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Quantum Thermodynamics

Ronnie Kosloff

arXiv:1305.2268v1quant-ph

TL;DR

Quantum thermodynamics asks how thermodynamic laws emerge from quantum mechanics and how their descriptions remain consistent. The paper uses open quantum systems and reduced dynamics to construct local thermodynamic partitions and examine all four laws. It concludes that apparent violations generally reflect flaws in approximations, especially master-equation derivations.

  • Problem

    The paper addresses how thermodynamic laws emerge from quantum mechanics while the two theories describe the same subject from different foundations.

  • Method

    The review uses open quantum systems to construct local partitions, reduced dynamics, and quantum models of thermodynamic processes.

  • Results

    Quantum thermodynamics applies down to a single particle, with quantum and thermodynamical adiabatic behavior linked and deviations producing friction and reduced efficiency.

  • Takeaways & Limitations

    Thermodynamics can serve as a consistency check for approximate quantum theories.

Abstract

from arXiv · show

Quantum thermodynamics addresses the emergence of thermodynamical laws from quantum mechanics. The link is based on the intimate connection of quantum thermodynamics with the theory of open quantum systems. Quantum mechanics inserts dynamics into thermodynamics giving a sound foundation to finite-time-thermodynamics. The emergence of the 0-law I-law II-law and III-law of thermodynamics from quantum considerations is presented. The emphasis is on consistence between the two theories which address the same subject from different foundations. We claim that inconsistency is the result of faulty analysis pointing to flaws in approximations.

I. INTRODUCTION

Quantum thermodynamics studies thermodynamical processes in quantum dynamics and seeks a consistent account of thermodynamic laws from quantum mechanics. Open quantum systems provide local partitions and dynamical descriptions connecting quantum models with finite-time thermodynamics.

  • I. INTRODUCTION: Quantum thermodynamics connects thermodynamic laws with their quantum dynamical origin, addressing two theories that developed separately.The review emphasizes consistency between thermodynamics and quantum mechanics.
  • I. INTRODUCTION: Open quantum systems separate a system from its environment and provide local reduced dynamics for thermodynamic descriptions.The system is represented by a mixed state when system-bath entanglement is possible, with observables obtained from expectation values.
  • I. INTRODUCTION: Quantum heat engines and refrigerators incorporate dynamics through reciprocating cycles, continuous engines, and system-bath heat-transfer segments.Finite-power operation requires non-equilibrium heat transfer and optimization of time allocation across cycle segments.
  • I. INTRODUCTION: Finite-speed adiabatic operation produces quantum friction because the state cannot remain diagonal in the instantaneous Hamiltonian basis.Infinitely slow operation satisfies adiabatic conditions but yields zero power, motivating faster operation despite losses.
  • I. INTRODUCTION: The review examines quantum manifestations of the 0-law, I-law, II-law, and III-law, including equilibrium, energy conservation, entropy generation, and unattainability of absolute zero.The III-law is considered through both vanishing equilibrium entropy and the dynamical unattainability principle.
  • I. INTRODUCTION: Reduced dynamics is derived from global system-bath evolution using partial traces, weak-coupling and fast-bath assumptions, or completely positive Markovian semigroups.The LGKS generator is the standard form for Markovian reduced dynamics.

III. THE 0-LAW

The 0-law emerges from thermodynamic partitions and equilibrium conditions in locally described quantum systems. The same framework also exposes ambiguities in assigning heat and power when relying on non-unique generators.

