L-GKS equations are connected through weak links have been thought to violate the second law of thermodynamics.{{cite journal | last1=Levy | first1=Amikam | last2=Kosloff | first2=Ronnie | s2cid=118498868 | title=The local approach to quantum transport may violate the second law of thermodynamics | journal=Europhysics Letters | volume=107 | issue=2 | date=2014-07-01 | issn=0295-5075 | doi=10.1209/0295-5075/107/20004 | arxiv=1402.3825 | bibcode=2014EL....10720004L | page=20004}} In 2018 has been shown that, by correctly taking into account all work and energy contributions in the full system, local master equations are fully coherent with the second law of thermodynamics.{{Cite journal |last1=De Chiara |first1=Gabriele |last2=Landi |first2=Gabriel |last3=Hewgill |first3=Adam |last4=Reid |first4=Brendan |last5=Ferraro |first5=Alessandro |last6=Roncaglia |first6=Augusto J |last7=Antezza |first7=Mauro |date=2018-11-16 |title=Reconciliation of quantum local master equations with thermodynamics |url=https://iopscience.iop.org/article/10.1088/1367-2630/aaecee |journal=New Journal of Physics |volume=20 |issue=11 |pages=113024 |doi=10.1088/1367-2630/aaecee |issn=1367-2630|arxiv=1808.10450 |bibcode=2018NJPh...20k3024D }}

Thermodynamic adiabatic processes have no entropy change. Typically, an external control modifies

the state. A quantum version of an adiabatic process can be modeled by an externally controlled time dependent

Hamiltonian $H(t)$. If the system is isolated, the dynamics are unitary, and therefore, $S\_\backslash text\{vn\}$ is a constant. A quantum adiabatic process is defined by the energy entropy $S\_E$ being constant. The quantum adiabatic condition is therefore equivalent to no net change in the population of the instantaneous energy levels. This implies that the Hamiltonian should commute with itself at different times: $[H(t),H(t\text{'})]\; =\; 0$.

When the adiabatic conditions are not fulfilled, additional work is required to reach the final control value. For an isolated system, this work is recoverable, since the dynamics is unitary and can be reversed. In this case, quantum friction can be suppressed using shortcuts to adiabaticity as demonstrated in the laboratory using a unitary Fermi gas in a time-dependent trap.{{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|pages=eaar5909|doi=10.1126/sciadv.aar5909|pmid= 29719865|pmc= 5922798|arxiv=1711.00650| bibcode=2018SciA....4.5909D }}

The coherence stored in the off-diagonal elements of the density operator carry the required information to recover the extra energy cost and reverse the dynamics. Typically, this energy is not recoverable, due to interaction with a bath that causes energy dephasing. The bath, in this case, acts like a measuring apparatus of energy. This lost energy is the quantum version of friction.{{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 }}{{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 | page=260601|arxiv=1407.3441| bibcode=2014PhRvL.113z0601P }}

There are seemingly two independent formulations of the third law of thermodynamics. Both were originally stated by Walther Nernst. The first formulation is known as the Nernst heat theorem, and can be phrased as:

• The entropy of any pure substance in thermodynamic equilibrium approaches zero as the temperature approaches zero.

The second formulation is dynamical, known as the unattainability principle{{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}}

• It is impossible by any procedure, no matter how idealized, to reduce any assembly to absolute zero temperature in a finite number of operations.

At steady state the second law of thermodynamics implies that the total entropy production is non-negative.

When the cold bath approaches the absolute zero temperature,

it is necessary to eliminate the entropy production divergence at the cold side

when $T\_\backslash text\{c\}\; \backslash rightarrow\; 0$, therefore

\dot S_\text{c} \propto - T_\text{c}^{\alpha}~~~,~~~~\alpha \geq 0~~.

For $\backslash alpha=0$ the fulfillment of the second law depends on the entropy production of the other baths, which should compensate for the negative entropy production of the cold bath.

The first formulation of the third law modifies this restriction. Instead of $\backslash alpha\; \backslash geq\; 0$ the third law imposes $\backslash alpha\; >\; 0$, guaranteeing that at absolute zero the entropy production at the cold bath is zero: $\backslash dot\; S\_\backslash text\{c\}\; =\; 0$. This requirement leads to the scaling condition of the heat current $\{\; J\}\_\backslash text\{c\}\; \backslash propto\; T\_\backslash text\{c\}^\{\backslash alpha+1\}$.

