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Cited by 286Advanced Thermodynamics and Statistical MechanicsQuantum and electron transport phenomenaThermal properties of materials

The local approach to quantum transport may violate the second law of thermodynamics

Open access
Levy, Amikam, Kosloff, Ronnie·2014·EPL (Europhysics Letters)
DOI 10.1209/0295-5075/107/20004
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Abstract

Clausius statement of the second law of thermodynamics reads: Heat will flow spontaneously from a hot to cold reservoir. This statement should hold for transport of energy through a quantum network composed of small subsystems each coupled to a heat reservoir. When the coupling between nodes is small, it seems reasonable to construct a local master equation for each node in contact with the local reservoir. The energy transport through the network is evaluated by calculating the energy flux after the individual nodes are coupled. We show by analyzing the most simple network composed of two quantum nodes coupled to a hot and cold reservoir, that the local description can result in heat flowing from cold to hot reservoirs, even in the limit of vanishing coupling between the nodes. A global derivation of the master equation which prediagonalizes the total network Hamiltonian and within this framework derives the master equation, is always consistent with the second law of thermodynamics.

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Written from the paper’s full text.

TL;DR

The local master-equation approach to quantum transport can predict heat flowing from a cold to a hot reservoir, violating the second law, even when the node-to-node coupling vanishes. A global master equation derived after prediagonalizing the full network Hamiltonian remains consistent with the second law.

Key findings

  • In the local approach, the steady-state hot-bath heat current is \(J_h=(e^{\beta_c\omega_c}-e^{\beta_h\omega_h})F\), with \(F>0\). For \(\omega_c/T_c<\omega_h/T_h\), \(J_h\) becomes negative, meaning heat flows from the cold bath to the hot bath and the second law is violated even at vanishingly small \(A\)-\(B\) coupling.
  • The global approach prediagonalizes \(\hat H_0+\hat H_{AB}\), yielding normal modes \(d_\pm\) with frequencies \(\omega_\pm\), and includes an effective coupling of subsystem \(A\) to the cold bath and \(B\) to the hot bath. Its heat current is positive for all physical parameters and, by construction, obeys Spohn’s inequality.
  • Local observables such as the population \(n_A=\langle \hat a^\dagger \hat a\rangle\) converge to the thermal population for small coupling, but non-local observables such as the heat current \(J_h\) deviate qualitatively and quantitatively between local and global descriptions.
  • In steady state, the local approach over-correlates the nodes: \(\mathrm{cor}(x_A,p_B)\) and \(\mathrm{cor}(p_A,x_B)\) remain finite locally but vanish globally; neither approach generates entanglement in the two-mode Gaussian steady state.

Methods

  • Theoretical study of the simplest two-node quantum network: subsystems \(A\) and \(B\) coupled by a swap Hamiltonian \(\hat H_{AB}=\epsilon(\hat a^\dagger \hat b+\hat a\hat b^\dagger)\), with \(A\) coupled to a hot bath at \(T_h\) and \(B\) to a cold bath at \(T_c\). Nodes are treated as harmonic oscillators or two-level systems, depending on commutation relations.
  • Two LGKS master equations are compared: a local one that ignores \(\hat H_{AB}\) in the dissipators, and a global one that first diagonalizes \(\hat H_0+\hat H_{AB}\) and then derives the weak-coupling master equation in the eigenbasis. Steady-state heat flow is computed as \(J_h=\mathrm{Tr}[(L_h\rho_s)(\hat H_0+\hat H_{AB})]\).
  • Bath rates are specified for a 3D phonon bath with linear dispersion, \(\gamma_l(\Omega)=\kappa\Omega^3[1-e^{-\beta_l\Omega}]^{-1}\). Figures compare local and global steady-state populations and currents versus coupling strength and \(\omega_h\), with example parameters including \(T_c=10,\omega_c=5,\epsilon=10^{-4},\kappa=10^{-7}\) (Fig. 2); \(T_h=12,T_c=10,\omega_h=10,\omega_c=5,\kappa=10^{-4}\) (Fig. 3); and \(T_h=12,T_c=10,\omega_c=5,\epsilon=10^{-3},\kappa=10^{-7}\) (Fig. 4).

