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arxiv: 2604.06153 · v1 · submitted 2026-04-07 · ❄️ cond-mat.mes-hall

Solving the Peierls-Boltzmann transport equation with matrix product states

Sangyeop Lee , Hirad Alipanah , Juan Jos\'e Mendoza-Arenas This is my paper

Pith reviewed 2026-05-10 18:30 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords Peierls-Boltzmann transport equationmatrix product statesphonon transporttensor networksfinite volume methodcrystalline silicon
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The pith

Matrix product states solve the Peierls-Boltzmann transport equation for silicon with high accuracy and sublinear cost

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

The paper establishes that matrix product states can be configured to represent the solution of the high-dimensional Peierls-Boltzmann transport equation so that correlations between tensors remain local. The configuration orders tensors by scattering events, uses a dimensionless form of the phonon distribution, and connects the real-space and modal-space chains at their coarsest grids in the center of the MPS. With this arrangement a density-matrix-renormalization-group-style solver applied to a finite-volume discretization reproduces reference solutions for crystalline silicon across ballistic, quasi-ballistic, and diffusive regimes even after truncation to a compression ratio of 10^{-3}. The resulting computational cost grows sublinearly with grid size and is roughly an order of magnitude lower than a conventional sparse-matrix finite-volume solve.

Core claim

An MPS representation of the solution to the discretized Peierls-Boltzmann equation, built with scattering-event ordering, dimensionless scaling, and optimal index ordering that joins coarsest real and modal grids at the center, allows truncation to a compression ratio of 10^{-3} while reproducing reference solutions with high fidelity for crystalline silicon in ballistic, quasi-ballistic, and diffusive regimes; the computational cost scales sublinearly with the number of grid points and yields roughly an order-of-magnitude reduction in runtime compared with the finite-volume method using sparse matrix operations.

What carries the argument

The matrix product state (MPS) of the phonon distribution function, ordered according to scattering events with real-space and modal-space chains joined at their coarsest indices in the center, which localizes inter-tensor correlations enough to permit aggressive low-rank compression.

If this is right

  • Accurate solutions are obtained for crystalline silicon in ballistic, quasi-ballistic, and diffusive transport regimes.
  • Computational cost scales sublinearly with the number of grid points in both real and modal spaces.
  • Truncation to a compression ratio of 10^{-3} reproduces reference solutions with high fidelity.
  • Roughly an order of magnitude reduction in computational time is achieved compared with the finite-volume method using sparse matrix operations.

Where Pith is reading between the lines

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

  • The same locality principle may allow tensor-network methods to be applied to other high-dimensional kinetic equations that share similar phase-space structure.
  • Finer grids than those tested here could become practical, revealing finer details of quasi-ballistic phonon transport.
  • The approach suggests a route to simulating phonon transport in complex device geometries where conventional methods become prohibitively expensive.

Load-bearing premise

That the chosen MPS index ordering based on scattering events together with dimensionless variables creates sufficiently local correlations to allow strong compression without destroying the accuracy of the transport solution.

What would settle it

A direct numerical comparison on the same finite-volume grid for silicon showing that the MPS solution truncated at compression ratio 10^{-3} deviates substantially from the full reference solution in the diffusive regime would falsify the claim of sufficient accuracy.

Figures

Figures reproduced from arXiv: 2604.06153 by Hirad Alipanah, Juan Jos\'e Mendoza-Arenas, Sangyeop Lee.

Figure 1
Figure 1. Figure 1: FIG. 1. A one-dimensional real-space domain considered in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. An MPS representation of the distribution function, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Tensor diagram of the DMRG-like method for solv [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Entanglement entropy of MPS constructed with [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Schmidt spectrum of MPS constructed with different [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Various MPS configurations. Blue and grey tensors [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison of various MPS configurations in terms of (a) entanglement entropy and (b) bond dimension ( [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. TNFVM results for ballistic, quasi-ballistic, and diffusive cases with different [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Computational cost with respect to the number of [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
read the original abstract

The Peierls-Boltzmann transport equation (PBE), which governs non-equilibrium phonon transport, suffers from the curse of dimensionality due to its high-dimensional phase space including both real and modal spaces. We explore the use of matrix product states (MPS) for numerical simulation of the PBE. We show that an MPS configuration based on scattering events combined with a dimensionless form of the solution can drastically increase the locality of correlations between tensors in the MPS representation, enhancing its effectiveness in dimension reduction. We further examine the effects of index ordering in an MPS and find that the highest locality is achieved when tensor chains associated with both real and modal spaces are connected from the coarsest grid to each other in the center of the MPS. Using this optimal configuration and a solver inspired by the density matrix renormalization group, we solve the PBE discretized by a finite volume method (FVM). The solution is obtained for crystalline silicon across ballistic, quasi-ballistic, and diffusive transport regimes. An MPS truncated to the compression ratio of $10^{-3}$ suffices to reproduce reference solutions with high fidelity. The computational cost scales sublinearly with the number of grid points in both real and modal spaces, achieving roughly an order of magnitude reduction in computational time compared to the FVM with sparse matrix operation.

