Pith. sign in

REVIEW 3 major objections 6 minor 1 cited by

Implicit unified gas kinetic particle method for steady-state solution of multiscale phonon transport

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The IUGKP method solves the steady-state phonon Boltzmann equation at all Knudsen numbers by sampling particles according to exact mean free paths and accelerating low-Knudsen convergence with a macroscopic prediction step, reaching…

desk verdict Solid extension of UGKWP for steady-state phonon transport, but the headline speedups are not fully reproducible as reported. read the letter →

arxiv 2506.09465 v1 pith:56RHXUXJ submitted 2025-06-11 physics.comp-ph

Hongyu Liu , Xiaojian Yang , Chuang Zhang , Xing Ji , Kun Xu This is my paper · ORCID
classification physics.comp-ph MSC 82C4082C80 PACS 44.10.+i
keywords phononBoltzmanntransportequationunifiedgas-kineticparticlemethodimplicitschememultiscaleheatconductionnullcollisionKnudsennumbersteady-statesolverMonteCarlo
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the steady-state phonon Boltzmann transport equation can be solved much faster by a particle method that samples the distribution function directly from the integral solution of the BGK equation, rather than stepping particles with short numerical time steps. At every iteration, each particle's free path is drawn from the exact mean-free-path distribution, so at large Knudsen numbers particles cross many cells per step and converge quickly. At small Knudsen numbers, where mean free paths are short, the method adds a macroscopic temperature prediction step based on inexact Newton iteration to guide the next particle resampling. A null-collision procedure sets a global time scale from the smallest relaxation time, fixing the spatio-temporal inconsistency that appears when the relaxation time varies in space. Numerical tests on one-dimensional films, discontinuous-relaxation-time problems, and two-dimensional square and rectangular domains report speedups of one to two orders of magnitude over explicit UGKWP, with solutions matching analytical, UGKS, and DUGKS references.

What carries the argument

The central object is the integral solution of the steady phonon BGK equation, written as a convex combination of neighboring equilibrium states weighted by the survival and collision probability densities. The method turns that identity into a particle sampling algorithm: each particle draws a free path $\lambda = -u\tau\ln(\eta)$ from the exponential distribution, real-collision particles are resampled from the wave-component energy $W^h$, and a null-collision accept-reject procedure with acceptance probability $\sigma(x)/\sigma_{\max} = \tau_{\min}/\tau(x)$ maintains a consistent global time scale. For low Knudsen numbers, the macroscopic energy conservation equation $\nabla\cdot q = 0$ is converted into a temperature-increment prediction $\nabla\cdot[\kappa\nabla(\Delta T)]$, whose solution supplies the equilibrium state for the next particle resampling step.

What would settle it

Run the one-dimensional Kn=0.001 film case with Eq. (32) implemented literally with its printed sign: if the update is anti-diffusive, the temperature will not converge to the analytical profile, and the 680-fold speedup in Table 1 cannot have come from that printed equation. Also, in the uniform-temperature test of Section 3.2, the null-collision procedure should keep the temperature exactly uniform at every iteration; any drift in that field would falsify the claim that the method resolves spatio-temporal inconsistency.

Watch

Extended reading notes

Core claim

The paper's central claim is that a steady-state particle solver for phonon transport can be built by treating the distribution function at a point as a weighted combination of equilibrium distributions from surrounding cells, with weights given by the cumulative distribution function of the free-path length. Particles emitted from a cell travel a distance sampled as $-u\tau\ln(\eta)$, reaching a collision point where they either conform to the local equilibrium or, when relaxation times vary in space, undergo a real or null collision selected by a local acceptance probability proportional to $\tau_{\min}/\tau(x)$. For small Knudsen numbers, the particle update is coupled to a macroscopic temperature prediction so that the next equilibrium state is sampled from an energy field advanced by an inexact Newton step. The result reported is convergence to steady state in one to two orders of magnitude fewer steps and less CPU time than explicit UGKWP, with accuracy matching analytical solutions in one dimension and UGKS and DUGKS solutions in multiscale and two-dimensional cases.

