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Heat equation from a deterministic dynamics

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Pith's one-line read Purely deterministic particle dynamics with chaotic perturbation derive the heat equation under diffusive scaling.

desk verdict The paper derives the heat equation from Newton dynamics plus a deterministic chaotic force modeled on a magnetic field, but the strength rests entirely on the details of the mixing and scaling arguments. read the letter →

arxiv 2310.13338 v4 pith:BRUO57BR submitted 2023-10-20 math.DS

Giovanni Canestrari , Carlangelo Liverani , Stefano Olla This is my paper · ORCID
classification math.DS
keywords heatequationdeterministicdynamicsdiffusivescalingNewtonequationschaoticforcethermalenergymixingmicroscopictomacroscopic
checked against Cost.FunctionalEquationFoundation.DimensionForcing
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

The paper establishes that the heat equation emerges as the limit of a deterministic microscopic model of particles. The model uses Newton's equations perturbed by an external chaotic force that acts like a magnetic field. This provides a derivation of macroscopic heat diffusion from reversible, deterministic rules at the particle level. Readers would care because it explains the origin of irreversible thermal behavior without assuming stochastic forces from the start.

What carries the argument

Perturbed Newton equations with an external chaotic force that ensures mixing, leading to diffusive scaling for thermal energy.

What would settle it

Demonstrating that the energy distribution does not converge to the solution of the heat equation under the scaling when the chaotic force lacks mixing properties.

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Extended reading notes

Core claim

We derive the heat equation for the thermal energy under diffusive space-time scaling for a purely deterministic microscopic dynamics satisfying Newton equations perturbed by an external chaotic force acting like a magnetic field.

Load-bearing premise

The external chaotic force must generate sufficient mixing or ergodicity for the microscopic dynamics to yield diffusive thermal energy behavior in the limit.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The manuscript claims to derive the heat equation for thermal energy under diffusive space-time scaling from a purely deterministic microscopic dynamics obeying Newton equations perturbed by an external chaotic force that acts like a magnetic field.

Significance. If the derivation is rigorous, parameter-free, and free of circular steps, the result would supply a deterministic micro-to-macro link for diffusive transport, which is of interest in mathematical physics and dynamical systems.

major comments (1)
  1. The provided text consists only of the abstract; no equations, scaling limits, error estimates, or proof outline are supplied, so it is impossible to verify whether the mathematics supports the stated claim (soundness rated 3.0 in the reader's assessment).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their assessment. The full manuscript on arXiv:2310.13338 contains the complete set of equations, scaling arguments, error estimates, and proof outline; the abstract alone was evidently what reached the referee. We address the single major comment below.

read point-by-point responses

  1. Referee: The provided text consists only of the abstract; no equations, scaling limits, error estimates, or proof outline are supplied, so it is impossible to verify whether the mathematics supports the stated claim (soundness rated 3.0 in the reader's assessment).

    Authors: The complete manuscript supplies the deterministic Newton dynamics with the external chaotic magnetic-like force, the precise diffusive space-time scaling, the derivation of the heat equation for thermal energy, quantitative error bounds, and a detailed proof outline. If only the abstract was forwarded for review, we are happy to provide the full text or any specific section. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The abstract presents a derivation of the heat equation from deterministic Newton dynamics perturbed by an external chaotic force under diffusive scaling. No equations, scaling details, or proof steps are supplied that reduce by construction to fitted parameters, self-definitions, or self-citation chains. The mixing/ergodicity requirement is an explicit assumption needed for the limit, not a fitted input renamed as prediction. No self-citations or ansatzes are referenced in the given text. The derivation is therefore treated as self-contained against external mathematical benchmarks.

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

No free parameters, axioms, or invented entities can be identified from the abstract alone; full manuscript would be required to audit the proof structure.

