Thermodynamics
The classicalCarnot heat engine
Branches Classical Statistical Chemical Quantum thermodynamics Equilibrium / Non-equilibrium
Laws Zeroth First Second Third
Systems Closed system Open system Isolated system State Equation of state Ideal gas Real gas State of matter Phase (matter) Equilibrium Control volume Instruments Processes Isobaric Isochoric Isothermal Adiabatic Isentropic Isenthalpic Quasistatic Polytropic Free expansion Reversibility Irreversibility Endoreversibility Cycles Heat engines Heat pumps Thermal efficiency
System properties Note:Conjugate variables initalics Property diagrams Intensive and extensive properties Process functions Work Heat Functions of state Temperature / Entropy (introduction) Pressure / Volume Chemical potential / Particle number Vapor quality Reduced properties
Material properties Property databases Specific heat capacity  ${\displaystyle c=}$ ${\displaystyle T}$ ${\displaystyle \mathrm{\partial }S}$ ${\displaystyle N}$ ${\displaystyle \mathrm{\partial }T}$ Compressibility  ${\displaystyle \beta =-}$ ${\displaystyle 1}$ ${\displaystyle \mathrm{\partial }V}$ ${\displaystyle V}$ ${\displaystyle \mathrm{\partial }p}$ Thermal expansion  ${\displaystyle \alpha =}$ ${\displaystyle 1}$ ${\displaystyle \mathrm{\partial }V}$ ${\displaystyle V}$ ${\displaystyle \mathrm{\partial }T}$
Equations Carnot's theorem Clausius theorem Fundamental relation Ideal gas law Maxwell relations Onsager reciprocal relations Bridgman's equations Table of thermodynamic equations
Potentials Free energy Free entropy Internal energy ${\displaystyle U(S,V)}$ Enthalpy ${\displaystyle H(S,p)=U+pV}$ Helmholtz free energy ${\displaystyle A(T,V)=U-TS}$ Gibbs free energy ${\displaystyle G(T,p)=H-TS}$
History Culture History General Entropy Gas laws "Perpetual motion" machines Philosophy Entropy and time Entropy and life Brownian ratchet Maxwell's demon Heat death paradox Loschmidt's paradox Synergetics Theories Caloric theory Vis viva("living force") Mechanical equivalent of heat Motive power Key publications An Inquiry Concerning the Source ... Friction On the Equilibrium of Heterogeneous Substances Reflections on the Motive Power of Fire Timelines Thermodynamics Heat engines Art Education Maxwell's thermodynamic surface Entropy as energy dispersal
Scientists Bernoulli Boltzmann Bridgman Carathéodory Carnot Clapeyron Clausius de Donder Duhem Gibbs von Helmholtz Joule Kelvin Lewis Massieu Maxwell von Mayer Nernst Onsager Planck Rankine Smeaton Stahl Tait Thompson van der Waals Waterston
Other Nucleation Self-assembly Self-organization
Category
v t e

Part of a series of articles about
Quantum mechanics
${\displaystyle i\hslash \frac{d}{dt}|\mathrm{\Psi }⟩=\hat{H}|\mathrm{\Psi }⟩}$ Schrödinger equation
Introduction Glossary History
Background Classical mechanics Old quantum theory Bra–ket notation Hamiltonian Interference
Fundamentals Complementarity Decoherence Entanglement Energy level Measurement Nonlocality Quantum number State Superposition Symmetry Tunnelling Uncertainty Wave function Collapse
Experiments Bell's inequality CHSH inequality Davisson–Germer Double-slit Elitzur–Vaidman Franck–Hertz Leggett inequality Leggett–Garg inequality Mach–Zehnder Popper Quantum eraser Delayed-choice Schrödinger's cat Stern–Gerlach Wheeler's delayed-choice
Formulations Overview Heisenberg Interaction Matrix Phase-space Schrödinger Sum-over-histories (path integral)
Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger
Interpretations Bayesian Consciousness causes collapse Consistent histories Copenhagen de Broglie–Bohm Ensemble Hidden-variable Many-worlds Objective-collapse Quantum logic Superdeterminism Relational Transactional
Advanced topics Relativistic quantum mechanics Quantum field theory Quantum information science Quantum computing Quantum chaos EPR paradox Density matrix Scattering theory Quantum statistical mechanics Quantum machine learning
Scientists Aharonov Bell Bethe Blackett Bloch Bohm Bohr Born Bose de Broglie Compton Dirac Davisson Debye Ehrenfest Einstein Everett Fock Fermi Feynman Glauber Gutzwiller Heisenberg Hilbert Jordan Kramers Lamb Landau Laue Moseley Millikan Onnes Pauli Planck Rabi Raman Rydberg Schrödinger Simmons Sommerfeld von Neumann Weyl Wien Wigner Zeeman Zeilinger
v t e