Load-bearing premise

The low-Knudsen speedup rests on the macro-prediction step in Section 3.3 being a well-defined update that pushes temperature toward the correct steady state, but as printed the sign in Eq. (32) contradicts the linear operator in Eq. (31), and the conductivity $\kappa$ is never specified in terms of the relaxation time, so the guiding mechanism is not fully defined.

Editorial extensions

If this is right

  • The approach yields steady-state phonon transport solutions one to two orders of magnitude faster than explicit UGKWP across Knudsen numbers from 0.001 to 10, with the largest reported speedups in the ballistic and diffusive limits.
  • Because the particle count per cell stays fixed, particles can be stored in a simple contiguous array, removing the linked-list overhead of the explicit wave-particle method.
  • The null-collision treatment makes spatially varying and discontinuous relaxation-time distributions stable and accurate, so the method applies to domains with large contrasts in local Knudsen number.
  • The same scheme reproduces reference solutions in ballistic, transition, and diffusive regimes of the gray phonon BTE, providing a single solver that covers all scales without switching models.
  • The paper states the method is extendable to photon transport, rarefied gas flow, and electron transport, since the construction follows from the BGK equation rather than from phonon-specific physics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the paper's core recipe is general: replace time-stepping with equilibrium-to-equilibrium free-path sampling and use the macroscopic moment equation as a preconditioner for the particle update; if it works for phonons, the same pattern should carry over to neutral-gas and radiative-transfer UGKWP solvers.
  • Editorial extension: a frequency-dependent or ab initio phonon BTE could be handled by sampling each particle's relaxation time from the material's spectral data, but the macro prediction step would then need to be a frequency-integrated energy equation, so the reported speedup would need revalidation in that setting.
  • Editorial extension: the printed sign inconsistency between Eqs. (31) and (32) and the unspecified form of $\kappa$ mean that a reader cannot reproduce the low-Knudsen acceleration from the text alone; deriving $\kappa$ from the BGK relaxation model and testing the update direction would settle which version of the macro prediction actually produced the reported speedups.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes an implicit unified gas-kinetic particle (IUGKP) method for steady-state solutions of the phonon Boltzmann transport equation under the BGK relaxation-time approximation. The method builds on the integral solution of the steady BGK equation, reinterpreting the distribution at any point as a probability-weighted sum of neighboring equilibrium states. At large Knudsen numbers, particles are freely streamed according to their exact mean free path, while at small Knudsen numbers an inexact-Newton macroscopic prediction is used to accelerate convergence to the diffusive limit. A null-collision treatment is introduced to handle spatially varying relaxation times by sampling candidate collisions at a uniform maximum rate and accepting them with the local true-collision probability. Numerical tests in one and two dimensions, including constant and discontinuous relaxation-time cases, are compared with analytical solutions, UGKS, and DUGKS. The paper claims speedups of one to two orders of magnitude over explicit UGKWP, as summarized in Tables 1 through 6 and Figures 8 through 14.

Significance. If the efficiency claims are reproducible, IUGKP would be a valuable addition to the toolkit for multiscale phonon-transport simulation, particularly for problems spanning ballistic to diffusive regimes. The paper's strengths include a self-contained derivation of the null-collision collision-rate recovery (Eqs. 21-27), which is internally consistent, and convincing accuracy comparisons in the 1D constant-tau cases against analytical solutions. The method is not circular: no target result is used as an input, no fitted constants hide the physics, and speedups are measured against explicit UGKWP with accuracy checked against independent references. The main weakness is that the central efficiency claim is not yet fully reproducible because the steady-state stopping criterion is not defined and the per-step CPU speedup Re is asserted rather than measured. These gaps are local and fixable within the manuscript's scope, but they currently prevent verification of the one-to-two-order speedup claim.

