pith. sign in

arxiv: 2604.10536 · v2 · submitted 2026-04-12 · ❄️ cond-mat.stat-mech · physics.class-ph

Heat Conduction in Momentum-Conserving Fluids: From quasi-2D to 3D systems

Rongxiang Luo , Jiaqi Wen , Juncheng Guo This is my paper

Pith reviewed 2026-05-10 15:53 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.class-ph
keywords heat conductionmomentum-conserving fluidsdimensional crossoverhydrodynamic regimethermal conductivitymolecular dynamicsmesoscopic fluidsautocorrelation function
0
0 comments X

The pith

Momentum-conserving fluids show heat conductivity diverging logarithmically in quasi-2D but remaining finite in 3D.

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

The paper simulates a mesoscopic fluid that conserves momentum to track how heat flows as the system changes from nearly flat to fully three-dimensional. It separates the behavior into ballistic, kinetic, and hydrodynamic regimes and shows that only the last one produces a clear difference by dimension. In the hydrodynamic regime the simulations find that conductivity grows with the logarithm of system size in quasi-two dimensions while it saturates in three dimensions, exactly as scaling theory predicts. This crossover matters because it explains why heat transport can look anomalous in thin layers yet normal in bulk fluids.

Core claim

Nonequilibrium and equilibrium molecular dynamics simulations with multiparticle collision dynamics identify three transport regimes across quasi-2D to 3D systems. In the hydrodynamic regime the quasi-2D case exhibits logarithmically divergent thermal conductivity together with a heat-current autocorrelation that decays as 1/t, while the 3D case shows finite conductivity and an autocorrelation decaying as t to the minus 3/2. These scalings quantitatively match theoretical predictions and establish a dimensional crossover from 2D-like anomalous transport to 3D Fourier behavior.

What carries the argument

The hydrodynamic regime scalings of thermal conductivity with system size and the associated decay of the heat current autocorrelation function, which differ by dimensionality in momentum-conserving fluids.

If this is right

  • Quasi-2D momentum-conserving systems display anomalous heat transport driven by long-lived hydrodynamic correlations.
  • Three-dimensional systems recover normal Fourier conduction with size-independent conductivity.
  • The same model reproduces both the ballistic and kinetic regimes before the hydrodynamic crossover appears.
  • These dimensional differences provide guidance for thermal design in micro- and nanoscale devices that span thin layers to bulk.

Where Pith is reading between the lines

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

  • Real colloidal or granular fluids that conserve momentum could be used to test the predicted logarithmic growth directly.
  • The crossover may help explain why some thin-film materials show size-dependent heat flow while thicker samples do not.
  • Extending the same simulation approach to other conserved quantities could reveal analogous dimensional transitions in momentum or particle transport.

Load-bearing premise

The multiparticle collision dynamics model and the chosen system sizes are large enough to reach the true hydrodynamic regime without finite-size artifacts or model biases that would change the observed scalings.

What would settle it

A measurement showing that thermal conductivity remains finite with increasing system size in a quasi-2D momentum-conserving fluid, or diverges in a three-dimensional one, would contradict the claimed crossover.

Figures

Figures reproduced from arXiv: 2604.10536 by Jiaqi Wen, Juncheng Guo, Rongxiang Luo.

Figure 1
Figure 1. Figure 1: FIG. 1. Dependence of the thermal current [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