The second formulation, known as the unattainability principle can be rephrased as;{{cite journal | last1=Levy | first1=Amikam | last2=Alicki | first2=Robert | last3=Kosloff | first3=Ronnie | s2cid=24251763 | title=Quantum refrigerators and the third law of thermodynamics | journal=Physical Review E | volume=85 | issue=6 | date=2012-06-26 | issn=1539-3755 | doi=10.1103/physreve.85.061126 | pmid=23005070 | page=061126|arxiv=1205.1347| bibcode=2012PhRvE..85f1126L }}

• No refrigerator can cool a system to absolute zero temperature at finite time.

The dynamics of the cooling process is governed by the equation:

{ J}_\text{c}(T_\text{c}(t)) = -c_V(T_\text{c}(t))\frac{dT_\text{c}(t)}{dt}~~.

where $c\_V(T\_\backslash text\{c\})$ is the heat capacity of the bath. Taking $\{\; J\}\_\backslash text\{c\}\; \backslash propto\; T\_\backslash text\{c\}^\{\backslash alpha+1\}$ and $c\_V\; \backslash sim\; T\_\backslash text\{c\}^\{\backslash eta\}$ with $\{\backslash eta\}\; \backslash geq\; 0$, we can quantify this formulation by evaluating the characteristic exponent $\backslash zeta$ of the cooling process,

\frac{dT_\text{c}(t)}{dt} \propto -T_\text{c}^{\zeta}, ~~~~~ T_\text{c} \to 0, \;\;\quad {\zeta=\alpha-\eta+1}

This equation introduces the relation between the characteristic exponents $\backslash zeta$ and $\backslash alpha$. When then the bath is cooled to zero temperature in a finite time, which implies a violation of the third law. It is apparent from the last equation, that the unattainability principle is more restrictive than the Nernst heat theorem.

The basic idea of quantum typicality is that the vast majority of all pure states featuring a common expectation value of some generic observable at a given time will yield very similar expectation values of the same observable at any later time. This is meant to apply to Schrödinger type dynamics in high dimensional Hilbert spaces. As a consequence individual dynamics of expectation values are then typically well described by the ensemble average.{{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 | page=110403|arxiv=0902.0927| bibcode=2009PhRvL.102k0403B }}

Quantum ergodic theorem originated by John von Neumann is a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of termed normal typicality, i.e. the statement that, for typical large systems, every initial wave function $\backslash psi\_0$ from an energy shell is ‘normal’: it evolves in such a way that $\backslash psi\_t$ for most t, is macroscopically equivalent to the micro-canonical density matrix.{{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}}

The second law of thermodynamics can be interpreted as quantifying state transformations which are statistically unlikely so that they become effectively forbidden. The second law typically applies to systems composed of many particles interacting; Quantum thermodynamics resource theory is a formulation of thermodynamics in the regime where it can be applied to a small number of particles interacting with a heat bath. For processes which are cyclic or very close to cyclic, the second law for microscopic systems takes on a very different form than it does at the macroscopic scale, imposing not just one constraint on what state transformations are possible, but an entire family of constraints. These second laws are not only relevant for small systems, but also apply to individual macroscopic systems interacting via long-range interactions, which only satisfy the ordinary second law on average. By making precise the definition of thermal operations, the laws of thermodynamics take on a form with the first law defining the class of thermal operations, the zeroth law emerging as a unique condition ensuring the theory is nontrivial, and the remaining laws being a monotonicity property of generalised free energies.{{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}}{{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 | page=143001|doi-access=free|arxiv=1505.07835| bibcode=2016JPhA...49n3001G }}

Thermodynamic systems typically conserve quantities—known as charges—such as energy and particle number. These charges are often implicitly assumed to commute. This assumption underlies, for example, the derivation of thermal state forms, the Eigenstate Thermalization Hypothesis, and transport coefficients. However, key quantum phenomena, including uncertainty relations, arise precisely from the noncommutation of observables. How does this noncommutation affect thermodynamic behaviour?{{Cite journal |last=Majidy |first=Shayan |last2=Braasch |first2=William F. |last3=Lasek |first3=Aleksander |last4=Upadhyaya |first4=Twesh |last5=Kalev |first5=Amir |last6=Yunger Halpern |first6=Nicole |date= |title=Noncommuting conserved charges in quantum thermodynamics and beyond |url=https://www.nature.com/articles/s42254-023-00641-9 |journal=Nature Reviews Physics |language=en |volume=5 |issue=11 |pages=689–698 |doi=10.1038/s42254-023-00641-9 |issn=2522-5820|arxiv=2306.00054 }}