major comments (3)
  1. [Section 4, Eq. (34), Tables 1-6] The number of iterations required to 'reach steady state' is never tied to a convergence criterion. No residual tolerance, temperature-change threshold, or other stopping rule is specified for either the explicit UGKWP or the current method. Since the speedup in Eq. (34) is proportional to the ratio of iteration counts, N_UGKWP and N_current are not reproducible without this definition, and the two methods may be terminated at different accuracy levels. Please specify the convergence measure, apply it identically to both methods, and report the achieved values. This is load-bearing because the headline contribution is the acceleration.
  2. [Section 4.1, Table 1 and Eq. (34)] The per-step CPU-time ratio Re=4 is asserted in the text ('Each step in the current method is about 4 times faster') without any wall-clock or CPU-time measurements. Because the total speedup is the product of the iteration-count ratio and Re, an unmeasured Re can dominate the reported speedups and make them misleading. Please report actual runtimes or a measured per-step timing ratio for at least the 1D cases, and clarify how Re is obtained in the 2D cases.
  3. [Section 3.3, Eqs. (31)-(33)] The macro-prediction equation is underspecified. The conductivity κ in Eq. (31) is never defined in terms of the physical parameters of the model (τ, C, |V_g|), so the equation cannot be implemented from the text. In addition, Eq. (33) identifies ∇·q with the change in the real-collision-particle energy between iterations, but no derivation or justification is given for why the residual of the steady BTE equals this particular discrete energy change. These points need to be clarified or derived before the low-Kn acceleration step can be reproduced and verified.
minor comments (6)
  1. [Section 3.3, Eqs. (31)-(32)] The relationship between Eqs. (31) and (32) should be stated explicitly: Eq. (32) follows from setting R + Q_tilde(ΔT)=0, which gives ∇·q = ∇·[κ∇(ΔT)]. As printed, the sign difference may appear inconsistent to readers; one sentence clarifying the Newton combination would remove the apparent contradiction.
  2. [Section 3.3] The update formula uses C_v while the rest of the paper uses C for the volumetric specific heat; unify the notation.
  3. [Section 4.3] The sentence 'illustrating that the UGKWP method can automatically recover the heat transfer physics in different scales' appears to refer to the current method, not UGKWP; please correct this typo.
  4. [Figure 8] Figure 8 contains typos ('V1 V2', '3rd ietration') and the subplot labels are not self-explanatory; please clean up the figure and add a legend or caption describing each panel's Knudsen number.
  5. [Tables 4 and 6] The notation 'UGKWP 1200 + 200' is ambiguous. Please clarify whether the statistical-averaging steps are included in both the UGKWP and current-method counts, and how the overall speedup is computed from these numbers.
  6. [Section 4, Eq. (34)] The symbol Re is used in Eq. (34) but is only defined later in the text; define it at first use to make the formula self-contained.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the method's derivation is self-contained, though the macro-prediction sign inconsistency and undefined stopping criterion are correctness/reproducibility concerns, not circular reasoning.