Quantum thermodynamics^[1][2] is the study of the relations between two independent physical theories:thermodynamics andquantum mechanics. The two independent theories address the physical phenomena of light and matter.In 1905,Albert Einstein argued that the requirement of consistency betweenthermodynamics andelectromagnetism^[3] leads to the conclusion that light is quantized, obtaining the relation${\displaystyle E=h\nu }$. This paper is the dawn ofquantum theory. In a few decadesquantum theory became established with an independent set of rules.^[4] Currently quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics. It differs fromquantum statistical mechanics in the emphasis on dynamical processes out of equilibrium. In addition, there is a quest for the theory to be relevant for a single individual quantum system.

There is an intimate connection of quantum thermodynamics with the theory ofopen quantum systems.^[5] 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 globalHamiltonian. For the combined systembath scenario, the global Hamiltonian can be decomposed into:$${\displaystyle H=H_{\text{S}}+H_{\text{B}}+H_{\text{SB}}}$$where${\displaystyle H_{\text{S}}}$ is the system Hamiltonian,${\displaystyle H_{\text{B}}}$ is the bath Hamiltonian and${\displaystyle H_{\text{SB}}}$ is the system-bath interaction.

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

${\displaystyle {\rho }_{\text{S}}(t)={\mathrm{Tr}}_{\text{B}}({\rho }_{\text{SB}}(t)).}$

Reduced dynamics is an equivalent description of thesystem dynamics utilizing only system operators.AssumingMarkov property for the dynamics the basic equation of motion for an open quantum system is theLindblad equation (GKLS):^[6][7]$${\displaystyle {\dot{\rho }}_{\text{S}}=-\frac{i}{\hslash }[H_{\text{S}},{\rho }_{\text{S}}]+L_{\text{D}}({\rho }_{\text{S}})}$$${\displaystyle H_{\text{S}}}$ is a (Hermitian)Hamiltonian part and${\displaystyle L_{\text{D}}}$:$${\displaystyle L_{\text{D}}({\rho }_{\text{S}})=\sum _n\left[V_n{\rho }_{\text{S}}{V}_n^{†}-\frac{1}{2}\left({\rho }_{\text{S}}{V}_n^{†}{V}_n+V_n^{†}{V}_n{\rho }_{\text{S}}\right)\right]}$$is the dissipative part describing implicitly through system operators${\displaystyle V_n}$ the influence of the bath on the system.TheMarkov property imposes that the system and bath are uncorrelated at all times${\displaystyle {\rho }_{\text{SB}}={\rho }_s\otimes {\rho }_{\text{B}}}$. The L-GKS equation is unidirectional and leads any initial state${\displaystyle {\rho }_{\text{S}}}$ to a steady state solution which is an invariant of the equation of motion${\displaystyle {\dot{\rho }}_{\text{S}}(t\to \mathrm{\infty })=0}$.^[5]

TheHeisenberg picture supplies a direct link to quantum thermodynamic observables. The dynamics of a system observable represented by the operator,${\displaystyle O}$, has the form:$${\displaystyle \frac{dO}{dt}=\frac{i}{\hslash }[H_{\text{S}},O]+L_{\text{D}}^{\ast }(O)+\frac{\mathrm{\partial }O}{\mathrm{\partial }t}}$$where the possibility that the operator,${\displaystyle O}$ is explicitly time-dependent, is included.

