pith. sign in

arxiv: 1907.02126 · v1 · pith:L43NLF54new · submitted 2019-07-03 · ⚛️ physics.chem-ph · cond-mat.stat-mech

A Nonequilibrium Variational Polaron Theory to Study Quantum Heat Transport

ChangYu Hsieh , Junjie Liu , Chenru Duan , Jianshu Cao This is my paper

Pith reviewed 2026-05-25 09:14 UTC · model grok-4.3

classification ⚛️ physics.chem-ph cond-mat.stat-mech
keywords quantum heat transportvariational polaron transformationOhmic bathspin-boson modelinfrared divergencecross-bath correlationsNIBA
0
0 comments X

The pith

A nonequilibrium variational polaron transformation with effective temperature accurately calculates heat currents in Ohmic baths by treating infrared divergence and cross-bath correlations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a nonequilibrium variational polaron transformation based on an ansatz for the nonequilibrium steady state with an effective temperature. This extends the polaron framework to Ohmic baths by addressing the infrared divergence in low-frequency modes that limited prior methods to super-Ohmic cases. The approach combines the transformed master equation with full counting statistics to include cross-bath correlations explicitly. It yields more accurate heat current values than the NIBA formalism for the nonequilibrium spin-boson model and demonstrates effects such as current turnover and rectification.

Core claim

We propose a nonequilibrium variational polaron transformation, based on an ansatz for nonequilibrium steady state (NESS) with an effective temperature, to study quantum heat transport at the nanoscale. By combining the variational polaron transformed master equation with the full counting statistics, the method extends the polaron framework beyond super-Ohmic baths. It treats the infrared divergence in low-frequency bath modes and includes cross-bath correlation effects, providing more accurate heat current calculations than the NIBA formalism for Ohmic bath models in the spin-boson model.

What carries the argument

nonequilibrium variational polaron transformation parameterized by a single effective temperature for the nonequilibrium steady state

Load-bearing premise

The nonequilibrium steady state can be described by a variational polaron transformation parameterized by a single effective temperature.

What would settle it

Direct comparison of predicted heat currents against numerically exact solvers such as hierarchical equations of motion for an Ohmic spin-boson model at low temperature where infrared effects matter.

Figures

Figures reproduced from arXiv: 1907.02126 by ChangYu Hsieh, Chenru Duan, Jianshu Cao, Junjie Liu.

Figure 1
Figure 1. Figure 1: Schematics of a quantum heat transfer model in which an open system (yellow circle, a two-level system for example) is simultaneously coupled to two heat reservoirs held at two different temperature TL and TR. The temperature gradient induces heat current directed from hot to cold reservoir. This non-equlibrium transport setup can be mapped to an effective equilibrum setup, so that the non-equlibrium stead… view at source ↗
Figure 2
Figure 2. Figure 2: Cross-bath correlation effects on FL(ω), Eq. (21), for an unbiased NESB model with parameters: ωc = 10∆, αL = 0.05 and TL = ∆. (a): variation of αR while TR = 0.75∆ is fixed. (b): variation of TR while αR = 0.05 is held fixed. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The heat current as a function of coupling strength. The NESB model parameters are ∆/ωc = 1/16,ε/ωc = 1/4, TL/ωc = 0.275, TR/ωc = 0.225t. The spectral density is super-ohmic with s=3 and a rational cutoff. 3.2 Unified Heat Current Calculation for Ohmic Spectral Density Next, we turn to the NESB cases in which heat reservoirs are featured with Ohmic spectral densities. The NE-PTRE reduces exactly to NIBA re… view at source ↗
Figure 4
Figure 4. Figure 4: The heat current as a function of coupling strength. (a) model parameters: ∆ = ωc/30, TL = 1.4∆, TR = 1.2∆ and ε = 0. (b) model parameters: ∆ = 0.02ωc, TL = 1.0ωc, TR = 0.9ωc and ε = 0. For the Ohmic bath models, NIBA and NE-PTRE are equivalent. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The heat current as a function of ∆ with fixed ωc. (a) Model parameters: α = 0.03, TL = 0.15ωc, TR = 0.14ωc. (b) Model parameters: α = 0.03, TL = 0.4ωc, TR = 0.3ωc. For the Ohmic bath models, NIBA and NE-PTRE are equivalent. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Thermal rectification ratio as a function of αR/αL asymmetry. The model parameters are ωc = 10∆, ε = 0, δT = |TL −TR| = 0.4∆ and Tavg = (TL +TR)/2 = ∆. The “red cross” and red dashed lines are VPTRE results for case (1): α = αL +αR = 0.05 and case (2): α = αL +αR = 0.20, respectively. The “blue cross” and blue dashed lines are NIBA results for the same case (1) and case (2), respectively. For Ohmic bath mo… view at source ↗
read the original abstract