full rationale

I walked the claimed derivation chain. The distribution function is built from the integral solution of the steady BGK equation, Eq. (10), leading to the convex-combination form Eq. (13) and the particle resampling/free-flight algorithm. This is a direct mathematical consequence of the BGK model, not a restatement of the target solution. The null-collision treatment derives the true-collision PDF f(s) = σ(s)exp(-∫σ du) from the acceptance-rejection construction, which is an independent first-principles derivation rather than an input fitted to the benchmark results. The macro-prediction acceleration uses the residual of ∇·q = 0 and solves for ΔT via a linear operator; although Eq. (31) and Eq. (32) have an inconsistent sign and κ is not explicitly tied to τ, this affects whether the update is well-defined and stable, not whether the result is circular. The speedup claims compare iteration counts against explicit UGKWP with an assumed per-step ratio Re = 4; the lack of a defined steady-state stopping criterion makes the numbers hard to reproduce, but the iteration counts are not fitted to produce a target speedup, so this is a reproducibility gap rather than a reduction of the prediction to its inputs. Self-citations, e.g., Refs. [37], [39], and [40], are used as methodological lineage for UGKWP and implicit UGKWP, not as external proof of this paper's acceleration result, and no uniqueness theorem is imported from the authors' prior work. The central accuracy comparisons are made against analytical solutions, UGKS, and DUGKS, which are independent references. Accordingly, no circular step can be exhibited from the paper's own equations, and the score is low.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The method adds no fitted physics constants. Its free parameters are numerical control choices (Nref, stopping threshold) and an unspecified κ for the acceleration operator. The physical input is the gray BGK model with a spatially varying relaxation time, and the mathematical machinery is standard exponential free-path sampling plus a macroscopic preconditioner.

free parameters (3)
  • Per-cell particle sample count Nref = 8000 in 1D; 200 or 100 in 2D
    Chosen by hand to balance statistical noise against cost. It affects the accuracy and statistical quality of the reported results but is not fitted to the analytical solution.
  • Macro-prediction conductivity κ = not specified
    The paper never states how κ relates to the relaxation time τ in Eqs. (31) and (32), so the preconditioner for the low-Kn acceleration is not uniquely determined from the text.
  • Steady-state stopping threshold = not specified
    All speedup ratios in Tables 1 through 6 depend on when each method is declared converged, but no residual tolerance or stopping rule is given.
assumptions (5)
  • domain assumption Gray BGK model with constant group velocity and a single relaxation time at each spatial point
    Invoked in Section 2; the method solves this model, not full frequency-dependent phonon transport.
  • domain assumption Boundary distribution functions are at equilibrium, so Eq. (13) expresses the solution as a convex combination of equilibrium states
    Stated in Section 2 before Eq. (13); if boundaries are not diffuse or equilibrium, the reconstruction changes.
  • ad hoc to paper The operator ∇·(-κ∇ΔT) approximates the Jacobian of the steady BTE residual for the inexact Newton update
    Introduced in Section 3.3 as a chosen linear operator; no proof that this preconditioner is convergent for all Kn is given.
  • standard math The null-collision acceptance-rejection scheme with global τmin reproduces the correct position-dependent free-path distribution
    Derived in Eqs. (19) through (27); the derivation is standard for exponential sampling with acceptance-rejection.
  • standard math Finite-particle Monte Carlo expectations are unbiased with acceptable variance at the reported Nref
    Assumed for the particle averages in Eqs. (16) and (28); no variance or seed information is reported.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Implicit unified gas kinetic particle method for steady-state solution of multiscale phonon transport." pith.science (2026). https://pith.science/paper/56RHXUXJ

@misc{pith2026250609465,
  author       = {Pith},
  title        = {Pith review of: Implicit unified gas kinetic particle method for steady-state solution of multiscale phonon transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/56RHXUXJ}},
  note         = {Machine review of arXiv:2506.09465}
}
read the original abstract

This paper presents a highly efficient implicit unified gas-kinetic particle (IUGKP) method for obtaining steady-state solutions of multi-scale phonon transport. The method adapts and reinterprets the integral solution of the BGK equation for time-independent solutions. The distribution function at a given point is determined solely by the surrounding equilibrium states, where the corresponding macroscopic quantities are computed through a weighted sum of equilibrium distribution functions from neighboring spatial positions. From a particle perspective, changes in macroscopic quantities within a cell result from particle transport across cell interfaces. These particles are sampled according to the equilibrium state of their original cells, accounting for their mean free path as the traveling distance. The IUGKP method evolves the solution according to the physical relaxation time scale, achieving high efficiency in large Knudsen number regimes. To accelerate convergence for small Knudsen numbers, an inexact Newton iteration method is implemented, incorporating macroscopic equations for convergence acceleration in the near-diffusive limit. The method also addresses spatial-temporal inconsistency caused by relaxation time variations in physical space through the null-collision concept. Numerical tests demonstrate the method's excellent performance in accelerating multi-scale phonon transport solutions, achieving speedups of one to two orders of magnitude. The IUGKP method proves to be an efficient and accurate computational tool for simulating multiscale non-equilibrium heat transfer, offering significant advantages over traditional methods in both numerical performance and physical applicability.