Using nonequilibrium and equilibrium molecular dynamics simulations, we investigate heat conduction in a momentum-conserving mesoscopic fluid modeled by multiparticle collision dynamics. Across quasi-two-dimensional (q-2D) to three-dimensional (3D) systems, we identify three distinct transport regimes: (i) a \emph{ballistic regime}, where thermal conductivity scales linearly with system size ($\kappa \sim L$) and the total heat current autocorrelation function $C(t)$ remains constant; (ii)~a \emph{kinetic regime}, characterized by size-independent $\kappa$ and exponentially decaying $C(t)$, demonstrating that normal heat conduction dominated by kinetic effects is far more ubiquitous than previously observed in 1D systems; and (iii)~a \emph{hydrodynamic regime}, where the q-2D system exhibits logarithmically divergent conductivity ($ \kappa \sim \ln L $ ) with $ C(t) \sim t^{-1} $ , while the 3D system displays finite $ \kappa $ and $ C(t) \sim t^{-3/2} $. Our results, observed in the hydrodynamic regime, quantitatively validate the scaling predictions for heat transport and reveal a clear dimensional crossover -- from 2D-like anomalous transport to 3D Fourier behavior. These results lay a foundation for understanding thermal transport in q-2D to 3D systems and have practical implications for the design of micro- and nanoscale thermal devices.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript uses nonequilibrium and equilibrium molecular dynamics simulations with the multiparticle collision dynamics (MPCD) model to study heat conduction in momentum-conserving fluids, spanning quasi-2D to 3D systems. It identifies three regimes—ballistic (κ ∼ L, constant C(t)), kinetic (size-independent κ, exponential C(t) decay), and hydrodynamic (κ ∼ ln L with C(t) ∼ t^{-1} in q-2D; finite κ with C(t) ∼ t^{-3/2} in 3D)—and claims that the hydrodynamic-regime results quantitatively validate theoretical scaling predictions while demonstrating a dimensional crossover from anomalous to normal (Fourier) transport.

Significance. If the hydrodynamic regime is rigorously established and the reported scalings hold without model-specific or finite-size artifacts, the work would supply useful numerical evidence supporting analytic predictions for low-dimensional anomalous heat transport and the 2D-to-3D crossover. The MPCD approach enables larger-scale simulations than fully atomistic MD, which is a methodological strength for accessing hydrodynamic behavior; the independent simulation tests of prior predictions (rather than self-referential derivations) add credibility.

major comments (2)
  1. [Abstract and hydrodynamic-regime identification] Abstract and hydrodynamic-regime results: the central claim that the simulations 'quantitatively validate' the scalings (κ ∼ ln L and C(t) ∼ t^{-1} in q-2D; finite κ and C(t) ∼ t^{-3/2} in 3D) rests on the unverified assumption that the largest simulated L lies deep inside the true hydrodynamic limit. No explicit checks—such as data collapse versus L/ℓ_mfp, variation of MPCD collision frequency or mean-free path, or demonstration that viscous lengths are exceeded—are described, leaving open the possibility that the reported behaviors remain contaminated by kinetic-regime remnants or model cutoffs.
  2. [Methods and regime-classification sections] Methods and regime-classification sections: the criteria used to demarcate the kinetic-to-hydrodynamic crossover (e.g., system-size thresholds, parameter independence tests, or goodness-of-fit metrics for the logarithmic versus constant κ) are not supplied with sufficient quantitative detail, including error bars, fitting ranges, or robustness against MPCD parameter changes; this directly undermines the load-bearing assertion that the observed dimensional crossover is asymptotic rather than an intermediate-window artifact.
minor comments (2)
  1. [Abstract] The abstract asserts 'quantitative validation' without referencing the specific fitting procedures or statistical controls that appear later; a brief cross-reference would improve clarity.
  2. [Figures] Figure captions and axis labels should explicitly indicate the MPCD parameters (collision frequency, particle density) and the range of L values used for each regime to allow readers to assess hydrodynamic convergence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We appreciate the positive assessment of the methodological strengths of the MPCD approach and the independent validation of theoretical predictions. We address each major comment below and will incorporate revisions to strengthen the presentation of the hydrodynamic regime and regime classification.

read point-by-point responses

  1. Referee: [Abstract and hydrodynamic-regime identification] Abstract and hydrodynamic-regime results: the central claim that the simulations 'quantitatively validate' the scalings (κ ∼ ln L and C(t) ∼ t^{-1} in q-2D; finite κ and C(t) ∼ t^{-3/2} in 3D) rests on the unverified assumption that the largest simulated L lies deep inside the true hydrodynamic limit. No explicit checks—such as data collapse versus L/ℓ_mfp, variation of MPCD collision frequency or mean-free path, or demonstration that viscous lengths are exceeded—are described, leaving open the possibility that the reported behaviors remain contaminated by kinetic-regime remnants or model cutoffs.