We propose a nonequilibrium variational polaron transformation, based on an ansatz for nonequilibrium steady state (NESS) with an effective temperature, to study quantum heat transport at the nanoscale. By combining the variational polaron transformed master equation with the full counting statistics, we have extended the applicability of the polaron-based framework to study nonequilibrium process beyond the super-Ohmic bath models. Previously, the polaron-based framework for quantum heat transport reduces exactly to the non-interacting blip approximation (NIBA) formalism for Ohmic bath models due to the issue of the infrared divergence associated with the full polaron transformation. The nonequilibrium variational method allows us to appropriately treat the infrared divergence in the low-frequency bath modes and explicitly include cross-bath correlation effects. These improvements provide more accurate calculation of heat current than the NIBA formalism for Ohmic bath models. We illustrate the aforementioned improvements with the nonequilibrium spin-boson model in this work and quantitatively demonstrate the cross-bath correlation, current turnover, and rectification effects in quantum heat transfer.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes a nonequilibrium variational polaron transformation based on an effective-temperature ansatz for the nonequilibrium steady state (NESS). Combined with the variational polaron master equation and full counting statistics, the method is claimed to extend polaron-based treatments of quantum heat transport to Ohmic baths by regularizing infrared divergences in low-frequency modes and incorporating cross-bath correlations, yielding heat currents more accurate than those from the non-interacting blip approximation (NIBA). The improvements are illustrated for the nonequilibrium spin-boson model, including current turnover and rectification.

Significance. If the single-effective-temperature ansatz is shown to be sufficient, the approach would provide a practical extension of polaron methods to nonequilibrium Ohmic regimes, enabling quantitative study of correlation effects and transport phenomena not accessible to NIBA.

major comments (2)
  1. [Abstract] Abstract (paragraph describing the proposal): The central ansatz that the NESS can be captured by a variational polaron transformation parameterized by a single effective temperature is load-bearing for the accuracy claim. No derivation, error bound, or independent test is supplied showing that this one-parameter family reproduces the multi-bath correlations required for systematic improvement over NIBA when T_L ≠ T_R.
  2. [Abstract] Abstract: The statement that the method 'provides more accurate calculation of heat current than the NIBA formalism for Ohmic bath models' is presented without any numerical comparison, error bars, or benchmark against exact results; the abstract supplies only the formal claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph describing the proposal): The central ansatz that the NESS can be captured by a variational polaron transformation parameterized by a single effective temperature is load-bearing for the accuracy claim. No derivation, error bound, or independent test is supplied showing that this one-parameter family reproduces the multi-bath correlations required for systematic improvement over NIBA when T_L ≠ T_R.

    Authors: The single-effective-temperature ansatz is obtained by applying the variational principle to minimize a suitable nonequilibrium functional that incorporates the bath spectral densities and the system-bath coupling; the resulting transformation explicitly retains cross-bath correlation terms that are absent in the standard polaron mapping. While a rigorous a-priori error bound is not derived, the manuscript presents numerical benchmarks for the nonequilibrium spin-boson model at T_L ≠ T_R that quantify the improvement in heat current relative to NIBA and demonstrate the role of the retained cross correlations. We will add a concise paragraph in the revised introduction and methods section that spells out the variational condition and the numerical evidence for multi-bath effects. revision: partial

  2. Referee: [Abstract] Abstract: The statement that the method 'provides more accurate calculation of heat current than the NIBA formalism for Ohmic bath models' is presented without any numerical comparison, error bars, or benchmark against exact results; the abstract supplies only the formal claim.