Figures

Figures reproduced from arXiv: 2506.09465 by the authors.

Figure 1
Figure 1. Illustrating the integral solution of the particle perspective. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Computational domain and boundary conditions for the uniform heat conduction case. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Initial particle distribution for the uniform heat conduction case. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Initial particle distribution for the uniform heat conduction case. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Spatial-time inconsistent transport property. [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Two types of particles in space. (1) If it is the first calculation step, sample N particles in each cell, with each particle following the local equilibrium distribution (w ini = W N , w ini is the initial energy for each particle) and its free path being sampled; if …
Figure 7
Figure 7. Figure 7: A schematic diagram of one-dimensional heat conduction in a dielectric film. [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Comparison of 1D heat conduction results across a film, from Kn = 0.001 to Kn = 10.0. [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Comparison of 1D heat conduction results across a film, discontinuous [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: multi-scale case. For the convergence rate, the UGKWP method requires 3800 steps to reach steady state, whereas the current method only needs 60 steps. Moreover, the current method’s per-step speed is approximately four times faster than that of UGKWP. This translates…
Figure 11
Figure 11. Figure 11: Computational domain and boundary conditions of Heat transfer in the 2D square domain. [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: Comparison of 2D square heat transfer results, respectively corresponding: Kn = 0.01, Kn = [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: Computational domain and boundary conditions of Heat transfer in the 2D rectangle domain. [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: Comparison of 2D rectangular heat transfer results. Left: Kn = 10.0, Right: 1.0. [PITH_FULL_IMAGE:figures/full_fig_p022_14.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. UGKWP and IUGKP methods for Multi-Scale Phonon Transport with Dispersion and Polarization

    physics.comp-ph 2025-06 conditional novelty 6.0 of 10

    The authors extend their unified gas-kinetic wave-particle method to frequency-dependent phonon transport with dispersion and polarization, adding an implicit steady-state variant and adaptive frequency sampling that ...

Reference graph

Works this paper leans on

43 extracted references · 9 canonical work pages · cited by 1 Pith paper

  1. [1]

    Nanoscale energy transport and conversion a parallel treatment of electrons molecules phonons and photons mit pappalardo series in mechanical engineering by gang chen

    Parallel Treatment Of Electrons Molecules Phonons. Nanoscale energy transport and conversion a parallel treatment of electrons molecules phonons and photons mit pappalardo series in mechanical engineering by gang chen

  2. [2]

    Colloquium: phononic thermal properties of two-dimensional materials

    Xiaokun Gu, Yujie Wei, Xiaobo Yin, Baowen Li, and Ronggui Yang. Colloquium: phononic thermal properties of two-dimensional materials. Rev. Mod. Phys., 90:041002, Nov 2018

    2018

  3. [3]

    Interfacial thermal resistance: Past, present, and future

    Jie Chen, Xiangfan Xu, Jun Zhou, and Baowen Li. Interfacial thermal resistance: Past, present, and future. Rev. Mod. Phys., 94:025002, Apr 2022

    2022

  4. [4]

    Review of multiscale simulation in submicron heat transfer

    Jayathi Y Murthy, Sreekant VJ Narumanchi, Pascual-Gutierrez Jose’A, Tianjiao Wang, Chunjian Ni, and Sanjay R Mathur. Review of multiscale simulation in submicron heat transfer. International Journal for Multiscale Computational Engineering, 3(1), 2005

    2005

  5. [5]