    Authors: We agree that explicit verification that the largest systems lie deep within the hydrodynamic regime is essential to support the quantitative validation claim. Our current identification is based on the emergence of the expected theoretical scalings (logarithmic in q-2D, finite in 3D) together with the corresponding current autocorrelation decays, which are inconsistent with kinetic-regime behavior. However, we acknowledge that additional checks would eliminate any ambiguity regarding possible crossover artifacts. In the revised manuscript we will add (i) conductivity data plotted against L/ℓ_mfp to demonstrate collapse or saturation in the hydrodynamic window, (ii) results obtained by varying the MPCD collision frequency (and thus the mean free path) while keeping other parameters fixed, and (iii) estimates showing that the simulated lengths exceed the relevant viscous hydrodynamic lengths. These additions will directly address the concern of kinetic contamination. revision: yes

  2. Referee: [Methods and regime-classification sections] Methods and regime-classification sections: the criteria used to demarcate the kinetic-to-hydrodynamic crossover (e.g., system-size thresholds, parameter independence tests, or goodness-of-fit metrics for the logarithmic versus constant κ) are not supplied with sufficient quantitative detail, including error bars, fitting ranges, or robustness against MPCD parameter changes; this directly undermines the load-bearing assertion that the observed dimensional crossover is asymptotic rather than an intermediate-window artifact.

    Authors: We accept that the demarcation criteria require more quantitative detail. In the revised manuscript we will expand the Methods and Results sections to specify (i) the precise system-size thresholds at which each regime is identified, (ii) error bars on all conductivity values and on the fitted slopes (logarithmic or constant), (iii) the exact fitting ranges used to distinguish κ ∼ ln L from size-independent κ, and (iv) additional robustness tests performed by changing the MPCD collision rule parameters. These quantitative elements will make the regime classification reproducible and will strengthen the evidence that the observed 2D-to-3D crossover is not an intermediate-size artifact. revision: yes

Circularity Check

0 steps flagged

No circularity: simulations test external hydrodynamic predictions

full rationale

The manuscript reports direct nonequilibrium and equilibrium MPCD simulations that measure conductivity scalings (κ ∼ L ballistic, size-independent kinetic, κ ∼ ln L or finite in hydrodynamic) and C(t) decays. These are presented as numerical observations that quantitatively match independent theoretical scaling predictions for momentum-conserving fluids, rather than being derived from the simulation outputs themselves or from self-citations. No equations reduce fitted parameters to predictions by construction, no uniqueness theorems are imported from the authors' prior work, and regime identification rests on observed behaviors rather than self-referential definitions. The work is self-contained against external benchmarks and contains no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are introduced or detailed in the abstract; the study relies on standard molecular dynamics techniques applied to an established mesoscopic fluid model.

pith-pipeline@v0.9.0 · 5568 in / 1104 out tokens · 24348 ms · 2026-05-10T15:53:21.035533+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

63 extracted references · 63 canonical work pages

  1. [1]

    1), the scaling j ∼ L− 1 (or size-independent κ) is observed only in the 3D system. As the system ap- proaches the q-2D case, the thermal current ultimately 3 /s61 /s32 /s73 /s110/s116 /s101 /s103 /s114 /s97 /s98/s108 /s101 /s32/s99 /s97 /s115 /s101 /s49/s48 /s49 /s49/s48 /s50 /s49/s48 /s51 /s49/s48 /s45 /s50 /s49/s48 /s45 /s49 /s49/s48 /s48 /s61 /s32/s48...

