Transition from population- to coherence-dominated nondiffusive thermal transport

Bradley J. Siwick; Laurenz Kremeyer; Samuel Huberman

arxiv: 2512.13616 · v3 · submitted 2025-12-15 · ❄️ cond-mat.mtrl-sci · cond-mat.mes-hall

Transition from population- to coherence-dominated nondiffusive thermal transport

Laurenz Kremeyer , Bradley J. Siwick , Samuel Huberman This is my paper

Pith reviewed 2026-05-16 22:03 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.mes-hall

keywords phonon transportWigner transport equationnondiffusive thermal transportphonon coherencesize effectsthermal conductivityCsPbBr3La2Zr2O7

0 comments

The pith

The Wigner Transport Equation predicts that phonon coherences cause nondiffusive heat transport in low-conductivity crystals at sub-micron scales.

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

This paper develops a method using the Wigner Transport Equation to track both phonon populations and coherences under arbitrary heat sources in materials where standard Boltzmann approaches do not apply. The equation accounts for tunnelling between overlapping phonon bands, which becomes relevant in low thermal conductivity crystals with large unit cells or strong anharmonicity. Applied to CsPbBr3 and La2Zr2O7 with first-principles inputs, the scheme shows that thermal conductivity deviates from bulk values at lengths from hundreds of nanometers to a few microns at room temperature. These scales are accessible to current experiments, so the predictions can be tested directly. The work therefore supplies a concrete way to calculate when coherence effects overtake population-driven diffusion in real materials.

Core claim

Via solutions to the Wigner Transport Equation, the dynamics of phonon populations and coherences are obtained as a function of an arbitrary heat source, yielding predictions of significant deviations from bulk thermal conductivity in CsPbBr3 and La2Zr2O7 at length scales of hundreds of nanometers to a few microns at room temperature.

What carries the argument

The Wigner Transport Equation, which incorporates tunnelling between overlapping phonon bands and evolves both populations and coherences.

If this is right

Size effects appear in thermal conductivity at experimentally accessible lengths in these materials.
Dynamical thermal conductivities can be computed from first-principles phonon data.
Coherence contributions become dominant over population diffusion below a few microns.
The transition length scale lies within reach of current nanoscale measurement techniques.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same coherence mechanism may govern transport in other strongly anharmonic or large-unit-cell insulators.
Nanostructure design in thermoelectric or thermal-management devices could be adjusted to exploit or suppress the coherence regime.
Time-dependent heat sources beyond steady state would follow the same population-coherence evolution scheme.

Load-bearing premise

The Wigner Transport Equation is the right framework for low thermal conductivity materials with large primitive cells or strong anharmonicity, and the first-principles phonon data accurately represent the real materials.

What would settle it

Direct measurement of thermal conductivity in thin films or nanostructures of CsPbBr3 or La2Zr2O7 with thicknesses between 100 nm and 5 microns, compared against the predicted size-dependent deviations from bulk values.

Figures

Figures reproduced from arXiv: 2512.13616 by Bradley J. Siwick, Laurenz Kremeyer, Samuel Huberman.

**Figure 2.** Figure 2: b, where the temperature is fixed at 500 K. The intersecting line of the two planes in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Phonon dispersion of La [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

read the original abstract

Deviations from diffusive heat transport in high thermal conductivity crystalline insulators are generally understood within the framework of the phonon Boltzmann Transport Equation. However, for low thermal conductivity materials with large primitive cells or strong anharmonicity, the recently developed Wigner Transport Equation is more appropriate as it includes tunnelling between overlapping phonon bands. In this work, via solutions to the Wigner Transport Equation, we develop a scheme to obtain the dynamics of the phonon populations and coherences as a function of an arbitrary heat source. The approach is applied to predict size effects and dynamical thermal conductivities in CsPbBr$_\text{3}$ and La$_\text{2}$Zr$_\text{2}$O$_\text{7}$ using first-principles data as input. We predict significant deviations from the bulk thermal conductivity in these materials at length scales on the order of hundreds of nanometers to a few microns at room temperature, well within the reach of direct observation using current experimental techniques.

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.