    Electrons and phonons: the theory of transport phenomena in solids

    John M Ziman. Electrons and phonons: the theory of transport phenomena in solids. Oxford university press, 2001

    2001

  6. [6]

    A comparative study of submicron phonon transport using the boltzmann transport equation and the lattice boltzmann method

    Ankur Chattopadhyay and Arvind Pattamatta. A comparative study of submicron phonon transport using the boltzmann transport equation and the lattice boltzmann method. Numerical Heat Transfer, Part B: Fundamentals, 66(4):360–379, 2014

    2014

  7. [7]

    Boltzmann transport equation for thermal transport in electronic materials and devices

    Matthew C Barry, Nitish Kumar, and Satish Kumar. Boltzmann transport equation for thermal transport in electronic materials and devices. Annual Review of Heat Transfer, 24:131–172, 2022

    2022

  8. [8]

    Boltzmann transport equation based modeling of phonon heat conduction: Progress and challenges

    Sandip Mazumder. Boltzmann transport equation based modeling of phonon heat conduction: Progress and challenges. Annual Review of Heat Transfer, 24:71–130, 2022

    2022

Show all 43 references
  1. [9]

    Progress of discrete unified gas-kinetic scheme for multiscale flows

    Zhaoli Guo and Kun Xu. Progress of discrete unified gas-kinetic scheme for multiscale flows. Adva. Aerodyn., 3(1):6, Feb 2021

    2021

  2. [10]

    Peraud, Colin D

    Jean-Philippe M. Peraud, Colin D. Landon, and Nicolas G. Hadjiconstantinou. Monte carlo methods for solving the Boltzmann equation. Annu. Rev. Heat Transfer, 17:205–265, 2014

    2014

  3. [11]

    Monte Carlo transient phonon transport in sil- icon and germanium at nanoscales

    David Lacroix, Karl Joulain, and Denis Lemonnier. Monte Carlo transient phonon transport in sil- icon and germanium at nanoscales. Physical Review B—Condensed Matter and Materials Physics, 72(6):064305, 2005. 23

    2005

  4. [12]

    Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media

    Knut Stamnes, S-Chee Tsay, Warren Wiscombe, and Kolf Jayaweera. Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media. Applied optics, 27(12):2502–2509, 1988

    1988

  5. [13]

    Finite volume method for radiative heat transfer using unstructured meshes

    Jayathi Y Murthy and SR Mathur. Finite volume method for radiative heat transfer using unstructured meshes. Journal of thermophysics and heat transfer, 12(3):313–321, 1998

    1998

  6. [14]

    Sadayappan, and Arpit Mittal

    Syed Ashraf Ali, Gautham Kollu, Sandip Mazumder, P. Sadayappan, and Arpit Mittal. Large-scale parallel computation of the phonon Boltzmann transport equation. Int. J. Therm. Sci, 86:341 – 351, 2014

    2014

  7. [15]

    V. A. Fiveland and J. P. Jessee. Acceleration schemes for the discrete ordinates method.J. Thermophys. Heat Transfer., 10(3):445–451, 1996

    1996

  8. [16]

    Molecular Gas Dynamics And The Direct Simulation Of Gas Flows

    G A Bird. Molecular Gas Dynamics And The Direct Simulation Of Gas Flows. Oxford University Press, 05 1994

    1994

  9. [17]

    R. B. Peterson. Direct Simulation of Phonon-Mediated Heat Transfer in a Debye Crystal. Journal of Heat Transfer, 116(4):815–822, 11 1994

    1994

  10. [18]

    Monte Carlo study of phonon transport in solid thin films including dispersion and polarization

    Sandip Mazumder and Arunava Majumdar. Monte Carlo study of phonon transport in solid thin films including dispersion and polarization. J. Heat Transfer, 123(4):749–759, 2001

    2001

  11. [19]

    Monte carlo study of phonon heat conduction in silicon thin films including contributions of optical phonons