    work page

  2. [2]

    Lepri, ed., From Statistical Physics to Nanoscale Heat Transfer, Lecture Notes in Physics, Vol

    S. Lepri, ed., From Statistical Physics to Nanoscale Heat Transfer, Lecture Notes in Physics, Vol. 921 (Springer, New York, 2016)

    work page 2016

  3. [3]

    Zhang, Y

    Z. Zhang, Y. Ouyang, Y. Cheng, J. Chen, N. Li, and G. Zhang, Size-dependent phononic thermal transport in low-dimensional nanomaterials, Physics Reports 860, 1 (2020)

    work page 2020

  4. [4]

    Benenti, G

    G. Benenti, G. Casati, K. Saito, and R. Whitney, Funda- mental aspects of steady-state conversion of heat to work at the nanoscale, Physics Reports 694, 1 (2017)

    work page 2017

  5. [5]

    N. Li, J. Ren, L. Wang, G. Zhang, P. H¨ anggi, and B. Li, Colloquium: Phononics: Manipulating heat flow with electronic analogs and beyond, Rev. Mod. Phys. 84, 1045 (2012)

    work page 2012

  6. [6]

    X. Gu, Y. Wei, X. Yin, B. Li, and R. Yang, Colloquium: Phononic thermal properties of two-dimensional materi- als, Rev. Mod. Phys. 90, 041002 (2018)

    work page 2018

  7. [7]

    J. Chen, X. Xu, J. Zhou, and B. Li, Interfacial thermal resistance: Past, present, and future, Rev. Mod. Phys. 94, 025002 (2022)

    work page 2022

  8. [8]

    Lepri, R

    S. Lepri, R. Livi, and A. Politi, Heat conduction in chain s of nonlinear oscillators, Phys. Rev. Lett. 78, 1896 (1997)

    work page 1997

  9. [9]

    Dhar, Heat conduction in a one-dimensional gas of elastically colliding particles of unequal masses, Phys

    A. Dhar, Heat conduction in a one-dimensional gas of elastically colliding particles of unequal masses, Phys. Rev. Lett. 86, 3554 (2001)

    work page 2001

  10. [10]

    Narayan and S

    O. Narayan and S. Ramaswamy, Anomalous heat conduc- tion in one-dimensional momentum-conserving systems, Phys. Rev. Lett. 89, 200601 (2002)

    work page 2002

  11. [11]

    Casati and T

    G. Casati and T. c. v. Prosen, Anomalous heat conduc- tion in a one-dimensional ideal gas, Phys. Rev. E 67, 5 015203 (2003)

    work page 2003

  12. [12]

    Li and J

    B. Li and J. Wang, Anomalous heat conduction and anomalous diffusion in one-dimensional systems, Phys. Rev. Lett. 91, 044301 (2003)

    work page 2003

  13. [13]

    Denisov, J

    S. Denisov, J. Klafter, and M. Urbakh, Dynamical heat channels, Phys. Rev. Lett. 91, 194301 (2003)

    work page 2003

  14. [14]

    Cipriani, S

    P. Cipriani, S. Denisov, and A. Politi, From anomalous energy diffusion to levy walks and heat conductivity in one-dimensional systems, Phys. Rev. Lett. 94, 244301 (2005)

    work page 2005

  15. [15]

    Delfini, S

    L. Delfini, S. Lepri, R. Livi, and A. Politi, Anoma- lous kinetics and transport from 1d self-consistent mode- coupling theory, Journal of Statistical Mechanics: Theory and Experiment 2007, P02007 (2007)

    work page 2007

  16. [16]

    Wang and T

    L. Wang and T. Wang, Power-law divergent heat conduc- tivity in one-dimensional momentum-conserving nonlin- ear lattices, Europhysics Letters 93, 54002 (2011)

    work page 2011

  17. [17]

    van Beijeren, Exact results for anomalous transport in one-dimensional hamiltonian systems, Phys