Limitations

  • Stated: the global approach breaks down on resonance when \(\omega_h=\omega_c\) and \(\epsilon<\kappa\), because the secular approximation is not justified since the Bohr frequencies \(\omega_\pm\) are not well separated.
  • Stated: for TLS nodes the expressions are more intricate, so the global analysis is restricted to harmonic nodes.
  • Stated: the local approach is incorrect for strong subsystem coupling; in the weak-coupling limit local observables converge, but non-local observables such as heat currents remain erroneous.
  • Evident from the text: the demonstration uses the simplest two-node network, and the global harmonic-node analysis does not cover the full TLS case or experimental validation.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1209/0295-5075/107/20004

Cites

27 of 27 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 6 cited sources checked · 0 supported · 0 flagged

  • Findingown data

    The calculated steady-state heat flow from the hot bath in the global approach is positive for all physical choices of parameters.

    “The calculated steady-state heat flow from the hot bath is given by see eq. (13) above which is positive for all physical choice of parameters.”
    Quote in the paperSubstantial claim
  • Findingown data

    In the local approach, for ωc/Tc < ωh/Th, heat flows from the cold bath to the hot bath, violating the second law of thermodynamics, even at vanishingly small coupling between the nodes.

    “We show by analyzing the most simple network composed of two quantum nodes coupled to a hot and cold reservoir, that the local description can result in heat flowing from cold to hot reservoirs, even in the limit of vanishing coupling between the nodes.”
    • Also stated in: § 20004-p1 · plain
      “For ωc Tc < ω h T h heat will flow from the cold bath to the hot bath, thus the second law is violated even at vanishing small AB coupling, cf.fig. 2.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The Clausius statement of the second law of thermodynamics holds: heat flows spontaneously from a hot to a cold reservoir.
  • Findingnone given

    Equations (8) fulfill the dynamical version of the first law of thermodynamics: the sum of all energy (heat) currents at steady state is zero, J_h + J_c = 0.

    “Equations ( 8) fulfill the dynamical version of the first law of thermodynamics: The sum of all energy (heat) currents at steady state is zero, J h + J c = 0.”
    Quote in the paperNot a substantial claim
  • Findingreasoning

    The direction of heat flow in the local approach depends on the choice of parameters.

    “It is apparent from eq. ( 9), that the direction of heat flow depends on the choice of parameters.”
    Quote in the paperSubstantial claim
  • Findingcitation

    A global derivation of the master equation that prediagonalizes the total network Hamiltonian is always consistent with the second law of thermodynamics.

    “A global derivation of the master equation which prediagonalizes the total network Hamiltonian and within this framework derives the master equation, is always consistent with the second law of thermodynamics.”
    • Also stated in: § 20004-p1 · plain
      “The global MME, by construction, obeys Spohn's inequality and therefore is consistent with the second law of thermodynamics [19].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [19] source

      Not checked in the free run.

  • Classified as probably not a substantial claim. 2
  • May combine two claims. 7

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given

    In both approaches, the correlations cor(xA,xB) and cor(pA,pB) are equal for small coupling.

    “In both approaches cor(x A , x B ) and cor(p A , p B ) are equal for small .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingcitation

    The only domain where the global approach breaks down is on resonance, when ωh = ωc and coupling < κ, because the secular approximation is not justified.

    “The only domain where the global approach breaks down is on resonance, when ω h = ω c and < κ.At this point, the secular approximation is not justified since the two Bohr frequencies ω ± are not well separated”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [21] Rivas A., 2010, New J. Phys. source

      Not checked in the free run.

  • May combine two claims.

    Findingown data

    In the global approach, the correlations cor(xA,pB) and cor(pA,xB) vanish at steady state, whereas in the local approach they remain finite, so the local approach over-correlates the nodes compared to the global approach.

    “The two additional correlations, cor(x A , p B ) and cor(p A , x B ), vanish at steady state in the global approach, where in the local approach they remain finite.Thus, in the local approach the nodes are over correlated compared to the global approach.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingcitation

    In steady state, none of the approaches generate entanglement; the two-mode Gaussian state is separable according to the separability criterion for continuous variable systems.

