{"id":"b0992a10-1fc4-44f5-b661-c60d0f53d8b8","arxiv_id":"2502.03362","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Replacing energy-conserving delta functions with self-consistent Lorentzian-broadened spectral functions brings predicted boron arsenide thermal conductivity closer to experiment and challenges four-phonon dominance.","lead":"Phonon calculations usually require each scattering event to conserve energy exactly, which the authors argue violates the fluctuation-dissipation theorem. They replace strict energy conservation with self-consistent, lifetime-broadened spectral functions and show thermal conductivity predictions for boron arsenide move closer to experiment.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Markovian validity is tested only in the energy-conserving (small-linewidth) regime, so the FD self-consistent BAs result—where linewidths are much larger—remains unvalidated.","rationale":"The reader's weakest_assumption correctly identifies the Markovian simplification and the neglect of the real part of the memory kernel as the key risk. My concern sharpens this: the paper's own validity test for the Markovian approximation (SM S5) is performed in the energy-conserving (EC) regime, where linewidths are small by construction, and therefore cannot validate the FD regime where linewidths are significantly broadened. This is an internal inconsistency in the evidence, not merely a disagreement with the community consensus. If a full non-Markovian FD calculation reveals a significant difference, then the BAs thermal conductivity result and the conclusion that the EC four-phonon agreement is 'a coincidence' would not be supported. Conversely, if the non-Markovian FD calculation agrees with the Markovian FD result, the central claim becomes substantially stronger. Thus the concern is load-bearing but addressable, consistent with the reader's CONDITIONAL verdict. I do not see grounds for REJECT because the theoretical framework is well-posed and the missing test is a clear computational task, not a logical contradiction.","tokens_in":98,"tokens_out":2354,"duration_ms":118478,"concrete_test":"Perform a full frequency-dependent self-consistent calculation for BAs at 300 K (and optionally 600 K) without the Markovian simplification: solve Eq. (2) using Γλ(ω) from Eq. (3) with Sλ′λ″(ω) computed via the non-Markovian convolution of Eq. (4), and include the real part Δλ(ω) obtained by Kramers–Kronig transformation. Feed the resulting frequency-dependent spectral functions into the single-mode thermal conductivity formula (SM Eq. S38) with the same MLIP and numerical settings. Compare κ_nonMarkovian_FD to the Markovian FD value reported in the main text. If the difference exceeds ~10–20% (or if Δλ shifts mode frequencies enough to alter the phase space), the Markovian simplification is not valid in the FD regime and the central BAs claim would need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central numerical claim is that self-consistent three-phonon linewidths under the fluctuation-dissipation condition (FD) reproduce the experimental thermal conductivity of BAs, making the four-phonon correction appear coincidental. This claim depends on the Markovian approximation: the memory kernel is collapsed to Γλ(Ωλ) and the spectral function to a sum of Lorentzians with no frequency shift Δλ (main text after Eq. 5; SM S2.B). The Markovian approximation is a controlled approximation only when linewidths are small compared to the relevant frequency scales and when the real part of the kernel is negligible. The authors test this approximation in SM S5, but the test (Eqs. S38–S40) is performed with strict energy conservation (EC), i.e., without FD self-consistency. Under EC, BAs linewidths are small because the acoustic–optical gap forbids many three-phonon processes, so the Markovian limit is expected to be accurate. In the FD calculation, however, the entire point is that those processes are 'unblocked', producing substantially larger linewidths (see Fig. 2c, main text). The small-linewidth assumption used in deriving the Markovian S-function (SM Eq. S32) and the neglect of Δλ may then break down. The self-consistency itself could amplify this: using broadened Lorentzians in Γ increases linewidths further, potentially making the approximation worse. Without a non-Markovian FD calculation, one cannot tell whether the BAs result is a physical prediction or an artifact of an unjustified simplification. The paper acknowledges that the SM S5 tests are 'without FD and without collective effects', but does not quantify the error this introduces in the FD regime. This gap is load-bearing because the BAs thermal conductivity reduction, and hence the 'four-phonon coincidence' conclusion, hinges on the magnitude and shape of the FD linewidths.