    Arpit Mittal and Sandip Mazumder. Monte carlo study of phonon heat conduction in silicon thin films including contributions of optical phonons. J. Heat Transfer, 132(5):052402, 2010

    2010

  12. [20]

    P´ eraud and Nicolas G

    Jean-Philippe M. P´ eraud and Nicolas G. Hadjiconstantinou. Efficient simulation of multidimen- sional phonon transport using energy-based variance-reduced monte carlo formulations. Phys. Rev. B, 84:205331, Nov 2011

    2011

  13. [21]

    Adjoint-based deviational Monte Carlo methods for phonon transport calculations

    Jean-Philippe M P´ eraud and Nicolas G Hadjiconstantinou. Adjoint-based deviational Monte Carlo methods for phonon transport calculations. Physical Review B, 91(23):235321, 2015

    2015

  14. [22]

    Silva, D

    B.H. Silva, D. Lacroix, M. Isaiev, and L. Chaput. Monte carlo simulation of phonon transport from ab-initio data with nano-k. Computer Physics Communications, 294:108954, 2024

    2024

  15. [23]

    Aref, Gary F

    Abhishek Pathak, Avinash Pawnday, Aditya Prasad Roy, Amjad J. Aref, Gary F. Dargush, and Dipan- shu Bansal. MCBTE: A variance-reduced monte carlo solution of the linearized boltzmann transport equation for phonons. Comput. Phys. Commun., 265:108003, 2021

    2021

  16. [24]

    A unified gas-kinetic scheme for continuum and rarefied flows

    Kun Xu and Juan-Chen Huang. A unified gas-kinetic scheme for continuum and rarefied flows. Journal of Computational Physics, 229(20):7747–7764, 2010

    2010

  17. [25]

    Discrete unified gas kinetic scheme for multiscale heat transfer based on the phonon boltzmann transport equation

    Zhaoli Guo and Kun Xu. Discrete unified gas kinetic scheme for multiscale heat transfer based on the phonon boltzmann transport equation. International Journal of Heat and Mass Transfer, 102:944–958, 2016

    2016

  18. [26]

    A discrete unified gas kinetic scheme for phonon boltzmann trans- port equation accounting for phonon dispersion and polarization

    Xiao-Ping Luo and Hong-Liang Yi. A discrete unified gas kinetic scheme for phonon boltzmann trans- port equation accounting for phonon dispersion and polarization. International Journal of Heat and Mass Transfer, 114:970–980, 2017

    2017

  19. [27]

    Discrete unified gas kinetic scheme for multiscale heat transfer with arbitrary temperature difference

    Chuang Zhang and Zhaoli Guo. Discrete unified gas kinetic scheme for multiscale heat transfer with arbitrary temperature difference. International Journal of Heat and Mass Transfer, 134:1127–1136, 2019

    2019

  20. [28]

    Unified implicit kinetic scheme for steady multiscale heat transfer based on the phonon Boltzmann transport equation

    Chuang Zhang, Zhaoli Guo, and Songze Chen. Unified implicit kinetic scheme for steady multiscale heat transfer based on the phonon Boltzmann transport equation. Phys. Rev. E, 96:063311, Dec 2017

    2017

  21. [29]

    Acceler- ation strategy of source iteration method for the stationary phonon boltzmann transport equation

    Chuang Zhang, Samuel Huberman, Xinliang Song, Jin Zhao, Songze Chen, and Lei Wu. Acceler- ation strategy of source iteration method for the stationary phonon boltzmann transport equation. International Journal of Heat and Mass Transfer, 217:124715, 2023

    2023

  22. [30]

    Synthetic iterative scheme for thermal applications in hotspot systems with large temperature variance

    Chuang Zhang, Qin Lou, and Hong Liang. Synthetic iterative scheme for thermal applications in hotspot systems with large temperature variance. International Journal of Heat and Mass Transfer, 236:126374, 2025