Desk Editor's Note private letter to a colleague

The paper gives a workable scheme to pull phonon populations and coherences from the Wigner equation for any heat source and applies it to forecast coherence-driven size effects in CsPbBr3 and La2Zr2O7 at experimentally reachable scales.

read the letter

Dear colleague, the main thing to know is that this paper develops a scheme to extract both population and coherence dynamics from solutions to the Wigner Transport Equation under arbitrary heat sources, then uses first-principles inputs to predict when coherence starts to dominate and shifts the effective thermal conductivity with sample size in CsPbBr3 and La2Zr2O7. They flag deviations from bulk values at lengths from a few hundred nanometers to a couple of microns at room temperature. That part is concrete and points to effects that current experiments could test directly. The approach extends the Wigner framework in a practical direction rather than just re-applying it to a new case. The materials chosen are relevant for low-conductivity applications, and the reported length scales are realistic for observation. The softer spot is the missing validation. The abstract and stress-test note give no direct comparison showing that the scheme recovers the standard Boltzmann transport limit or matches measured bulk conductivity for these compounds. Since the inputs come from first-principles calculations on strongly anharmonic systems with large cells, errors in the phonon dispersions or anharmonic matrix elements would move the predicted crossover lengths. Without that check, the size at which coherence takes over remains sensitive to input uncertainty. This is aimed at researchers already working with quantum phonon transport models who need a way to handle general sources and coherence in size-dependent problems. A reader looking for new numerical predictions in complex crystals will find usable results here, though they will want to inspect the implementation details. I would send it for peer review. The claims are specific enough that referees can evaluate the numerics and ask for the necessary benchmarks.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a numerical scheme based on solutions to the Wigner Transport Equation (WTE) to compute the time-dependent dynamics of phonon populations and coherences under arbitrary heat sources. Using first-principles phonon data as input, the approach is applied to CsPbBr₃ and La₂Zr₂O₇ to predict size-dependent deviations from bulk thermal conductivity arising from coherence (off-diagonal) terms, with significant nondiffusive effects claimed at length scales of hundreds of nanometers to a few microns at room temperature.

Significance. If the central predictions hold after validation, the work would be significant for extending phonon transport theory beyond the Boltzmann equation to low-κ materials with large unit cells or strong anharmonicity. The explicit treatment of coherence effects and the scheme for arbitrary sources could enable modeling of dynamical thermal transport in complex crystals, with the claimed length scales being experimentally accessible. The use of first-principles inputs is a positive feature, though the absence of internal benchmarks against known limits reduces immediate impact.

major comments (2)

[§2 and §4] §2 (WTE formulation) and §4 (application to CsPbBr₃/La₂Zr₂O₇): the manuscript does not demonstrate that the WTE solutions recover the known diffusive (Boltzmann) limit at large length scales or match measured bulk κ values for these materials. Without this check, the predicted transition to coherence-dominated transport at 100 nm–few μm remains sensitive to possible systematic errors in the input dispersions, velocities, and anharmonic matrix elements.
[§3] §3 (numerical scheme): the propagation of populations and coherences for arbitrary heat sources is presented without reported error analysis, convergence tests with respect to k-point sampling, or comparison to analytic limits for simple cases. This is load-bearing because any bias in the coherence terms directly affects the claimed length-scale predictions.

minor comments (2)

[Figure 1] Figure 1 and associated text: the definition of the heat-source term and its coupling to the WTE should be clarified with an explicit equation reference to avoid ambiguity in how arbitrary sources are implemented.
[Abstract and §1] The abstract and introduction use 'significant deviations' without quantifying the percentage change in effective κ; adding a table or plot of κ_eff vs. length scale would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript to incorporate additional validation.

read point-by-point responses

Referee: [§2 and §4] §2 (WTE formulation) and §4 (application to CsPbBr₃/La₂Zr₂O₇): the manuscript does not demonstrate that the WTE solutions recover the known diffusive (Boltzmann) limit at large length scales or match measured bulk κ values for these materials. Without this check, the predicted transition to coherence-dominated transport at 100 nm–few μm remains sensitive to possible systematic errors in the input dispersions, velocities, and anharmonic matrix elements.