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This Letter develops a self-consistent, fluctuation-dissipation (FD) framework for phonon linewidths within the authors' Mori-Zwanzig/mode-coupling theory of anharmonic lattice dynamics. The central formal result is that the standard delta-function expression for the three-phonon self-energy (Eq. 5) follows from inserting non-interacting spectral functions into the exact spectral convolution (Eq. 4), and that replacing the deltas by Lorentzian convolutions with self-consistently determined linewidths (Eq. 6 in the Markovian limit) restores consistency with the fluctuation-dissipation theorem at the level of individual scattering events. Applying this scheme to boron arsenide with an MLIP-based ab initio force-constant pipeline, the authors find that self-consistent three-phonon linewidths alone bring the thermal conductivity into the experimental range, and argue that the agreement of the energy-conserving (EC) three-phonon-plus-four-phonon calculation with experiment is coincidental. Silicon and silver iodide are presented as control systems with, respectively, negligible and moderate FD corrections.","tokens_in":17824,"tokens_out":26659,"duration_ms":254836,"significance":"If correct, the paper makes three contributions: a formal justification for previously proposed self-consistent phonon broadening schemes; the identification of long Lorentzian tails (rather than self-consistency per se) as the mechanism that activates acoustic-optical scattering across the BAs band gap; and a sharp, falsifiable prediction that four-phonon scattering is not required to explain the measured BAs conductivity. The calculations are parameter-free in the sense that matters: the linewidths are fixed points of a self-consistent integral equation, not fits to experiment, and the experimental values enter only as benchmarks. The MLIP is validated against DFT (force RMSE 0.016 eV/Å), the EC results reproduce published calculations, and the Si control behaves as expected for a gapless harmonic system, which strengthens the causal argument that the BAs effect stems from the gap. The main risk is that the numerical BAs claim rests on the Markovian and small-linewidth reductions in exactly the regime (large self-consistent linewidths) where those reductions are least controlled, as detailed in the major comments.","major_comments":[{"comment":"The validity test of the Markovian approximation is performed only in the energy-conserving (EC) regime; SM S5 states explicitly that the non-Markovian thermal conductivity is computed 'without the fluctuation dissipation theorem' and 'maintaining strict energy conservation'. Under EC the BAs three-phonon linewidths are small because the acoustic-optical gap forbids the relevant processes, so the small-linewidth conditions behind the Markovian reduction (Eqs. S29 and S32) hold by construction. The FD calculation is precisely the regime in which the linewidths become substantially larger (main text Fig. 2c) and in which the self-consistent feedback of Eq. (6) can broaden the spectral functions further. The central numerical claim of the Letter therefore rests on an approximation whose validity is untested in the regime that matters most. I ask that the authors either perform a non-Markovian FD test, for example by solving the frequency-dependent convolution in Eqs. (3) and (4) with G< built from the full susceptibility of Eq. (2) on a subset of modes or a coarser q-grid, or provide a quantitative bound on the Markovian error at the self-consistent FD linewidths.","section":"SM S5, Eqs. (S38)-(S40), Figs. S8-S9"},{"comment":"Independent of the memory-kernel question, the reduction to Eq. (6) involves two further approximations that are uncontrolled in the FD regime. First, the real part Δλ(ω) of the memory kernel is dropped: the full susceptibility of Eq. (2) contains the 2ωΔλ(ω) term in the denominator, whereas Eq. (S29) and the iteration cycle retain only Γλ = Γλ(Ωλ); since Γλ and Δλ are Kramers-Kronig partners, a substantially broadened FD linewidth implies a non-negligible shift. Second, the Bose factors n(±Ω')n(±Ω'')/n(ω) in Eq. (S32) are replaced by their values at the Lorentzian peaks under the 'small enough linewidth' assumption; at 100-300 K the scale kBT over which n(ω) varies can be comparable to the detunings and widths that control the acoustic-optical unblocking. These errors enter directly into the self-consistent linewidths that produce the BAs agreement, so they should be quantified, for example by comparing Eq. (6) with the direct convolution of