  • III. THE 0-LAW: Thermodynamic partitions idealize subsystems while allowing heat flow across an isothermal boundary without destroying subsystem descriptions.Consistency with globally structured quantum mechanics is therefore non-trivial.
  • III. THE 0-LAW: The LGKS generator is closely linked to a thermodynamic framework when local observables describe a system without system-bath entanglement.This local description corresponds thermodynamically to extensivity of observables.
  • III. THE 0-LAW: The KMS condition for a tensor-product state implies the 0-law for a network of coupled systems.For thermal equilibrium, the Gibbs state is stationary, and under stated mild conditions it is unique while initial states relax toward equilibrium.
  • III. THE 0-LAW: For a closed passive universe, unitary evolution keeps the total energy expectation constant.This is the global quantum statement underlying energy conservation.
  • III. THE 0-LAW: The dynamical I-law balances local energy changes against heat currents from baths and power from external sources.The paper identifies Markovian equations for heat currents and power as dynamical versions of the I-law.
  • III. THE 0-LAW: LGKS representations are non-unique, and transformations preserving dynamics can change the heat current, requiring additional restrictions for a consistent I-law.The ambiguity stems from how system-bath interaction energy is assigned to the system Hamiltonian.

A. The dynamical generator in the weak system-bath coupling limit

The review derives a thermodynamically consistent reduced generator in the weak system-bath coupling limit. Markovian and frequency-separation approximations produce a Davies/LGKS dynamics with thermal equilibrium properties and well-defined heat flow.

  • A. The dynamical generator in the weak system-bath coupling limit: Weak system-bath coupling neglects interaction energy while allowing energy transfer across a tensor-product system-bath partition.This is presented as the quantum version of an isothermal partition.
  • A. The dynamical generator in the weak system-bath coupling limit: The interaction Hamiltonian couples system and bath operators through λ ˆHint = λˆS ⊗ˆB, with the bath assumed stationary.The coupling strength is represented by λ.
  • A. The dynamical generator in the weak system-bath coupling limit: The reduced system dynamics is obtained by tracing over the bath and expanding the interaction-picture evolution in powers of the coupling.Renormalizing terms include Lamb-shift corrections, while the lowest-order Born approximation initiates the cumulant expansion.
  • A. The dynamical generator in the weak system-bath coupling limit: The Markov approximation and decomposition into Hamiltonian Bohr frequencies yield a Davies generator as a special LGKS case.The Davies construction removes off-diagonal frequency terms by coarse-graining fast oscillations in time.
  • A. The dynamical generator in the weak system-bath coupling limit: G(ω) ≥ 0 is required for complete positivity of the resulting Markovian master equation.The positivity follows from Bochner’s theorem.
  • A. The dynamical generator in the weak system-bath coupling limit: The Davies generator fixes the Hamiltonian-dissipator partition and shows that arbitrary control Hamiltonians cannot generally be combined with a given LGKS generator.Erroneous master-equation derivations can violate thermodynamic laws.
  • A. The dynamical generator in the weak system-bath coupling limit: The KMS condition yields a Gibbs stationary state and, under mild conditions, relaxation toward a unique equilibrium consistent with the 0-law.The thermal generator is constructed from the bath’s equilibrium correlations.
  • A. The dynamical generator in the weak system-bath coupling limit: Weak coupling preserves autonomous system observables while restoring consistent definitions of heat flow and external power.This connects the reduced dynamics directly to the quantum thermodynamic I-law.

B. Thermal generators for periodic driving fields

For periodically driven open systems, the dissipative generator must incorporate the time dependence of the Hamiltonian. Floquet theory provides an effective Hamiltonian and Fourier decomposition for deriving the periodic generator and its steady state.

  • Periodic driving requires incorporating temporal Hamiltonian changes into the dissipative generator, especially when changes are fast.For slow changes, an adiabatic approach using the temporary Hamiltonian is available; fast changes complicate the derivation.
  • Periodically driven systems include periodic heat engines and power-driven refrigerators under a Markovian assumption.
  • The derivation uses a renormalised Hamiltonian that is periodic in time and an effective Hamiltonian defined from the periodic propagator’s spectrum.The effective Hamiltonian’s eigenvalues are called quasi-energies.
  • Floquet theory replaces the ordinary Fourier decomposition with a double Fourier decomposition involving the drive frequency and differences between quasi-energies.The drive frequency is Ω = 2π/τ, while the relevant frequency set contains differences ϵ_k − ϵ_l.
  • The interaction-picture generator is assembled from Fourier components, and its solutions include periodic steady states defined by states annihilated by the transformed generator.
  • For multiple couplings and heat baths, the generator is represented as a sum of corresponding terms.