The noncommutation of conserved charges has been shown to challenge standard assumptions: it can invalidate conventional derivations of the thermal state,{{Cite journal |last=Yunger Halpern |first=Nicole |last2=Faist |first2=Philippe |last3=Oppenheim |first3=Jonathan |last4=Winter |first4=Andreas |date=2016-07-07 |title=Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges |url=https://www.nature.com/articles/ncomms12051 |journal=Nature Communications |language=en |volume=7 |issue=1 |pages=12051 |doi=10.1038/ncomms12051 |issn=2041-1723|arxiv=1512.01189 }} increase entanglement,{{Cite journal |last=Majidy |first=Shayan |last2=Lasek |first2=Aleksander |last3=Huse |first3=David A. |last4=Yunger Halpern |first4=Nicole |date=2023-01-03 |title=Non-Abelian symmetry can increase entanglement entropy |url=https://journals.aps.org/prb/abstract/10.1103/PhysRevB.107.045102 |journal=Physical Review B |volume=107 |issue=4 |pages=045102 |doi=10.1103/PhysRevB.107.045102|arxiv=2209.14303 }} induce critical dynamics,{{Cite journal |last=Majidy |first=Shayan |last2=Agrawal |first2=Utkarsh |last3=Gopalakrishnan |first3=Sarang |last4=Potter |first4=Andrew C. |last5=Vasseur |first5=Romain |last6=Halpern |first6=Nicole Yunger |date=2023-08-17 |title=Critical phase and spin sharpening in SU(2)-symmetric monitored quantum circuits |url=https://journals.aps.org/prb/abstract/10.1103/PhysRevB.108.054307 |journal=Physical Review B |volume=108 |issue=5 |pages=054307 |doi=10.1103/PhysRevB.108.054307|arxiv=2305.13356 }} alter entropy production,{{Cite journal |last=Manzano |first=Gonzalo |last2=Parrondo |first2=Juan M.R. |last3=Landi |first3=Gabriel T. |date=2022-01-06 |title=Non-Abelian Quantum Transport and Thermosqueezing Effects |url=https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.3.010304 |journal=PRX Quantum |volume=3 |issue=1 |pages=010304 |doi=10.1103/PRXQuantum.3.010304|arxiv=2011.04560 }} and conflict with the eigenstate thermalization hypothesis,{{Cite journal |last=Murthy |first=Chaitanya |last2=Babakhani |first2=Arman |last3=Iniguez |first3=Fernando |last4=Srednicki |first4=Mark |last5=Yunger Halpern |first5=Nicole |date=2023-04-06 |title=Non-Abelian Eigenstate Thermalization Hypothesis |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.130.140402 |journal=Physical Review Letters |volume=130 |issue=14 |pages=140402 |doi=10.1103/PhysRevLett.130.140402|arxiv=2206.05310 }} among other effects.

A central open question remains: evidence suggests that noncommuting charges can both hinder and enhance thermalization, revealing a conceptual tension at the heart of this growing field.{{Cite journal |last=Majidy |first=Shayan |date=2024-09-20 |title=Noncommuting charges can remove non-stationary quantum many-body dynamics |url=https://www.nature.com/articles/s41467-024-52588-9 |journal=Nature Communications |language=en |volume=15 |issue=1 |pages=8246 |doi=10.1038/s41467-024-52588-9 |issn=2041-1723|arxiv=2403.13046 }}

Nanoscale allows for the preparation of quantum systems in physical states without classical analogs. There, complex out-of-equilibrium scenarios may be produced by the initial preparation of either the working substance or the reservoirs of quantum particles, the latter dubbed as "engineered reservoirs".