Bose-weighted Lorentzian spectral functions over the full frequency range for a sample of representative modes.","section":"SM S2.B, Eqs. (S29)-(S32); main text Eq. (6)"},{"comment":"The complete FD treatment including four-phonon interactions appears as the 'fd 3ph+4ph' curve in SM Fig. S2, but the corresponding numbers are never quoted or discussed in the main text. The reader therefore cannot determine whether the fully converged FD 3ph+4ph result agrees with experiment or falls below it; should it fall below, the relationship between the virtual-phonon processes generated by the self-consistent three-phonon channel (the diagram of Fig. 1a is itself an effective four-phonon process) and the explicit fourth-order vertex Γ4ph of Eq. (S36) would need to be examined for double counting. In addition, the claim that the EC 3ph+4ph agreement with experiment is 'actually a coincidence' requires uncertainty estimates for the EC calculations, which are not provided; the Monte-Carlo noise visible in the FD convergence curve (Fig. S10) suggests that the EC numbers carry comparable scatter. Please report the numerical values of all six curves of Fig. S2 at 300 K with their variances and discuss the FD 3ph+4ph outcome explicitly.","section":"Fig. 2b and SM Fig. S2"},{"comment":"The headline statement that the delta functions of Eq. (5) 'violate the fluctuation-dissipation theorem' is stronger than the derivation supports. SM S2.A obtains Eq. (5) by inserting delta-function spectral functions into the formally exact convolution (Eq. 4), and the Bose factors of Eq. (5) (via the identity S26) preserve the detailed-balance structure of the rates; this is the standard lowest-order self-energy evaluated on free propagators, which is FDT-consistent at that order for an equilibrium system. What the manuscript actually establishes is the need for self-consistency between the spectral functions used to evaluate the memory kernel and the spectral function that the kernel generates, with the delta functions as the zero-broadening limit of that cycle. I recommend qualifying the FDT-violation wording, or demonstrating a concrete failure of the detailed-balance condition for the delta-function S.","section":"Eqs. (4) and (5), SM S2.A"}],"minor_comments":[{"comment":"The Markovian-limit substitution in Eq. (S41) appears to have a prefactor inconsistency: with χ''λ normalized as in Eq. (S29), one has ∫ dω χ''λ(ω)χ''λ(ω) = 1/(2πΓλ), so the replacement should read c(Ωλ)/(2πΓλ). Please check the constant, as it underlies the heuristic argument that strongly non-Markovian modes contribute little to κ.","section":"SM S5, Eq. (S41)"},{"comment":"The sentence 'Simulation time was awarded by by PRACE on Discoverer' contains a duplicated 'by'.","section":"Acknowledgements"},{"comment":"The non-Markovian test is performed in the single-mode approximation and omits the collective (off-diagonal) transport contribution, which the text identifies as important in BAs; the 'we anticipate' statement should be replaced by an explicit estimate or a clearly stated limitation of the test.","section":"SM S5, Figs. S8-S9"},{"comment":"The convergence procedure (ten iterations, mean over the last five) has no stated tolerance, and the iteration-to-iteration evolution in Fig. S10 shows visible oscillation; please report the numerical variance over the averaged iterations and define in the captions what the shaded areas in Figs. 2b and S2-S5 represent.","section":"SM S6, Fig. S10"},{"comment":"The supplemental material is cited with the placeholder 'URL-will-be-inserted-by-publisher'; a stable identifier is needed for review.","section":"Ref. [7]"},{"comment":"The trained BAs MLIP and the self-consistent iteration script are not made available; releasing the potential and the numerical κ(T) data plotted in Fig. S2 would permit independent verification of the headline BAs result.","section":"SM S6"},{"comment":"The claims that the reasoning 'holds identically with higher orders' and that 'similar arguments will hold for all of the commonly used (perturbative) theories' are broader than the demonstrated derivation, which uses the mode-coupling decoupling of four-point correlations (SM Eq. S18); this generality should be substantiated or hedged.","section":"Main text, after Eq. (5)"}],"recommendation":"major_revision","confidential_remarks":"The headline numerical claim (FD 3ph reproduces BAs conductivity; EC 4ph agreement is coincidental) is important but currently depends on approximations validated only outside the regime of interest. I would advise the editor that acceptance should require either a non-Markovian FD test (even on a reduced grid) or an explicit delimiting of the BAs result as an untested prediction. The 'coincidence' framing should be backed by the FD 3ph+4ph numbers and EC error bars, which are currently missing. The manuscript builds heavily on the authors' own prior mode-coupling papers (refs. [5,6]); this is normal, but the availability of the trained potential and iteration data would materially help independent assessment. No issues with attribution or citation fairness were evident."