When${\displaystyle O=H_{\text{S}}}$ thefirst law of thermodynamics emerges:$${\displaystyle \frac{dE}{dt}=⟨\frac{\mathrm{\partial }{H}_{\text{S}}}{\mathrm{\partial }t}⟩+⟨L_{\text{D}}^{\ast }(H_{\text{S}})⟩}$$where power is interpreted as

$${\displaystyle P=⟨\frac{\mathrm{\partial }{H}_{\text{S}}}{\mathrm{\partial }t}⟩}$$

and the heat current^[8][9][10]

$${\displaystyle J=⟨L_{\text{D}}^{\ast }(H_{\text{S}})⟩.}$$

Additional conditions have to be imposed on the dissipator${\displaystyle L_{\text{D}}}$ to be consistent with thermodynamics.

First the invariant${\displaystyle {\rho }_{\text{S}}(\mathrm{\infty })}$ should become an equilibriumGibbs state. This implies that the dissipator${\displaystyle L_{\text{D}}}$ should commute with the unitary part generated by${\displaystyle H_{\text{S}}}$.^[5] 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,${\displaystyle L_{\text{D}}}$, in the weak system bathcoupling limit.^[11] In this limit, the interaction energy can be neglected. This approach represents a thermodynamic idealization: it allows energy transfer, while keeping a tensor product separationbetween the system and bath, i.e., a quantum version of anisothermal partition.

Markovian behavior involves a rather complicated cooperation between system and bath dynamics. This means that inphenomenological treatments, one cannot combine arbitrary system Hamiltonians,${\displaystyle H_{\text{S}}}$, 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${\displaystyle L_{\text{D}}}$. An important class of problems in quantum thermodynamics is periodically driven systems. Periodicquantum heat engines and power-drivenrefrigerators fall into this class.

A reexamination of the time-dependent heat current expression using quantum transport techniques has been proposed.^[12]

A derivation of consistent dynamics beyond the weak coupling limit has been suggested.^[13]

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.^[14][15]

Thesecond 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.^[16] 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 theentropy 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.^[17] 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 ofMaxwell's demon, which has been resolved byLeó Szilárd.^[18][19][20]

Theentropy of an observable is associated with the complete projective measurement of an observable,${\displaystyle ⟨A⟩}$, where the operator${\displaystyle A}$ has a spectral decomposition:$${\displaystyle A=\sum _j{\alpha }_j{P}_j,}$$ where${\displaystyle P_j}$ are the projection operators of theeigenvalue${\displaystyle {\alpha }_j.}$The probability of outcome${\displaystyle j}$ is${\displaystyle p_j=\mathrm{Tr}(\rho P_j).}$ The entropy associated with the observable${\displaystyle ⟨A⟩}$ is theShannon entropy with respect to the possible outcomes:$${\displaystyle S_A=-\sum _j{p}_j\ln p_j}$$

The most significant observable in thermodynamics is the energy represented by theHamiltonian operator${\displaystyle H,}$ and its associated energy entropy,${\displaystyle S_E.}$^[21]

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. Theentropy of this observable is termed theVon Neumann entropy and is equal to$${\displaystyle S_{\text{vn}}=-\mathrm{Tr}(\rho \ln \rho ).}$$As a consequence,${\displaystyle S_A\ge S_{\text{vn}}}$ for all observables. At thermal equilibrium the energyentropy is equal to thevon Neumann entropy:${\displaystyle S_E=S_{\text{vn}}.}$

${\displaystyle S_{\text{vn}}}$ is invariant to a unitary transformation changing the state. TheVon Neumann entropy${\displaystyle S_{\text{vn}}}$ is additive only for a system state that is composed of a tensor product of its subsystems:$${\displaystyle \rho ={\mathrm{\Pi }}_j\otimes {\rho }_j}$$

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$${\displaystyle \sum _n\frac{J_n}{T_n}\ge 0}$$

A dynamical version of the II-law can be proven, based onSpohn's inequality:^[8]$${\displaystyle \mathrm{Tr}\left(L_{\text{D}}\rho \left[\ln \rho (\mathrm{\infty })-\ln \rho \right]\right)\ge 0,}$$which is valid for any L-GKS generator, with a stationary state,${\displaystyle \rho (\mathrm{\infty })}$.^[5]

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 thesecond law of thermodynamics.^[22] 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 thesecond law of thermodynamics.^[23]

Thermodynamicadiabatic processes have no entropy change. Typically, an external control modifiesthe state. A quantum version of anadiabatic process can be modeled by an externally controlled time dependentHamiltonian${\displaystyle H(t)}$. If the system is isolated, the dynamics are unitary, and therefore,${\displaystyle S_{\text{vn}}}$ is a constant. A quantum adiabatic process is defined by the energy entropy${\displaystyle S_E}$ being constant. The quantumadiabatic 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:${\displaystyle [H(t),H(t^{\prime })]=0}$.