    Authors: We agree that the abstract should not assert improved accuracy without qualification. In the revised manuscript we will rephrase the relevant sentence to read that the method 'yields heat currents that are more accurate than NIBA, as shown by direct numerical comparison for the nonequilibrium spin-boson model with Ohmic baths.' revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on explicit new variational ansatz for NESS, independent of fitted inputs or self-citation chains

full rationale

The paper introduces a nonequilibrium variational polaron transformation parameterized by a single effective temperature as its central proposal. This ansatz is stated directly rather than derived from prior equations or data fits within the work. No load-bearing step reduces by construction to a fitted parameter renamed as prediction, nor does any uniqueness theorem or ansatz originate from self-citation. The claimed improvements over NIBA for Ohmic baths follow from applying the new transformation to the master equation and FCS, without the derivation collapsing to its own inputs. The method is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the nonequilibrium steady-state ansatz with a single effective temperature and on the assumption that the variational polaron transformation can be combined with the master equation without introducing uncontrolled errors. No free parameters beyond the variational temperature are mentioned; no new entities are postulated.

free parameters (1)
  • effective temperature
    Variational parameter introduced via the NESS ansatz to regularize the polaron transformation for nonequilibrium conditions.
axioms (1)
  • domain assumption A nonequilibrium steady state can be represented by a variational polaron transformation parameterized by an effective temperature
    Invoked in the proposal of the nonequilibrium variational method (abstract).

pith-pipeline@v0.9.0 · 5722 in / 1369 out tokens · 22415 ms · 2026-05-25T09:14:49.934357+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Nitzan, A, Chemical Dynamics in Condensed Phases (Oxford University Press, New York), 2006

    work page 2006

  2. [2]

    Nitzan, A.; Ratner, M. A. Electron Transport in Molecular Wire Junctions, Science, 2003, 300, 1384-1389

    work page 2003

  3. [3]

    A.; Pauly, F.; Cuevas, J

    Lee, W.; Kim, K.; Jeong, W.; Zotti, L. A.; Pauly, F.; Cuevas, J. C.; Reddy, P. Heat Dissipation in Atomic-scale Junctions, Nature 2013, 498, 209–212

    work page 2013

  4. [4]

    Colloquium : Phononics: Manipulating Heat Flow with Electronic Analogs and Beyond, Rev

    Li, N.; Ren, J.; Wang, L.; Zhang, G, H¨anggi, P.; Li, B. Colloquium : Phononics: Manipulating Heat Flow with Electronic Analogs and Beyond, Rev. Mod. Phys. 2012, 84, 1045–1066

    work page 2012

  5. [5]

    Quantum Heat Engines and Refrigerators: Continuous Devices,Annu

    Kosloff, R.; Levy, A. Quantum Heat Engines and Refrigerators: Continuous Devices,Annu. Rev. Phys. Chem. 2014, 65, 365–393

    work page 2014

  6. [6]

    Polaron Effects on the Performance of Light-harvesting Systems: A Quantum Heat Engine Perspective, New J

    Xu, D.; Wang, C.; Zhao, Y .; Cao, J. Polaron Effects on the Performance of Light-harvesting Systems: A Quantum Heat Engine Perspective, New J. Phys. 2016, 18, 023003/1–14

    work page 2016

  7. [7]

    Efficiency at Maximum Power of a Laser Quantum Heat Engine Enhanced by Noise-induced Coherence.’ Phys

    Dorfman, K.E.; Xu, D.; Cao, J. Efficiency at Maximum Power of a Laser Quantum Heat Engine Enhanced by Noise-induced Coherence.’ Phys. Rev. E 2018, 97, 042120/1–8

    work page 2018

  8. [8]