    2025

  23. [31]

    Unified gas-kinetic wave-particle methods

    Yajun Zhu, Chang Liu, Chengwen Zhong, and Kun Xu. Unified gas-kinetic wave-particle methods. ii. multiscale simulation on unstructured mesh. Physics of Fluids, 31(6), 2019

    2019

  24. [32]

    Unified gas-kinetic wave-particle methods iv: multi-species gas mixture and plasma transport

    Chang Liu and Kun Xu. Unified gas-kinetic wave-particle methods iv: multi-species gas mixture and plasma transport. Advances in Aerodynamics, 3:1–31, 2021. 24

    2021

  25. [33]

    Unified gas-kinetic wave-particle method for multiscale flow simulation of partially ionized plasma

    Zhigang Pu and Kun Xu. Unified gas-kinetic wave-particle method for multiscale flow simulation of partially ionized plasma. Journal of Computational Physics, page 113918, 2025

    2025

  26. [34]

    Unified gas kinetic wave particle method for polydisperse gas-solid particle multiphase flow

    Xiaojian Yang, Wei Shyy, and Kun Xu. Unified gas kinetic wave particle method for polydisperse gas-solid particle multiphase flow. Journal of Fluid Mechanics, 983:A37, 2024

    2024

  27. [35]

    Unified gas-kinetic wave-particle method for frequency-dependent radiation transport equation

    Xiaojian Yang, Yajun Zhu, Chang Liu, and Kun Xu. Unified gas-kinetic wave-particle method for frequency-dependent radiation transport equation. Journal of Computational Physics, 522:113587, 2025

    2025

  28. [36]

    Wave-Particle Based Multiscale Modeling and Simulation of Non- equilibrium Turbulent Flows

    Xiaojian Yang and Kun Xu. Wave-Particle Based Multiscale Modeling and Simulation of Non- equilibrium Turbulent Flows. arXiv preprint arXiv:2503.07207, 2025

    2025 arXiv

  29. [37]

    Unified gas-kinetic wave-particle method for multi-scale phonon transport

    Hongyu Liu, Xiaojian Yang, Chuang Zhang, Xing Ji, and Kun Xu. Unified gas-kinetic wave-particle method for multi-scale phonon transport. arXiv preprint arXiv:2505.09297, 2025

    2025 arXiv

  30. [38]

    Implicit unified gas-kinetic scheme for steady state solutions in all flow regimes

    Yajun Zhu, Chengwen Zhong, and Kun Xu. Implicit unified gas-kinetic scheme for steady state solutions in all flow regimes. Journal of Computational Physics, 315:16–38, 2016

    2016

  31. [39]

    An implicit unified gas-kinetic wave– particle method for radiative transport process

    Chang Liu, Weiming Li, Yanli Wang, Peng Song, and Kun Xu. An implicit unified gas-kinetic wave– particle method for radiative transport process. Physics of Fluids, 35(11), 2023

    2023

  32. [40]

    An implicit unified gas-kinetic particle method with large time steps for gray radiation transport

    Yuan Hu, Chang Liu, Huayun Shen, Gang Xiao, and Jinghong Li. An implicit unified gas-kinetic particle method with large time steps for gray radiation transport. Physics of Fluids, 36(11), 2024

    2024

  33. [41]

    A model for collision processes in gases

    Prabhu Lal Bhatnagar, Eugene P Gross, and Max Krook. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Physical Review, 94(3):511, 1954

    1954

  34. [42]

    Radiative transport and wall temperature slip in an absorbing planar medium

    Max A Heaslet and Robert F Warming. Radiative transport and wall temperature slip in an absorbing planar medium. International Journal of Heat and Mass Transfer, 8(7):979–994, 1965

    1965

  35. [43]

    Majumdar

    A. Majumdar. Microscale heat conduction in dielectric thin films. J. Heat Transfer, 115(1):7–16, 1993. 25

    1993

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.