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Quantum thermodynamics^[1]^[2] 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.

In 1905, Albert Einstein argued that consistency between thermodynamics and electromagnetism^[3] leads to the conclusion that light is quantized, obtaining the relation $E = h ν$ . 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.^[4]

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.^[5] The first university course titled "Quantum Thermodynamics" was offered at MIT in the spring of 1971 by George Hatsopoulos and Elias Gyftopoulos.^[6]

Dynamical view

There is an intimate connection between quantum thermodynamics and the theory of open quantum systems.^[5] 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:^[7]^[8] ${\dot{ρ}}_{S} = - \frac{i}{ℏ} [H_{S}, ρ_{S}] + L_{D} (ρ_{S}) .$

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 ${\dot{ρ}}_{S} (t \to \infty) = 0$ .^[5]

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,^[9]^[10]^[11] $J = ⟨ L_{D}^{*} (H_{S}) ⟩ .$

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.^[5]^[12] 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.^[13]^[14]

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.^[15] 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.^[16] 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.^[17]

A well-known illustration of the link between information and thermodynamics is Maxwell's demon, whose resolution connects entropy, information, and measurement.^[18]^[19]^[20]

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 $ρ (\infty)$ .^[9]

These thermodynamic constraints are also used to test quantum transport models. Some local master-equation models appeared to violate the second law,^[21] but later work showed that, when all energy and work contributions are accounted for, local master equations can be fully reconciled with thermodynamics.^[22]

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.^[5]

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.^[23]^[24] Such friction can be reduced by shortcuts to adiabaticity, which have been demonstrated experimentally in a unitary Fermi gas.^[25]

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:^[26] 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 $T_{c} \to 0$ .^[27]

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.^[28]

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.^[29]

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.^[30]^[31]

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.^[32]

Noncommuting charges can alter the form of thermal states,^[33] increase entanglement,^[34] modify entropy production and transport,^[35] and even challenge the eigenstate thermalization hypothesis.^[36] Recent work suggests that noncommuting charges may either hinder or enhance thermalization depending on the setting.^[37]

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.^[38]^[39]^[40]

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.^[41]

Table of contents (138 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

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Physics:Quantum Thermodynamics

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

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Table of contents (138 articles)

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