    H. van Beijeren, Exact results for anomalous transport in one-dimensional hamiltonian systems, Phys. Rev. Lett. 108, 180601 (2012)

    work page 2012

  18. [18]

    Spohn, Nonlinear fluctuating hydrodynamics for an- harmonic chains, Journal of Statistical Physics 154, 1191 (2014)

    H. Spohn, Nonlinear fluctuating hydrodynamics for an- harmonic chains, Journal of Statistical Physics 154, 1191 (2014)

    work page 2014

  19. [19]

    P. I. Hurtado and P. L. Garrido, A violation of universal - ity in anomalous fourier’s law, Scientific Reports 6, 38823 (2016)

    work page 2016

  20. [20]

    Hai-Bin and L

    L. Hai-Bin and L. Zhen, Anomalous energy diffusion and heat conduction in one-dimensional system, Chinese Physics B 19, 054401 (2010)

    work page 2010

  21. [21]

    Lepri, R

    S. Lepri, R. Livi, and A. Politi, On the anomalous ther- mal conductivity of one-dimensional lattices, Europhysics Letters 43, 271 (1998)

    work page 1998

  22. [22]

    Basile, C

    G. Basile, C. Bernardin, and S. Olla, Momentum con- serving model with anomalous thermal conductivity in low dimensional systems, Phys. Rev. Lett. 96, 204303 (2006)

    work page 2006

  23. [23]

    Luo and S

    R. Luo and S. Lepri, Multi-particle-collision simulat ion of heat transfer in low-dimensional fluids, Entropy 27, 10.3390/e27050455 (2025)

    work page doi:10.3390/e27050455 2025

  24. [24]

    L. Wang, D. He, and B. Hu, Heat conduction in a three- dimensional momentum-conserving anharmonic lattice, Phys. Rev. Lett. 105, 160601 (2010)

    work page 2010

  25. [25]

    Saito and A

    K. Saito and A. Dhar, Heat conduction in a three dimen- sional anharmonic crystal, Phys. Rev. Lett. 104, 040601 (2010)

    work page 2010

  26. [26]

    L. Yang, P. Grassberger, and B. Hu, Dimensional crossover of heat conduction in low dimensions, Phys. Rev. E 74, 062101 (2006)

    work page 2006

  27. [27]

    Luo, Heat conduction in two-dimensional momentum- conserving and -nonconserving gases, Phys

    R. Luo, Heat conduction in two-dimensional momentum- conserving and -nonconserving gases, Phys. Rev. E 102, 052104 (2020)

    work page 2020

  28. [28]

    Hisamoto, Y

    K. Hisamoto, Y. Kobayashi, T. Ikeda, and M. Yamakawa, Dimensional crossover of thermal transport in oligomer aqueous solutions: From three-dimensional to quasi-one- dimensional behavior, Phys. Rev. E 112, 055414 (2025)

    work page 2025

  29. [29]

    L. Dong, Q. Xi, D. Chen, J. Guo, T. Nakayama, Y. Li, Z. Liang, J. Zhou, X. Xu, and B. Li, Dimensional crossover of heat conduction in amor- phous polyimide nanofibers, National Science Review 5, 500 (2018) , https://academic.oup.com/nsr/article- pdf/5/4/500/31568240/nwy004.pdf

    work page 2018

  30. [30]

    Giardin` a, R

    C. Giardin` a, R. Livi, A. Politi, and M. Vassalli, Finit e thermal conductivity in 1d lattices, Phys. Rev. Lett. 84, 2144 (2000)

    work page 2000

  31. [31]

    O. V. Gendelman and A. V. Savin, Normal heat conduc- tivity of the one-dimensional lattice with periodic poten- tial of nearest-neighbor interaction, Phys. Rev. Lett. 84, 2381 (2000)

    work page 2000

  32. [32]