    Leitner, D. M. Quantum ergodicity and energy flow in molecules Advances in Physics (Taylor Francis), 2015

    work page 2015

  9. [9]

    Heat Flux Manipulation with Engineered Thermal Materials,Phys

    Narayana, S.; Sato, Y . Heat Flux Manipulation with Engineered Thermal Materials,Phys. Rev. Lett. 2012, 108, 214303/1–5

    work page 2012

  10. [10]

    Tuning the Aharonov-Bohm Effect with Dephasing in Nonequilibrium Transport, Phys

    Engelhardt, G.; Cao, J. Tuning the Aharonov-Bohm Effect with Dephasing in Nonequilibrium Transport, Phys. Rev. B 2019, 99, 075436/1–1

    work page 2019

  11. [11]

    Nonequilibrium Fluctuations, Fluctuation Theorems, and Counting Statistics in Quantum Systems, Rev

    Esposito, M.; Harbola, U.; Mukamel, S. Nonequilibrium Fluctuations, Fluctuation Theorems, and Counting Statistics in Quantum Systems, Rev. Mod. Phys. 2009, 81, 1665–1702

    work page 2009

  12. [12]

    Colloquium: Quantum Fluctuation Relations: Foundations and Applications, Rev

    Campisi, M.; H¨anggi, P.; Talkner, P. Colloquium: Quantum Fluctuation Relations: Foundations and Applications, Rev. Mod. Phys. 2011, 83, 771–791

    work page 2011

  13. [13]

    Frequency-dependent Current Noise in Quantum Heat Transfer with Full Counting Statistics

    Liu, J.; Hsieh, C.-Y .; Cao, J. Frequency-dependent Current Noise in Quantum Heat Transfer with Full Counting Statistics. J. Chem. Phys. 2018, 148, 234104/1–11. 14

    work page 2018

  14. [14]

    Bidirectional Counting of Single Electrons

    Fujisawa, T.; Hayashi, T.; Tomita, R.; Hirayama, Y . Bidirectional Counting of Single Electrons. Science 2006, 312, 1634–1636

    work page 2006

  15. [15]

    S.; Utsumi, Y .; Ihn, T.; Ensslin, K

    K¨ung, B.; Kung, B.; Rossler, C.; Beck, M.; Marthaler, M.; Golubev, D. S.; Utsumi, Y .; Ihn, T.; Ensslin, K. Irreversibility on the Level of Single-electron Tunneling, Phys. Rev. X 2012, 2, 011001/1–6

    work page 2012

  16. [16]

    C.; Gossard, A

    Gustavsson, S.; Leturcq, R.; Simovic, B.; Schleser, R.; Ihn, T.; Studerus, P.; Ensslin, K.; Driscoll, D. C.; Gossard, A. C. Counting Statistics of Single Electron Transport in a Quantum Dot, Phys. Rev. Lett. 2006, 96, 076605/1–4

    work page 2006

  17. [17]

    Nonequilibrium Energy Transfer at Nanoscale: A Unified Theory from Weak to Strong Coupling, Sci

    Wang, C.; Ren, J.; Cao, J. Nonequilibrium Energy Transfer at Nanoscale: A Unified Theory from Weak to Strong Coupling, Sci. Rep. 2015, 5, 11787

    work page 2015

  18. [18]

    Unifying Quantum Heat Transfer in a Nonequilibrium Spin-Boson Model with Full Counting Statistics, Phys

    Wang, C.; Ren, J.; Cao, J. Unifying Quantum Heat Transfer in a Nonequilibrium Spin-Boson Model with Full Counting Statistics, Phys. Rev. A 2017, 95, 023610/1–1

    work page 2017

  19. [19]

    K.; Moix, J.; Cao, J

    Lee, C. K.; Moix, J.; Cao, J. Accuracy of Second Order Perturbation Theory in the Polaron and Variational Polaron Frames, J. Chem. Phys. 2012, 136 204120/1–7

    work page 2012

  20. [20]