Authors: We agree that explicit verification of the diffusive limit is necessary to confirm the numerical implementation and input reliability. In the revised manuscript we will add a dedicated subsection demonstrating that, for large length scales (several microns and beyond), the WTE-derived thermal conductivity converges to the value obtained from the Boltzmann transport equation using the same first-principles phonon dispersions, velocities, and anharmonic matrix elements. We will also include direct comparisons of the computed bulk thermal conductivities for CsPbBr₃ and La₂Zr₂O₇ against available experimental literature values to address possible systematic errors. revision: yes
Referee: [§3] §3 (numerical scheme): the propagation of populations and coherences for arbitrary heat sources is presented without reported error analysis, convergence tests with respect to k-point sampling, or comparison to analytic limits for simple cases. This is load-bearing because any bias in the coherence terms directly affects the claimed length-scale predictions.

Authors: We acknowledge the importance of rigorous numerical validation. In the revision we will add an error analysis for the time-propagation scheme, including convergence tests with respect to k-point sampling density. We will also include benchmarks against analytic limits for simplified cases (e.g., the harmonic limit and the relaxation-time approximation) to confirm the accuracy of the coherence propagation before discussing the length-scale predictions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; forward WTE solutions from external first-principles inputs

full rationale

The paper develops a computational scheme to solve the Wigner Transport Equation for phonon populations and coherences driven by arbitrary heat sources, then applies it to first-principles phonon data for CsPbBr3 and La2Zr2O7 to predict size-dependent deviations from bulk conductivity. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or ansatz smuggled from prior work; the reported length-scale predictions are forward propagations of independent inputs. This is the normal case of a self-contained numerical prediction against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the Wigner equation is required for these materials and on the accuracy of first-principles phonon calculations; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The Wigner Transport Equation is more appropriate than the phonon Boltzmann Transport Equation for low thermal conductivity materials with large primitive cells or strong anharmonicity.
Explicitly stated in the abstract as the justification for the chosen framework.

pith-pipeline@v0.9.0 · 5478 in / 1383 out tokens · 55306 ms · 2026-05-16T22:03:00.090607+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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M. Simoncelli, N. Marzari, and F. Mauri, Wigner formu- lation of thermal transport in solids, Phys. Rev. X12, 041011 (2022)

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L.Kremeyer,greenWTE—PythonPackage,pypi.org/p/ greenWTE(2025), version 1.0.0

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Simoncelli, N

M. Simoncelli, N. Marzari, and F. Mauri, Materials cloud - wigner formulation of thermal transport in solids, 10.24435/materialscloud:g0-yc (2021). Appendix A: V ectorization operator The vectorization operatorvec(A)will stack the columns of an(m×n)matrixAon top of one another, resulting in an(mn×1)vector: vec(A) = [a1,1, . . . , am,1, . . . , am,2, . . ....

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We sub- sequently usedphono3pyto compute the phonon prop- erties required as input togreenWTEon a15×15×15 grid

Second- and third-order interatomic forces for silicon were taken from thephono3pyexamples [28]. We sub- sequently usedphono3pyto compute the phonon prop- erties required as input togreenWTEon a15×15×15 grid. InthecaseofCsPbBr 3 andLa 2Zr2O7, precomputed interatomic force constants from [29] were used to com- pute the phonon properties on8×8×5and19×19×19 ...

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3b and c

A zoom into the low- energy part of the spectrum is shown in Fig. 3b and c. One can directly compare the static case with the high- frequency case at 31GHz that was discussed in the main text. Whilethecharacteroftheindividualphononmodes does not change significantly, the population-dominated acoustic modes are suppressed in the high-frequency case and the...

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[1] [1]

The phonon disper- sion of La2Zr2O7 along the KΓLhigh-symmetry path is shown in Fig

This baseline is compared against the high-frequency case at 31GHz, whereκ P =κ C orχ= 0. The phonon disper- sion of La2Zr2O7 along the KΓLhigh-symmetry path is shown in Fig. 2c. Along with the dispersion, the change in transport character∆χbetween the static and the high-frequency case is encoded in the color of the phonon modes. At no point in the Brill...

work page 2021

[2] [2]

Y. S. Ju and K. E. Goodson, Phonon scattering in silicon films with thickness of order 100 nm, Applied Physics Letters74, 3005 (1999)

work page 1999

[3] [3]