    Zhong, Y

    Y. Zhong, Y. Zhang, J. Wang, and H. Zhao, Normal heat conduction in one-dimensional momentum conserv- ing lattices with asymmetric interactions, Phys. Rev. E 85, 060102 (2012)

    work page 2012

  33. [33]

    Z. Yi, Z. Yong, W. Jiao, and Z. Hong, Normal thermal conduction in lattice models with asymmetric harmonic interparticle interactions, Chinese Physics B 22, 070505 (2013)

    work page 2013

  34. [34]

    S. Chen, Y. Zhang, J. Wang, and H. Zhao, Key role of asymmetric interactions in low-dimensional heat trans- port, Journal of Statistical Mechanics: Theory and Ex- periment 2016, 033205 (2016)

    work page 2016

  35. [35]

    A. V. Savin and Y. A. Kosevich, Thermal conductivity of molecular chains with asymmetric potentials of pair interactions, Phys. Rev. E 89, 032102 (2014)

    work page 2014

  36. [36]

    S. Chen, J. Wang, G. Casati, and G. Benenti, Noninte- grability and the fourier heat conduction law, Phys. Rev. E 90, 032134 (2014)

    work page 2014

  37. [37]

    Chen, Q.-Y

    H. Chen, Q.-Y. Zhang, J. Wang, N. Li, and J. Chen, Normal energy and stretch diffusion in a one-dimensional momentum conserving lattice with nonlinear bounded ki- netic energy, Chinese Physics B 34, 094401 (2025)

    work page 2025

  38. [38]

    Zhao and W.-g

    H. Zhao and W.-g. Wang, Fourier heat conduction as a strong kinetic effect in one-dimensional hard-core gases, Phys. Rev. E 97, 010103 (2018)

    work page 2018

  39. [39]

    W. Fu, Z. Wang, Y. Wang, Y. Zhang, and H. Zhao, From near-integrable to far-from-integrable: A unified picture of thermalization and heat transport, Phys. Rev. Res. 8, 013314 (2026)

    work page 2026

  40. [40]

    L. Wang, B. Hu, and B. Li, Validity of fourier’s law in one-dimensional momentum-conserving lattices with asymmetric interparticle interactions, Phys. Rev. E 88, 052112 (2013)

    work page 2013

  41. [41]

    S. G. Das, A. Dhar, and O. Narayan, Heat conduction in the fermi-pasta-ulam chain, Journal of Statistical Physics 154, 204 (2014)

    work page 2014

  42. [42]

    Miron, J

    A. Miron, J. Cividini, A. Kundu, and D. Mukamel, Derivation of fluctuating hydrodynamics and crossover from diffusive to anomalous transport in a hard-particle gas, Phys. Rev. E 99, 012124 (2019)

    work page 2019

  43. [43]

    Lepri, R

    S. Lepri, R. Livi, and A. Politi, Too close to inte- grable: Crossover from normal to anomalous heat dif- fusion, Phys. Rev. Lett. 125, 040604 (2020)

    work page 2020

  44. [44]

    Malevanets and R

    A. Malevanets and R. Kapral, Mesoscopic model for sol- vent dynamics, The Journal of Chemical Physics 110, 8605 (1999)

    work page 1999

  45. [45]

    Gompper, T

    G. Gompper, T. Ihle, D. M. Kroll, and R. G. Win- kler, Multi-particle collision dynamics: A particle-base d mesoscale simulation approach to the hydrodynamics of complex fluids, in Advanced Computer Simulation Ap- proaches for Soft Matter Sciences III , edited by C. Holm and K. Kremer (Springer Berlin Heidelberg, Berlin, Hei- delberg, 2009) pp. 1–87

    work page 2009

  46. [46]

    J. T. Padding and A. A. Louis, Hydrodynamic inter- actions and brownian forces in colloidal suspensions: Coarse-graining over time and length scales, Phys. Rev. E 74, 031402 (2006)

    work page 2006

  47. [47]