C. Heat flows and power for periodically driven open systems

In periodically driven open systems, heat currents decompose into local Fourier components associated with exchanges between the system and each bath. The resulting dynamical energy-current formulation is consistent with the thermodynamic II-law.

  • Heat currents for periodic systems decompose into Fourier components for each bath, and the interaction-picture generator decomposes correspondingly.
  • Each local heat current describes energy exchange ω_q + qΩ with bath j for any initial state.
  • The heat current associated with bath j is the sum of its local heat currents over Floquet-frequency multiples and effective-Hamiltonian Bohr frequencies.
  • The periodically driven system’s dynamical I-law is formulated through energy currents between the system and baths.
  • The energy-current derivation for periodically driven systems is consistent with the II-law of thermodynamics.

V. THE II-LAW

The II-law is framed as irreversibility and positive entropy generation, with quantum formulations accounting either globally through unitary evolution or locally through subsystem and bath entropy changes. Quantum measurement, observable entropy, and entanglement clarify how entropy behaves under restricted access to subsystems.

  • The II-law states that dynamics are irreversible, equivalently breaking time-reversal symmetry, and that entropy change in the universe satisfies ΔS ≥ 0.The thermodynamic statement corresponds to spontaneous heat flow from a hot source to a cold sink, with additive entropy generation.
  • For a closed quantum system, the II-law can be derived from unitary evolution by comparing entropy before and after a change in the entire system.A dynamical formulation instead accounts locally for subsystem entropy changes and entropy generated in baths.
  • Quantum entropy is connected to the ability to measure and manipulate a system using information obtained from measurements.
  • The entropy of an observable is the Shannon entropy of projective-measurement outcomes, with probabilities p_i = tr{ρP_i}.The observable has spectral decomposition A = Σ_i α_i P_i, with P_i = |α_i⟩⟨α_i|; entropy units use k_B = 1.
  • The von Neumann entropy is obtained by minimizing observable entropy and is associated with an observable commuting with the quantum state.It is invariant under unitary transformations because unitary evolution preserves the eigenvalues of ρ.
  • Von Neumann entropy is additive only for tensor-product states, whereas entangled subsystems generally require local observable entropies.For entangled pure bipartite states, total von Neumann entropy is zero while each subsystem entropy is positive.

B. Quantum networks and quantum devices

Quantum networks are built from wires and tricycles connecting quantum systems and heat baths, while their currents obey thermodynamic laws. The paper emphasizes that consistent quantum thermodynamics requires carefully derived open-system dynamics, especially for the II-law.

  • Network structure: Quantum networks consist of interconnected quantum systems and baths at different temperatures, decomposed into wires and tricycle junctions.Wires connect two baths, while tricycles combine three currents; larger networks can be assembled from these elements.
  • Network structure: A wire transports energy between two baths, whereas a tricycle combines three currents and can function as a heat engine or heat-driven refrigerator.The figure identifies the tricycle as connected to three baths and the wire as connected to hot and cold baths.
  • Thermodynamic laws: Quantum networks obey the Clausius II-law: no process can solely transfer heat from a colder body to a hotter body.The statement is generalized to N coupled heat baths in steady state and supported dynamically using Spohn’s inequality for LGKS generators.
  • Thermodynamic laws: In steady state, heat currents in periodically driven systems satisfy the II-law, while the I-law constrains their total energy balance.The cited relations also identify averaged power, which is negative when the system acts as a heat engine.
  • Entropy generation: External power, infinite-temperature baths, white noise, and weak quantum measurements can produce zero entropy generation, making them thermodynamically equivalent as refrigerator driving sources.Work is a zero-entropy source, but when converted to heat in a driven system, consistency with the II-law requires dissipation into the bath.
  • Consistency conditions: Violations of the II-law can result from faulty master equations, so deriving the LGKS generator carefully is necessary for thermodynamic consistency.For tricycles, dressed-state derivations and appropriate treatment of driving can restore the II-law.