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea is genuinely new: replace energy-conserving delta functions with self-consistent Lorentzian convolutions, justify that replacement via fluctuation-dissipation, and show that for boron arsenide this shifts three-phonon thermal conductivity down to the experimental range. The formal machinery is coherent, and the BAs application is a concrete, falsifiable target.\n\nThe derivation from Mori-Zwanzig through the mode-coupling approximation to Eq. (6) is careful. The authors also handle prior work fairly: they credit earlier self-consistent linewidth schemes (refs. 19, 20) and explain why those schemes break the FDT connection. The supplementary tests on Si and AgI usefully bracket when FD matters—negligible for Si, sizable for AgI. That is honest, and the computational details (MLIP validation, convergence of the self-consistent cycle) are reported in enough detail to reproduce.\n\nThe soft spot is load-bearing. The numerical BAs conclusion rests on the Markovian approximation: the memory kernel is collapsed to a single number and the real part of the kernel is dropped. The validity test in SM S5 compares Markovian vs. non-Markovian results only under strict energy conservation, where BAs linewidths are small. In the FD calculation, the entire point is that the acoustic–optical gap is bridged and linewidths grow substantially (Fig. 2c)—exactly the regime where the Lorentzian convolution and the neglect of the frequency shift are least controlled. Self-consistency can amplify that error. The paper acknowledges the test is \"without FD and without collective effects\" but does not quantify the error in the FD regime. So the BAs result is a suggestive calculation, not a demonstrated prediction.\n\nConsequently, the claim that the EC 4-phonon agreement is a coincidence is overinterpretation. That conclusion depends on the FD linewidths being quantitatively right, and they are not yet tested at the point where it matters. I would not call this a fatal flaw—the formal framework stands on its own—but the headline numerical claim needs a stronger check.\n\nThis paper deserves a serious referee. The formal framework is a legitimate contribution even if the BAs numbers shift, and the BAs claim is important enough to referee. Send it to review, but referees should require a non-Markovian FD test, preferably including the real part of the memory kernel, before the central numerical conclusion is accepted.","headline":"A coherent formal argument for FD-consistent phonon linewidths, with a BAs application that is suggestive but rests on an untested Markovian approximation in the very regime where it matters.","tokens_in":18475,"tokens_out":2074,"would_cite":false,"duration_ms":22120,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Delta-function phonon scattering breaks a key thermodynamic law.","keywords":["phonon-phonon interactions","fluctuation-dissipation theorem","self-consistent linewidths","thermal conductivity","boron arsenide","Markovian approximation","memory kernel","virtual phonons"],"falsifier":"Compute the fully frequency-dependent self-consistent three-phonon linewidths in boron arsenide without the Markovian limit, keeping the full convolution and the real-part frequency shift $\\Delta_\\lambda(\\omega)$, and compare the resulting thermal conductivity with experiments at 300 K and 900 K; a significant departure from the Lorentzian Markovian result would falsify the numerical claim.","tokens_in":17346,"feed_emoji":"🔬","tokens_out":5159,"duration_ms":49262,"temperature":0.7,"pith_summary":"The paper argues that the Dirac delta functions enforcing energy conservation at each microscopic phonon-phonon collision are an approximation that violates the fluctuation-dissipation theorem. It replaces them with convolutions of phonon spectral functions, solved self-consistently, so that energy is conserved only between initial and final states and intermediate phonons are virtual. In the Markovian limit this reduces to Lorentzian-broadened scattering kernels with linewidth increments given by sums of phonon linewidths. Applied to boron arsenide, the self-consistent three-phonon linewidths reproduce experimental thermal conductivity, undercutting the prior conclusion that