    K.; Moix, J.; Cao, J

    Lee, C. K.; Moix, J.; Cao, J. Coherent Quantum Transport in Disordered Systems: A Unified Polaron Treatment of Hopping and Band-like Transport,” J. Chem. Phys. 2015, 142 164103/1–7

    work page 2015

  21. [21]

    Non-canonical Distribution and Non-equilibrium Transport Beyond Weak System-bath Coupling Regime: A Polaron Transformation Approach, Front

    Xu, D.; Cao, J. Non-canonical Distribution and Non-equilibrium Transport Beyond Weak System-bath Coupling Regime: A Polaron Transformation Approach, Front. Phys. 2016, 11, 111308/1–17

    work page 2016

  22. [22]

    Non-equilibrium Spin-Boson Model: Counting Statistics and the Heat Exchange Fluctuation Theorem, J

    Nicolin, L.; Segal, D. Non-equilibrium Spin-Boson Model: Counting Statistics and the Heat Exchange Fluctuation Theorem, J. Chem. Phys. 2011, 135, 164106/1–14

    work page 2011

  23. [23]

    Silbey, R.; Harris, R. A. Variational Calculation of the Dynamics of a Two Level System Interacting with a Bath,J. Chem. Phys. 1984, 80, 2615–17

    work page 1984

  24. [24]

    A.; Silbey, R

    Harris, R. A.; Silbey, R. Variational Calculation of the Tunneling System Interacting with a Heat Bath. II. Dynamics of an Asymmetric Tunneling System, J. Chem. Phys. 1985, 83, 1069–1074

    work page 1985

  25. [25]

    Reformulation of Steady State Nonequilibrium Quantum Statistical Mechanics,Phys

    Hershfield, S. Reformulation of Steady State Nonequilibrium Quantum Statistical Mechanics,Phys. Rev. Lett. 1993, 70, 2134–2137

    work page 1993

  26. [26]

    Nonequilibrium Density Matrix in Quantum Open Systems: Generalization for Simultaneous Heat and Charge Steady-state Transport, Phys

    Ness, H. Nonequilibrium Density Matrix in Quantum Open Systems: Generalization for Simultaneous Heat and Charge Steady-state Transport, Phys. Rev. E 2014, 90, 062119/1–10

    work page 2014

  27. [27]

    Improved Dyson Series Expansion for Steady-state Quantum Transport Beyond the Weak Coupling Limit: Divergences and Resolution, J

    Thingna, J.; Zhou, H.; Wang, J.-S. Improved Dyson Series Expansion for Steady-state Quantum Transport Beyond the Weak Coupling Limit: Divergences and Resolution, J. Chem. Phys. 2014, 141, 194101/1–13

    work page 2014

  28. [28]

    Nonequilibrium Density-matrix Description of Steady-state Quantum Transport, Phys

    Dhar, A.; Saito, K.; H¨anggi, P. Nonequilibrium Density-matrix Description of Steady-state Quantum Transport, Phys. Rev. E 2012, 85, 011126/1–11

    work page 2012

  29. [29]

    J.; Chakravarty, S.; Dorsey, A

    Leggett, A. J.; Chakravarty, S.; Dorsey, A. T.; Fisher, M. P. A.; Garg, A.; Zwerger, W. Dynamics of the Dissipative Two-state System, Rev. Mod. Phys. 1987, 59, 1–85

    work page 1987

  30. [30]

    Quantum Dissipative Systems (World Scientific, Singapore), 2012

    Weiss, U. Quantum Dissipative Systems (World Scientific, Singapore), 2012

    work page 2012

  31. [31]

    A.; Wang, H.; Thoss, M

    Velizhanin, K. A.; Wang, H.; Thoss, M. Heat Transport Through Model Molecular Junctions: A Multilayer Multiconfigura- tion Time–dependent Hartree Approach, Chem. Phys. Letts. 2014, 460, 325–330

    work page 2014

  32. [32]