C. Approach to steady state: Limit cycle

Quantum network dynamics approaches a steady limit cycle under suitable completely positive evolution. Quantum adiabaticity requires constant energy entropy, while noncommuting time-dependent controls produce friction unless specialized controls or noise are used.

  • C. Approach to steady state: Limit cycle: Completely positive evolution monotonically approaches an invariant state, yielding a steady state for quantum networks.A network generated by a sum of LGKS generators reaches steady state.
  • C. Approach to steady state: Limit cycle: A reciprocating quantum engine reaches a limit cycle because its cycle map is the product of completely positive maps for each segment.For an Otto refrigerator, the cycle map is U = UhUhcUcUch.
  • C. Approach to steady state: Limit cycle: Quantum adiabaticity is defined by constant energy entropy, equivalent to no net change in instantaneous energy-level populations.For isolated dynamics, von Neumann entropy remains constant, so energy entropy is the relevant measure.
  • C. Approach to steady state: Limit cycle: Noncommuting drift and control Hamiltonians make strictly adiabatic processes impossible, requiring slow control changes for approximate adiabaticity.Time-ordering corrections arise when [H0, HC(t)] ≠ 0.
  • C. Approach to steady state: Limit cycle: Quantum friction can be reduced by enforcing commutation with the instantaneous Hamiltonian, but phase-noise lubrication works only in a limited parameter window.Outside that window, noise causes additional heating; in refrigerators it can impose a minimum reachable temperature.
  • C. Approach to steady state: Limit cycle: Shortcuts to adiabaticity use special dynamical symmetries to connect diagonal initial and final energy states in finite time without friction.The control schedule is optimized to achieve the transformation in the shortest time.

VI. THE III-LAW

The paper examines static and dynamical formulations of the III-law and connects them to low-temperature cooling dynamics. The unattainability formulation imposes stricter constraints on heat currents, bath properties, and interaction scaling.

  • VI. THE III-LAW: The III-law has static and dynamical formulations: equilibrium entropy approaches zero at zero temperature, while absolute zero cannot be reached in finitely many operations.The dynamical formulation is the unattainability principle.
  • VI. THE III-LAW: Quantum considerations are used to examine the debated relationship between the two III-law formulations and the second law.The paper frames this as a consistency issue between thermodynamic statements.
  • VI. THE III-LAW: The second law requires eliminating entropy-production divergence at the cold bath as Tc →0.At steady state, total entropy production must remain non-negative.
  • VI. THE III-LAW: α > 0 ensures zero cold-bath entropy production at absolute zero and requires the heat current to scale as Jc ∼ T^(α+1).For α = 0, other baths must compensate the cold bath’s negative entropy production.
  • VI. THE III-LAW: The unattainability principle requires the cooling rate to vanish as Tc →0 and imposes constraints on system-bath interactions and cold-bath properties.This formulation is more restrictive than the static formulation.
  • VI. THE III-LAW: Optimized refrigerators tune the receiving-mode energy linearly with temperature, ωc ∼ Tc, making the III-law depend on heat conductivity and heat-capacity scaling.The result applies across the refrigerator models studied.

A. Harmonic oscillator cold heat bath

For harmonic cold baths, open-system modeling yields low-frequency cooling-rate scalings that connect bath dispersion and coupling to the III-law. Optimization tunes the receiving frequency proportionally to cold-bath temperature.