four-phonon processes dominate. The framework is universal but its numerical impact is system-dependent: negligible in silicon, substantial in silver iodide.","feed_headline":"Delta-function phonon scattering breaks a key thermodynamic law.","feed_subtitle":"A self-consistent three-phonon calculation reproduces boron arsenide's thermal conductivity without four-phonon processes.","key_machinery":"The memory kernel $K_\\lambda(t)$ of the generalized Langevin equation is the central object; its real part $\\Gamma_\\lambda(\\omega)$ is built from a convolution $S_{\\lambda'\\lambda''}(\\omega)$ of two lesser phonon correlation functions. The spectral function $\\chi''_\\lambda(\\omega)$ depends on $\\Gamma_\\lambda(\\omega)$ and its Hilbert-transform partner $\\Delta_\\lambda(\\omega)$. Replacing the memory-less delta approximation by Lorentzian convolutions with widths $\\Gamma_{\\lambda'}+\\Gamma_{\\lambda''}$ and iterating equations (2), (3), and (6) to convergence is the self-consistency mechanism that restores fluctuation-dissipation balance.","core_discovery":"The central claim is that the energy-conserving delta functions in the phonon self-energy stem from neglecting the memory kernel, which treats phonons as infinitely long-lived quasiparticles, and that this neglect breaks the fluctuation-dissipation relation. The correct treatment keeps the convolution of lesser correlation functions and enforces self-consistency between the memory kernel and the spectral function; in the Markovian limit, the delta functions become Lorentzians with widths given by sums of phonon linewidths. For boron arsenide, this self-consistent three-phonon calculation yields thermal conductivities in line with experiments, and the agreement of energy-conserving four-phonon calculations is argued to be a coincidence of an over-high three-phonon value and an over-large four-phonon correction.","pith_inferences":["If the violation is general, perturbative quasiparticle calculations that enforce on-shell energy conservation for intermediate states, such as electron-phonon and magnon-phonon problems, may need the same replacement; the paper notes that self-consistent electronic linewidths already resolve divergences in piezoelectric materials.","The Markovian assumption is the main limiting step for the quantitative boron-arsenide claim; a fully frequency-dependent self-consistent calculation with the Kramers-Kronig shift would determine whether the Lorentzian tails are quantitatively accurate.","A testable extension would apply the same self-consistent convolution to materials with very large acoustic-optical gaps other than boron arsenide, where strict energy-conservation selection rules are expected to overestimate thermal conductivity.","The paper's argument implies that energy-conservation tests in strongly anharmonic materials should be revisited with frequency-dependent spectral functions rather than delta function smearing, because the choice of Gaussian versus Lorentzian broadening can change the outcome."],"forward_implications":["In boron arsenide, the strict energy-conservation three-phonon thermal conductivity is too high, the strict energy-conservation four-phonon correction is too large, and the two errors roughly cancel; treating three-phonon scattering self-consistently with the fluctuation-dissipation condition reproduces the experimental values without invoking four-phonon dominance.","The apparent dominance of four-phonon processes in boron arsenide is overestimated; four-phonon interactions are not negligible, but they are not needed to explain the measured thermal conductivity.","Interactions between acoustic and optical phonons that are forbidden by selection rules under strict energy conservation are reopened through virtual-phonon-mediated processes, which expands the scattering phase space in large-band-gap materials.","The Markovian self-consistent scheme avoids the delicate numerical convergence parameters used to approximate delta functions, because the Lorentzian convolution provides a broader, physically motivated smearing.","The fluctuation-dissipation condition is universal, but its practical effect is system-dependent: it barely changes silicon, while it lowers the three-phonon thermal conductivity of silver iodide by 15% to 25% between 100 and 300 K.","The method extends beyond three-phonon processes: including four-phonon and isotope scattering in the self-consistent memory kernel preserves the fluctuation-dissipation structure while incorporating all scattering channels."],"supporting_citations":[{"why":"Establishes the conventional perturbative phonon-phonon scattering framework with energy-conservation delta functions that the