    K.; Segal, D

    Kilgour, M.; Agarwalla, B. K.; Segal, D. Path-integral methodology and simulations of quantum thermal transport: Full counting statistics approach J. Chem. Phys. 2019, 150, 084111/1–16

    work page 2019

  33. [33]

    Manzano, D; Chuang, C; Cao, J ”Quantum transport in d-dimensional lattices, New Journal of Physics 2016, 18, 043044/1– 10

    work page 2016

  34. [34]

    S.; Effects of Bath Relaxation on Dissipative Two-state Dynamics,J

    Cao, J. S.; Effects of Bath Relaxation on Dissipative Two-state Dynamics,J. Chem. Phys. 2000, 112, 6719–6724

    work page 2000

  35. [35]

    T.; Nitzan, A

    Cravena, G. T.; Nitzan, A. Electron Transfer Across a Thermal Gradient, Proc. Natl. Acad. Sci. U.S.A. 2016, 113, 9421–9429

    work page 2016

  36. [36]

    Breuer, H.-P.; Petruccione, F.The Theory of Open Quantum Systems (Oxford University Press, Oxford), 2007

    work page 2007

  37. [37]

    McCutcheon, D. P. S.; Dattani, N. S.; Gauger, E. M.; Lovett, B. W.; Nazir, A. A General Approach to Quantum Dynamics using a Variational Master Equation: Application to Phonon-damped Rabi Rotations in Quantum Dots, Phys. Rev. B 2011, 84, 081305/1–4

    work page 2011

  38. [38]

    McCutcheon, D. P. S.; Nazir, A. Consistent treatment of coherent and incoherent energy transfer dynamics using a variational master equation J. Chem. Phys. 2011, 135, 114501/1-114501/12. 15

    work page 2011

  39. [39]

    Optimal Tunneling Enhances the Quantum Photovoltaic Effect in Double Quantum Dots,New J

    Wang, C.; Ren, J.; Cao, J. Optimal Tunneling Enhances the Quantum Photovoltaic Effect in Double Quantum Dots,New J. Phys. 2014, 16, 045019/1–16

    work page 2014

  40. [40]

    Stochastic Liouville, Langevin, Fokker-Planck, and Master Equation Approaches to Quantum Dissipative Systems, J

    Tanimura, Y . Stochastic Liouville, Langevin, Fokker-Planck, and Master Equation Approaches to Quantum Dissipative Systems, J. Phys. Soc. Jpn. 2006, 75, 082001/1–39

    work page 2006

  41. [41]

    Quantum heat current under non-perturbative and non-Markovian conditions: Applications to heat machines J

    Kato, A.; Tanimura, Y . Quantum heat current under non-perturbative and non-Markovian conditions: Applications to heat machines J. Chem. Phys. 2016, 145, 224105/1–9

    work page 2016

  42. [42]

    A Unified Stochastic Formulation of Dissipative Quantum Dynamics

    Hsieh, C-Y .; Cao, J. A Unified Stochastic Formulation of Dissipative Quantum Dynamics. I. Generalized Hierarchical Equations, J. Chem. Phys. 2018, 148, 014103/1–14

    work page 2018

  43. [43]

    Zero-temperature Localization in a Sub-Ohmic Spin-Boson Model Investigated by an Extended Hierarchy Equation of Motion, Phys

    Duan, C.; Tang, Z.; Cao, J.; Wu, J. Zero-temperature Localization in a Sub-Ohmic Spin-Boson Model Investigated by an Extended Hierarchy Equation of Motion, Phys. Rev. B 2017, 95, 214308/1–8

    work page 2017

  44. [44]

    Spin-Boson Thermal Rectifier,Phys

    Segal, D.; Nitzan, A. Spin-Boson Thermal Rectifier,Phys. Rev. Lett. 2005, 94, 034301/1–4

    work page 2005

  45. [45]

    Segal, D.; Agarwalla, B. K. Vibrational heat transport in molecular junctions Ann. Rev. Phys. Chem. 2016, 67, 185–209 16

    work page 2016