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 usingshortcuts to adiabaticity as demonstrated in the laboratory using a unitary Fermi gas in a time-dependent trap.^[24]

Thecoherence 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.^[25][26]

There are seemingly two independent formulations of thethird law of thermodynamics. Both were originally stated byWalther Nernst. The first formulation is known as theNernst 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 theunattainability principle^[27]

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

At steady state thesecond law of thermodynamics implies that the totalentropy production is non-negative.

When the cold bath approaches the absolute zero temperature,it is necessary to eliminate theentropy production divergence at the cold sidewhen${\displaystyle T_{\text{c}}\to 0}$, therefore$${\displaystyle {\dot{S}}_{\text{c}}\propto -T_{\text{c}}^{\alpha },\alpha \ge 0.}$$For${\displaystyle \alpha =0}$ the fulfillment of thesecond law depends on theentropy production of the other baths, which should compensate for the negativeentropy production of the cold bath.

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

The second formulation, known as the unattainability principle can be rephrased as;^[28]

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

The dynamics of the cooling process is governed by the equation:$${\displaystyle J_{\text{c}}(T_{\text{c}}(t))=-c_V(T_{\text{c}}(t))\frac{d{T}_{\text{c}}(t)}{dt}.}$$where${\displaystyle c_V(T_{\text{c}})}$ is the heat capacity of the bath. Taking${\displaystyle J_{\text{c}}\propto T_{\text{c}}^{\alpha +1}}$ and${\displaystyle c_V\sim T_{\text{c}}^{\eta }}$ with${\displaystyle \eta \ge 0}$, we can quantify this formulation by evaluating the characteristic exponent${\displaystyle \zeta }$ of the cooling process,$${\displaystyle \frac{d{T}_{\text{c}}(t)}{dt}\propto -T_{\text{c}}^{\zeta },T_{\text{c}}\to 0,\zeta =\alpha -\eta +1}$$

This equation introduces the relation between the characteristic exponents${\displaystyle \zeta }$ and${\displaystyle \alpha }$. 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 theNernst 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.^[29]

Quantum ergodic theorem originated byJohn 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${\displaystyle {\psi }_0}$ from an energy shell is 'normal': it evolves in such a way that${\displaystyle {\psi }_t}$ for most t, is macroscopically equivalent to the micro-canonical density matrix.^[30]

Thesecond 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.^[31][32]

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, theEigenstate Thermalization Hypothesis, andtransport coefficients. However, key quantum phenomena, including uncertainty relations, arise precisely from the noncommutation of observables. How does this noncommutation affect thermodynamic behaviour?^[33]

The noncommutation of conserved charges has been shown to challenge standard assumptions: it can invalidate conventional derivations of the thermal state,^[34] increase entanglement,^[35] induce critical dynamics,^[36] alter entropy production,^[37] and conflict with the eigenstate thermalization hypothesis,^[38] 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.^[39]

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,^[40][41][42] while others rely solely on nonthermal classical probability distribution functions.^{[43][44][45][46]} Interesting phenomena may emerge from the use of engineered reservoirs such as efficiencies greater than the Otto limit,^[42] violations of Clausius inequalities,^[47] or simultaneous extraction of heat and work from the reservoirs.^[41]

• Quantum statistical mechanics
• Thermal quantum field theory

• Deffner, Sebastian; Campbell, Steve (2019).Quantum Thermodynamics: An introduction to the thermodynamics of quantum information. Morgan & Claypool Publishers.Bibcode:2019qtit.book.....D.doi:10.1088/2053-2571/ab21c6.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 "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)

• Quantum mechanics
• Thermodynamics
• Non-equilibrium thermodynamics
• Philosophy of thermal and statistical physics

• CS1 German-language sources (de)
• CS1 errors: ISBN date
• Articles with short description
• Short description is different from Wikidata