    Belushkin, R

    M. Belushkin, R. Livi, and G. Foffi, Hydrodynamics and the fluctuation theorem, Phys. Rev. Lett. 106, 210601 (2011). 6

    work page 2011

  48. [48]

    Benenti, G

    G. Benenti, G. Casati, and C. Mej ´ ıa-Monasterio, Ther- moelectric efficiency in momentum-conserving systems, New Journal of Physics 16, 015014 (2014)

    work page 2014

  49. [49]

    R. Luo, G. Benenti, G. Casati, and J. Wang, Thermo- dynamic bound on heat-to-power conversion, Phys. Rev. Lett. 121, 080602 (2018)

    work page 2018

  50. [50]

    R. Luo, G. Benenti, G. Casati, and J. Wang, Onsager reciprocal relations with broken time-reversal symmetry, Phys. Rev. Res. 2, 022009 (2020)

    work page 2020

  51. [51]

    R. Luo, Q. Zhang, G. Lin, and S. Lepri, Heat conduc- tion in low-dimensional electron gases without and with a magnetic field, Phys. Rev. E 111, 014116 (2025)

    work page 2025

  52. [52]

    J. L. Lebowitz and H. Spohn, Transport properties of the Lorentz gas: Fourier’s law, Journal of Statistical Physics 19, 633 (1978)

    work page 1978

  53. [53]

    Tehver, F

    R. Tehver, F. Toigo, J. Koplik, and J. R. Banavar, Ther- mal walls in computer simulations, Phys. Rev. E 57, R17 (1998)

    work page 1998

  54. [54]

    Ihle and D

    T. Ihle and D. M. Kroll, Stochastic rotation dynamics: A galilean-invariant mesoscopic model for fluid flow, Phys. Rev. E 63, 020201 (2001)

    work page 2001

  55. [55]

    Luo, Negative differential thermal resistance in one - dimensional hard-point gas models, Phys

    R. Luo, Negative differential thermal resistance in one - dimensional hard-point gas models, Phys. Rev. E 99, 032138 (2019)

    work page 2019

  56. [56]

    R. Luo, J. Guo, J. Zhang, and H. Yang, Negative differ- ential thermal resistance of fluids induced by heat baths, Phys. Rev. Res. 4, 043184 (2022)

    work page 2022

  57. [57]

    R. Luo, L. Huang, and S. Lepri, Heat conduction in a three-dimensional momentum-conserving fluid, Phys. Rev. E 103, L050102 (2021)

    work page 2021

  58. [58]

    R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics , 2nd ed., Springer Series in Solid-State Sciences, Vol. 31 (Springer-Verlag, Berlin, 1991)

    work page 1991

  59. [59]

    Lepri, R

    S. Lepri, R. Livi, and A. Politi, Thermal conduction in classical low-dimensional lattices, Physics Reports 377, 1 (2003)

    work page 2003

  60. [60]

    Dhar, Heat transport in low-dimensional systems, Ad- vances in Physics 57, 457 (2008)

    A. Dhar, Heat transport in low-dimensional systems, Ad- vances in Physics 57, 457 (2008)

    work page 2008

  61. [61]

    Zhao, Identifying diffusion processes in one- dimensional lattices in thermal equilibrium, Phys

    H. Zhao, Identifying diffusion processes in one- dimensional lattices in thermal equilibrium, Phys. Rev. Lett. 96, 140602 (2006)

    work page 2006

  62. [62]

    S. Chen, Y. Zhang, J. Wang, and H. Zhao, Finite-size effects on current correlation functions, Phys. Rev. E 89, 022111 (2014)

    work page 2014

  63. [63]

    Lepri, R

    S. Lepri, R. Livi, and A. Politi, Anomalous trans- port in the fermi-pasta-ulam-tsingou model: a review and open problems, (2026), arXiv:2602.15512 [cond- mat.stat-mech]

    work page arXiv 2026