  • A. Harmonic oscillator cold heat bath: The harmonic bath includes photon, phonon, and Bose-Einstein-condensate excitations and is modeled with linear refrigerator-bath coupling.Its Hamiltonian is represented using bosonic modes.
  • A. Harmonic oscillator cold heat bath: For a d-dimensional bosonic field with linear dispersion, the low-frequency cooling rate follows scaling determined by the mode density and coupling.The resulting cold-current scaling is derived from these low-frequency properties.
  • A. Harmonic oscillator cold heat bath: ωc ∼ Tc is the optimized receiving-frequency tuning for the continuous refrigerator, determining the final cold-current scaling.The same temperature-linear tuning appears in the periodically driven refrigerator, together with λ ∝ Tc.
  • A. Harmonic oscillator cold heat bath: The III-law constrains the interaction form with a bosonic bath, including the low-frequency behavior of its coupling and dispersion.Standard electromagnetic and acoustic baths have linear dispersion, while some exotic dispersions are excluded.
  • A. Harmonic oscillator cold heat bath: The III-law analysis assumes a physically stable harmonic bath with a ground state; an inverted oscillator potential can invalidate the law.The failure occurs when the cold bath lacks a ground state.

B. The existence of a ground state

The paper links the III-law to the existence of a stable ground state in bosonic baths and tests the connection for ideal Bose and Fermi gases. Cooling currents and temperature-decrease rates vanish toward zero temperature under the stated models.

  • B. The existence of a ground state: A stable open-quantum-system model requires a total Hamiltonian bounded from below and possessing a ground state.For linearly coupled bosonic baths, this condition becomes a constraint on the bosonic Hamiltonian.
  • B. The existence of a ground state: Infinite bosonic mode sets can produce divergent ground-state energies or prevent the canonical transformation from being unitary.Non-unitarity is associated with the van Hove phenomenon and non-existence of the ground state in Fock space.
  • B. The existence of a ground state: With linear dispersion, the low-frequency conditions required for a bosonic ground state are identical to those required by the III-law.The paper identifies this as consistency between ground-state existence and the III-law.
  • B. The existence of a ground state: For ideal Bose and Fermi gases, low-temperature changes in participating density and heat capacity modify both sides of the cooling dynamics consistently with the III-law.For Fermi gases, only a small fraction of fermions participates in scattering at low temperature.
  • B. The existence of a ground state: In three dimensions, the harmonic-bath and Bose-gas cases both show vanishing cooling current and temperature-decrease rate as Tc →0.The Bose gas cools faster near zero temperature, but its temperature-decrease rate is slower than the harmonic bath’s.

D. Thermoelectric refrigerators

Thermoelectric quantum devices modify the first law through electrical power while retaining the second law, and their cooling behavior can remain consistent with the third law under suitable bath scaling. The review places these results within a broader framework where thermodynamic consistency checks quantum models and finite-time isolation remains impossible.

  • Thermoelectric refrigerators: Electrical power modifies the first law of thermoelectric devices, while scattering theory leaves the second law unchanged because charge redistribution does not change entropy.The electrical contribution is expressed as P_j = V_jI_j.
  • Thermoelectric refrigerators: J_c scales as T_c^2 for the maximum heat current extracted from the cold bath.The scaling is stated for the number of scattering channels N_c.
  • Thermoelectric refrigerators: For fermions, c_V scales as T_c, yielding ζ = 1 and consistency with the third law.The exponent is identified through the relation between heat-current and heat-capacity scaling.
  • Thermoelectric refrigerators: Third-law exponents depend on cold-bath characteristics, specifically the scaling of heat conductivity relative to heat capacity, and refrigerator-model type does not determine them.The required bath ratio scales as T_c^ζ, with ζ > 1 required for the third law to hold.
  • Thermoelectric refrigerators: The review treats apparent violations of thermodynamics in quantum models as consequences of flawed approximations, especially uncontrolled master-equation derivations.Thermodynamics is proposed as a consistency check for approximate quantum theories.
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