    “It should be noted that in steady state none of the approaches generate entanglement.The two-mode Gaussian state is a separable state according to the separability criterion for continuous variable systems [22,23].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [22] source

      Not checked in the free run.

    • cites [23] source

      Not checked in the free run.

  • May combine two claims.

    Findingown data

    In the weak coupling limit, local observables converge to their correct value, but non-local observables such as heat currents are qualitatively and quantitatively erroneous in the local MME.

    “In the weak coupling limit, local observables converge to their correct value.The non-local observables such as heat currents are qualitatively and quantitatively erroneous in the local MME.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The local approach is not reliable even for parameters where the second law is obeyed (ωc/Tc > ωh/Th); deviations from the exact global approach appear in the favorable domain of small coupling.

    “The local approach is also not reliable even for parameters where the second law is obeyed: ωc Tc > ω h T h .Deviations from the exact global approach appear in the favorable domain of small , as seen in fig. 4”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The global approach creates an effective coupling of system A with the cold bath and of system B with the hot bath, which is absent in the local approach and is crucial for saving the second law.

    “But more importantly, it creates an effective coupling of the system A with the cold bath and of the system B with the hot bath.This indirect coupling absent in the local approach is crucial, and essentially saves the second law of thermodynamics.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Premisecitation

    The treatment introduced in [16] corresponds to the local approach described in the paper.

    “In fact, the treatment introduced in [16] corresponds to the local approach described above.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [16] Novotný T., 2002, Europhys. Lett. source

      Not checked in the free run.

  • May combine two claims.

    Interpretationown data

    The local approach is valid for local observables such as the population of each node, but not valid for non-local observables describing energy fluxes.

    “We will show that the local approach is only valid for local observables such as the population of each node, and is not valid for non-local observables describing energy fluxes.”
    • Also stated in: § 20004-p1 · plain
      “In the weak coupling limit, local observables converge to their correct value.The non-local observables such as heat currents are qualitatively and quantitatively erroneous in the local MME.”
    Quote in the paperSubstantial claim
9 of 9

  • The paper relies on this without support.

    The Clausius statement of the second law of thermodynamics holds: heat flows spontaneously from a hot to a cold reservoir.

    “Clausius statement of the second law of thermodynamics reads: Heat will flow spontaneously from a hot to cold reservoir.”
1 of 1

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The presence of an external field ensures that the local master equation does not result in any violations of the laws of thermodynamics.

    “In contrast to [58,59,63], which lack an external field, we find that the presence of such a field ensures that the local master equation does not result in any violations of the laws of thermodynamics.”

    From Markovian master equations for quantum thermal machines: Local versus global approach (2017) · cites this paper as [59]

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. Lindblad G., 1976, J. Phys. A: Math. Gen., 48, 119

Cited by

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Related

Authors
Levy, A., & Kosloff, R.
Affiliations
Hebrew University of Jerusalem (IL)
ORCID
Venue
EPL (Europhysics Letters)
Publisher
IOP Publishing
Published
2014-07-01
Volume
107
Issue
2
Pages
20004
ISSN
  • 0295-5075 (ISSN-L)
  • 1286-4854
DOI
10.1209/0295-5075/107/20004
OpenAlex
W2241679631
Cited by
286
References
27
Citations by year
2026: 12 · 2025: 25 · 2024: 19 · 2023: 25 · 2022: 34 · 2021: 31 · 2020: 32 · 2019: 35 · 2018: 29 · 2017: 17 · 2016: 13 · 2015: 11 · 2014: 3
FWCI
9.66
Type
journal article
Language
en
Open access
Licence
Locations
  • Europhysics Letters (EPL) (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • Springer Link (Chiba Institute of Technology) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and electron transport phenomena
  • Physical Sciences › Materials Science › Materials Chemistry › Thermal properties of materials
Keywords
second law of thermodynamics, quantum heat transport, quantum networks, energy flux, open quantum systems, Clausius statement

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