Liu and M

W. Liu and M. Asheghi, Thermal conductivity measure- ments of ultra-thin single crystal silicon layers, Journal of heat transfer128, 75 (2006)

work page 2006

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Cheaito, J

R. Cheaito, J. C. Duda, T. E. Beechem, K. Hattar, J. F. Ihlefeld, D. L. Medlin, M. A. Rodriguez, M. J. Campion, E. S. Piekos, and P. E. Hopkins, Experimental investiga- tion of size effects on the thermal conductivity of silicon- germanium alloy thin films, Physical review letters109, 195901 (2012)

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E. A. Scott, C. Perez, C. Saltonstall, D. P. Adams, V. Carter Hodges, M. Asheghi, K. E. Goodson, P. E. Hopkins, D. Leonhardt, and E. Ziade, Simultaneous thickness and thermal conductivity measurements of thinned silicon from 100 nm to 17µm, Applied Physics Letters118(2021)

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J. A. Johnson, A. Maznev, J. Cuffe, J. K. Eliason, A. J. Minnich, T. Kehoe, C. M. S. Torres, G. Chen, and K. A. Nelson, Direct measurement of room-temperature non- diffusive thermal transport over micron distances in a silicon membrane, Physical review letters110, 025901 (2013)

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Y. Hu, L. Zeng, A. J. Minnich, M. S. Dresselhaus, and G. Chen, Spectral mapping of thermal conductivity through nanoscale ballistic transport, Nature nanotech- nology10, 701 (2015)

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Cuffe, J

J. Cuffe, J. K. Eliason, A. A. Maznev, K. C. Collins, J. A. Johnson, A. Shchepetov, M. Prunnila, J. Ahopelto, C. M. Sotomayor Torres, G. Chen,et al., Reconstructing phonon mean-free-path contributions to thermal conduc- tivity using nanoscale membranes, Physical Review B91, 245423 (2015)

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L. Zeng, K. C. Collins, Y. Hu, M. N. Luckyanova, A. A. Maznev, S. Huberman, V. Chiloyan, J. Zhou, X. Huang, K. A. Nelson,et al., Measuring phonon mean free path distributions by probing quasiballistic phonon transport in grating nanostructures, Scientific reports5, 17131 (2015)

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Huberman, V

S. Huberman, V. Chiloyan, R. A. Duncan, L. Zeng, R. Jia, A. A. Maznev, E. A. Fitzgerald, K. A. Nelson, and G. Chen, Unifying first-principles theoretical predic- tions and experimental measurements of size effects in thermal transport in sige alloys, Physical Review Mate- rials1, 054601 (2017)

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Huberman, R

S. Huberman, R. A. Duncan, K. Chen, B. Song, V. Chiloyan, Z. Ding, A. A. Maznev, G. Chen, and K. A. Nelson, Observation of second sound in graphite at tem- peratures above 100 k, Science364, 375 (2019)

work page 2019

[12] [12]

Jeong, X

J. Jeong, X. Li, S. Lee, L. Shi, and Y. Wang, Transient hydrodynamic lattice cooling by picosecond laser irradi- ation of graphite, Physical Review Letters127, 085901 (2021)

work page 2021

[13] [13]

Z. Ding, K. Chen, B. Song, J. Shin, A. A. Maznev, K. A. Nelson, and G. Chen, Observation of second sound in graphite over 200 k, Nature communications13, 285 (2022)

work page 2022

[14] [14]

Kremeyer, T

L. Kremeyer, T. L. Britt, B. J. Siwick, and S. C. Hu- berman, Ultrafast electron diffuse scattering as a tool for studying phonon transport: Phonon hydrodynamics and second sound oscillations, Structural Dynamics11, 7 10.1063/4.0000224 (2024)

work page doi:10.1063/4.0000224 2024

[15] [15]

Simoncelli, N

M. Simoncelli, N. Marzari, and F. Mauri, Unified the- ory of thermal transport in crystals and glasses, Nature Physics15, 809 (2019)

work page 2019

[16] [16]

Chiloyan, S

V. Chiloyan, S. Huberman, Z. Ding, J. Mendoza, A. A. Maznev, K. A. Nelson, and G. Chen, Green’s functions of the boltzmann transport equation with the full scat- tering matrix for phonon nanoscale transport beyond the relaxation-time approximation, Physical Review B104, 245424 (2021)