There are different forms of engineered reservoirs. Some of them involve subtle quantum coherence or correlation effects,{{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 }}{{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 |page=052120 |doi=10.1103/PhysRevE.93.052120 |pmid=27300843 |issn=2470-0045|arxiv=1512.07881 |bibcode=2016PhRvE..93e2120M }}{{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 |page=240602 |doi=10.1103/PhysRevLett.122.240602 |pmid=31322364 |arxiv=1811.02917 |bibcode=2019PhRvL.122x0602D |issn=0031-9007}} while others rely solely on nonthermal classical probability distribution functions.{{Cite journal |last1=Pothier |first1=H. |last2=Guéron |first2=S. |last3=Birge |first3=Norman O. |last4=Esteve |first4=D. |last5=Devoret |first5=M. H. |date=1997-11-03 |title=Energy Distribution Function of Quasiparticles in Mesoscopic Wires |url=https://link.aps.org/doi/10.1103/PhysRevLett.79.3490 |journal=Physical Review Letters |language=en |volume=79 |issue=18 |pages=3490–3493 |doi=10.1103/PhysRevLett.79.3490 |bibcode=1997PhRvL..79.3490P |issn=0031-9007|url-access=subscription }}{{Cite journal |last1=Chen |first1=Yung-Fu |last2=Dirks |first2=Travis |last3=Al-Zoubi |first3=Gassem |last4=Birge |first4=Norman O. |last5=Mason |first5=Nadya |date=2009-01-23 |title=Nonequilibrium Tunneling Spectroscopy in Carbon Nanotubes |url=https://link.aps.org/doi/10.1103/PhysRevLett.102.036804 |journal=Physical Review Letters |language=en |volume=102 |issue=3 |page=036804 |doi=10.1103/PhysRevLett.102.036804 |pmid=19257380 |arxiv=0810.1308 |bibcode=2009PhRvL.102c6804C |issn=0031-9007}}{{Cite journal |last1=Altimiras |first1=C. |last2=le Sueur |first2=H. |last3=Gennser |first3=U. |last4=Cavanna |first4=A. |last5=Mailly |first5=D. |last6=Pierre |first6=F. |date=2010-11-23 |title=Tuning Energy Relaxation along Quantum Hall Channels |url=https://link.aps.org/doi/10.1103/PhysRevLett.105.226804 |journal=Physical Review Letters |language=en |volume=105 |issue=22 |page=226804 |doi=10.1103/PhysRevLett.105.226804 |pmid=21231413 |arxiv=1007.0974 |bibcode=2010PhRvL.105v6804A |issn=0031-9007}}{{Cite journal |last1=Bronn |first1=Nicholas |last2=Mason |first2=Nadya |date=2013-10-29 |title=Spatial dependence of electron interactions in carbon nanotubes |url=https://link.aps.org/doi/10.1103/PhysRevB.88.161409 |journal=Physical Review B |language=en |volume=88 |issue=16 |page=161409 |doi=10.1103/PhysRevB.88.161409 |arxiv=1306.4988 |bibcode=2013PhRvB..88p1409B |issn=1098-0121}} Interesting phenomena may emerge from the use of engineered reservoirs such as efficiencies greater than the Otto limit, violations of Clausius inequalities,{{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 |page=216801 |doi=10.1103/PhysRevLett.123.216801 |pmid=31809128 |arxiv=1811.02453 |bibcode=2019PhRvL.123u6801S |issn=0031-9007}} or simultaneous extraction of heat and work from the reservoirs.

• Quantum statistical mechanics
• Thermal quantum field theory

{{Reflist}}

• {{Cite book|url=https://iopscience.iop.org/book/978-1-64327-658-8|doi = 10.1088/2053-2571/ab21c6|title = Quantum Thermodynamics: An introduction to the thermodynamics of quantum information|year = 2019|last1 = Deffner|first1 = Sebastian|last2 = Campbell|first2 = Steve|publisher = Morgan & Claypool Publishers| bibcode=2019qtit.book.....D |isbn = 978-1-64327-658-8|s2cid = 195791624}}
• F. Binder, L. A. Correa, C. Gogolin, J. Anders, G. Adesso (eds.) (2018). Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions. Springer, {{isbn|978-3-319-99045-3}}.
• Jochen Gemmer, M. Michel, Günter Mahler (2009). Quantum thermodynamics: Emergence of Thermodynamic Behavior Within Composite Quantum Systems. 2nd edition, Springer, {{isbn|978-3-540-70509-3}}.
• Heinz-Peter Breuer, Francesco Petruccione (2007). The Theory of Open Quantum Systems. Oxford University Press, {{isbn|978-0-19-921390-0}}.

• Go to "[http://einsteinpapers.press.princeton.edu/vol2-trans/100 Concerning an Heuristic Point of View Toward the Emission and Transformation of Light]" to read an English translation of Einstein's 1905 paper. (Retrieved: 2014 Apr 11)

{{Branches of physics}}

Category:Quantum mechanics

Category:Thermodynamics

Category:Non-equilibrium thermodynamics

Category:Philosophy of thermal and statistical physics