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

1 major / 2 minor

Summary. The manuscript proposes solving the high-dimensional Peierls-Boltzmann transport equation (PBE) for phonon transport using matrix product states (MPS). It introduces an MPS configuration based on scattering events together with a dimensionless form of the solution, combined with an index ordering that connects coarsest grids from real and modal spaces at the MPS center. A DMRG-inspired solver is applied to the finite-volume-method (FVM) discretization, yielding solutions for crystalline silicon in ballistic, quasi-ballistic, and diffusive regimes. The paper reports that truncation at a compression ratio of 10^{-3} reproduces reference FVM solutions with high fidelity while the computational cost scales sublinearly with the number of grid points, achieving roughly an order-of-magnitude reduction in wall time relative to sparse-matrix FVM.

Significance. If the reported fidelity and scaling hold under the proposed configuration, the work demonstrates that tensor-network compression can mitigate the curse of dimensionality in phonon transport simulations. The explicit numerical support for sublinear scaling across transport regimes and the order-of-magnitude speedup constitute concrete evidence that appropriately chosen MPS representations can compress the discretized PBE solution space effectively.

major comments (1)
  1. [Results and MPS configuration sections] The central claim that the scattering-event MPS configuration plus dimensionless rescaling produces sufficiently rapid decay of correlations to permit 10^{-3} truncation across regimes (abstract and results) is load-bearing for the method's generality. The manuscript presents final accuracy and sublinear wall-time scaling but does not report singular-value spectra, entanglement entropy versus bond index, or bond-dimension growth versus grid size for the three regimes. Without these diagnostics it remains possible that the observed fidelity is regime-specific or arises from the operator representation rather than the claimed localization of the solution itself.
minor comments (2)
  1. [Abstract and Results] Quantitative error metrics (e.g., relative L2 or maximum deviation) used to define 'high fidelity' should be stated explicitly rather than left as a qualitative descriptor.
  2. [Methods] Implementation details of the DMRG-inspired solver (sweeping schedule, convergence criteria, handling of the FVM-to-MPS mapping) are only sketched; expanding them would aid reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for the detailed and insightful report. The suggestion to include additional diagnostics on the MPS structure is valuable, and we will revise the manuscript accordingly to provide stronger support for our claims.

read point-by-point responses

  1. Referee: [Results and MPS configuration sections] The central claim that the scattering-event MPS configuration plus dimensionless rescaling produces sufficiently rapid decay of correlations to permit 10^{-3} truncation across regimes (abstract and results) is load-bearing for the method's generality. The manuscript presents final accuracy and sublinear wall-time scaling but does not report singular-value spectra, entanglement entropy versus bond index, or bond-dimension growth versus grid size for the three regimes. Without these diagnostics it remains possible that the observed fidelity is regime-specific or arises from the operator representation rather than the claimed localization of the solution itself.

    Authors: We agree that these diagnostics would provide direct evidence for the rapid decay of correlations enabled by the scattering-event configuration and dimensionless rescaling. The current manuscript demonstrates the effectiveness through the high fidelity of the 10^{-3} truncated MPS solutions matching the reference FVM results across ballistic, quasi-ballistic, and diffusive regimes, along with sublinear scaling in grid size. However, to address the possibility of regime-specific behavior or operator-driven effects, we will include in the revised manuscript the singular-value spectra for the MPS in each regime, entanglement entropy as a function of bond index, and the required bond dimension as a function of grid size. These additions will confirm the locality of the solution and support the generality of the approach. revision: yes

Circularity Check

0 steps flagged

No circularity in numerical MPS solver for discretized PBE

full rationale

The paper presents a computational method: finite-volume discretization of the PBE followed by an MPS representation whose index ordering and scattering-event grouping, together with a dimensionless rescaling, are chosen to improve tensor locality. The central results (high-fidelity reproduction of independent FVM reference solutions across ballistic-to-diffusive regimes, sublinear scaling, and 10^{-3} truncation sufficiency) are obtained by running the solver and comparing outputs; they are not quantities defined in terms of themselves, fitted parameters renamed as predictions, or self-citation chains. No load-bearing step reduces by construction to an input, and the manuscript relies on external reference solutions and standard DMRG-inspired techniques rather than tautological derivations.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard numerical techniques for tensor networks and transport equations plus one practical truncation choice.

free parameters (1)
  • compression ratio = 10^{-3}
    Truncation threshold of 10^{-3} chosen to achieve high-fidelity match to reference solutions.
axioms (2)
  • domain assumption The Peierls-Boltzmann transport equation can be discretized by finite volume method on combined real and modal grids.
    Foundation for the numerical representation used throughout.
  • domain assumption The chosen index ordering and dimensionless form produce sufficiently local correlations for low-bond-dimension MPS to be accurate.
    Core premise that enables the dimension reduction and sublinear scaling.

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Reference graph

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