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Callaway, Model for lattice thermal conductivity at low temperatures, Phys

J. Callaway, Model for lattice thermal conductivity at low temperatures, Phys. Rev.113, 1046 (1959)

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P. B. Allen, Improved callaway model for lattice thermal conductivity, Phys. Rev. B88, 144302 (2013)

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Fugallo, M

G. Fugallo, M. Lazzeri, L. Paulatto, and F. Mauri, Ab initio variational approach for evaluating lattice thermal conductivity, Phys. Rev. B88, 045430 (2013)

work page 2013

[20] [20]

Cepellotti and N

A. Cepellotti and N. Marzari, Thermal transport in crys- tals as a kinetic theory of relaxons, Phys. Rev. X6, 041013 (2016)

work page 2016

[21] [21]

C. J. Glassbrenner and G. A. Slack, Thermal conduc- tivity of silicon and germanium from 3K to the melting point, Phys. Rev.134, A1058 (1964)

work page 1964

[22] [22]

Simoncelli, N

M. Simoncelli, N. Marzari, and F. Mauri, Wigner formu- lation of thermal transport in solids, Phys. Rev. X12, 041011 (2022)

work page 2022

[23] [23]

Caldarelli, M

G. Caldarelli, M. Simoncelli, N. Marzari, F. Mauri, and L. Benfatto, Many-body green’s function approach to lat- tice thermal transport, Phys. Rev. B106, 024312 (2022)

work page 2022

[24] [24]

Bencivenga, R

F. Bencivenga, R. Cucini, F. Capotondi, A. Battistoni, R. Mincigrucci, E. Giangrisostomi, A. Gessini, M. Man- fredda, I.P.Nikolov, E.Pedersoli, E.Principi, C.Svetina, P. Parisse, F. Casolari, M. B. Danailov, M. Kiskinova, and C. Masciovecchio, Four-wave mixing experiments with extreme ultraviolet transient gratings, Nature520, 205 (2015)

work page 2015

[25] [25]

Chergui, M

M. Chergui, M. Beye, S. Mukamel, C. Svetina, and C. Masciovecchio, Progress and prospects in nonlinear extreme-ultraviolet and x-ray optics and spectroscopy, Nat. Rev. Phys.5, 578 (2023)

work page 2023

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S. G. Volz, Thermal insulating behavior in crystals at high frequencies, Phys. Rev. Lett.87, 074301 (2001)

work page 2001

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L.Chaput,Directsolutiontothelinearizedphononboltz- mann equation, Phys. Rev. Lett.110, 265506 (2013)

work page 2013

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L.Kremeyer,greenWTE—PythonPackage,pypi.org/p/ greenWTE(2025), version 1.0.0

work page 2025

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A. Togo, L. Chaput, T. Tadano, and I. Tanaka, Imple- mentation strategies in phonopy and phono3py, J. Phys. Condens. Matter35, 353001 (2023)

work page 2023

[30] [30]

Simoncelli, N

M. Simoncelli, N. Marzari, and F. Mauri, Materials cloud - wigner formulation of thermal transport in solids, 10.24435/materialscloud:g0-yc (2021). Appendix A: V ectorization operator The vectorization operatorvec(A)will stack the columns of an(m×n)matrixAon top of one another, resulting in an(mn×1)vector: vec(A) = [a1,1, . . . , am,1, . . . , am,2, . . ....

work page doi:10.24435/materialscloud:g0-yc 2021

[31] [31]

We sub- sequently usedphono3pyto compute the phonon prop- erties required as input togreenWTEon a15×15×15 grid

Second- and third-order interatomic forces for silicon were taken from thephono3pyexamples [28]. We sub- sequently usedphono3pyto compute the phonon prop- erties required as input togreenWTEon a15×15×15 grid. InthecaseofCsPbBr 3 andLa 2Zr2O7, precomputed interatomic force constants from [29] were used to com- pute the phonon properties on8×8×5and19×19×19 ...

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[32] [32]

3b and c

A zoom into the low- energy part of the spectrum is shown in Fig. 3b and c. One can directly compare the static case with the high- frequency case at 31GHz that was discussed in the main text. Whilethecharacteroftheindividualphononmodes does not change significantly, the population-dominated acoustic modes are suppressed in the high-frequency case and the...

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