paper argues is incomplete.","marker":"[1]"},{"why":"Supplies the standard implementation of energy-conservation delta functions in phonon Boltzmann transport calculations, the method the paper contrasts with its self-consistent approach.","marker":"[2]"},{"why":"Defines the four-phonon scattering extension that the boron-arsenide result challenges.","marker":"[4]"},{"why":"Provides the mode-coupling theory of anharmonic lattice dynamics that yields the generalized Langevin equation and memory-kernel expression used in the derivation.","marker":"[5]"},{"why":"States the fluctuation-dissipation theorem, the principle the paper claims is violated by delta-function energy conservation.","marker":"[8]"},{"why":"Introduces the interpretation of second-order processes as interactions mediated by virtual phonons that are not subject to energy conservation.","marker":"[11]"},{"why":"Derives the selection rules that forbid acoustic-optical three-phonon interactions in boron arsenide under strict energy conservation, the constraint the fluctuation-dissipation treatment relaxes.","marker":"[15]"},{"why":"Provides the experimental thermal conductivity data for boron arsenide used to assess the self-consistent three-phonon calculation.","marker":"[21]"},{"why":"Attributes the reduced thermal conductivity of boron arsenide to four-phonon scattering, the conclusion the paper argues is a coincidental agreement.","marker":"[25]"}],"fun_headline_variants":["Self-consistent phonons fix a fluctuation-dissipation violation","Three-phonon theory matches experiments when made self-consistent","Delta functions in phonon theory break a thermodynamic law","Boron arsenide thermal conductivity fits without four-phonon terms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The numerical boron-arsenide result rests on the Markovian approximation: the memory kernel is reduced to a single number $\\Gamma_\\lambda(\\Omega_\\lambda)$ and the spectral function to a sum of Lorentzians with no frequency shift, while the non-Markovian validity tests are performed without the fluctuation-dissipation condition and without collective effects; if memory effects matter in boron arsenide, the self-consistent linewidths and the three-phonon conclusion would change.","fun_headline_variants_meta":{"raw":{"variants":["Self-consistent phonons fix a fluctuation-dissipation violation","Three-phonon theory matches experiments when made self-consistent","Delta functions in phonon theory break a thermodynamic law","Boron arsenide thermal conductivity fits without four-phonon terms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000178,"raw_usage":{"total_tokens":1278,"prompt_tokens":910,"completion_tokens":368,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":298}},"tokens_in":526,"tokens_out":368,"duration_ms":4026,"temperature":1.0,"reasoning_tokens":298,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T04:58:07.363451+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the fully frequency-dependent self-consistent three-phonon linewidths in boron arsenide without the Markovian limit, keeping the full convolution and the real-part frequency shift $\\Delta_\\lambda(\\omega)$, and compare the resulting thermal conductivity with experiments at 300 K and 900 K; a significant departure from the Lorentzian Markovian result would falsify the numerical claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the conventional perturbative phonon-phonon scattering framework with energy-conservation delta functions that the paper argues is incomplete."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the standard implementation of energy-conservation delta functions in phonon Boltzmann transport calculations, the method the paper contrasts with its self-consistent approach."},{"cited_title":"Castellano, J","cited_arxiv_id":null,"evidence_quote":"Provides the mode-coupling theory of anharmonic lattice dynamics that yields the generalized Langevin equation and memory-kernel expression used in the derivation."},{"cited_title":"Kubo, The fluctuation-dissipation theorem, Rep","cited_arxiv_id":null,"evidence_quote":"States the fluctuation-dissipation theorem, the principle the paper claims is violated by delta-function energy conservation."},{"cited_title":"Carruthers, Resonance in phonon-phonon scattering, Phys","cited_arxiv_id":null,"evidence_quote":"Introduces the interpretation of second-order processes as interactions mediated by virtual phonons that are not subject to energy conservation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the experimental thermal conductivity data for boron arsenide used to assess the self-consistent three-phonon calculation."}],"review_version":1}