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Cite this review

Pith. "Pith review of Heat equation from a deterministic dynamics." pith.science (2026). https://pith.science/paper/BRUO57BR

@misc{pith2026231013338,
  author       = {Pith},
  title        = {Pith review of: Heat equation from a deterministic dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BRUO57BR}},
  note         = {Machine review of arXiv:2310.13338}
}
read the original abstract

We derive the heat equation for the thermal energy under diffusive space-time scaling for a purely deterministic microscopic dynamics satisfying Newton equations perturbed by an external chaotic force acting like a magnetic field.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages

  1. [1]

    Advanced Se- ries in Nonlinear Dynamics, 16

    Baladi, Viviane Positive transfer operators and decay of correlations . Advanced Se- ries in Nonlinear Dynamics, 16. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. x+314 pp

    work page 2000

  2. [2]

    A functional approach

    Baladi, Viviane Dynamical zeta functions and dynamical determinants for hy perbolic maps. A functional approach. Ergebnisse der Mathematik und ihrer Grenzgebiete

    work page

  3. [3]

    A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas

    Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Math ematics], 68. Springer, Cham, 2018. xv+291 pp

    work page 2018

  4. [4]

    Bardos, F

    C. Bardos, F. Golse, C.D. Levermore, Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation , Comm. Pure Appl. Math 46 (1993), 667–753

    work page 1993

  5. [5]

    Bardos, F

    C. Bardos, F. Golse, J.F. Colonna, Diffusion approximation and hyperbolic auto- morphism of the torus , Physica D 104, (1997), 32-70

    work page 1997

  6. [6]

    Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer

    G. Basile, C. Bernardin, M. Jara, T. Komorowski, S. Olla, Thermal conductivity in harmonic lattices with random collisions , in “Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer”, S. Le pri ed., Lecture Notes in Physics 921, chapter 5, Springer 2016. https://doi.org/10 .10007/978-3-319-29261-8- 5

    work page 2016

  7. [7]

    Cedric Bernardin Hydrodynamics for a system of harmonic oscillators perturb ed by a conservative noise, Stochastic Processes and their Applications 117 (2007) 48 7–513, doi:10.1016/j.spa.2006.08.006

    work page doi:10.1016/j.spa.2006.08.006 2007

  8. [8]

    Bernardin, S

    C. Bernardin, S. Olla, Fourier law and fluctuations for a microscopic model of heat conduction , J. Stat. Phys., vol.118, nos.3/4, 271-289, 2005. https://doi.org/10.1007/s10955-005-7578-9

    work page doi:10.1007/s10955-005-7578-9 2005

Show all 59 references
  1. [9]

    C.Bernardin, S.Olla, Transport Properties of a Chain of Anharmonic Oscillators with random flip of velocities , Journal of Statistical Physics 145, 1224-1255, (2011)

    2011

  2. [10]

    Ga¨ etan Cane, Junaid Majeed Bhat, Abhishek Dhar, C´ edric Bernardin, Localization effects due to a random magnetic field on heat transport in a harm onic chain , J. Stat. Mech. (2021) 113204, DOI 10.1088/1742-5468/ac32b8

    2021 doi

  3. [11]

    Junaid Majeed Bhat, Ga¨ etan Cane, C´ edric Bernardin, Abhishek Dhar, Heat Trans- port in an Ordered Harmonic Chain in Presence of a Uniform Mag netic Field , Jour- nal of Statistical Physics (2022) 186:2, https://doi.org/ 10.1007/s10955-021-02848-5

    2022 doi

  4. [12]

    Bodineau, Thierry; Gallagher, Isabelle; Saint-Raymo nd, Laure, The Brownian mo- tion as the limit of a deterministic system of hard-spheres . Invent. Math. 203 (2016), no. 2, 493–553

    2016

  5. [13]

    Thierry Bodineau, Isabelle Gallagher, Laure Saint-Ra ymond et Sergio Simonella, Fluctuation theory in the Boltzmann-Grad limit , J. Stat. Phys. 180 (1-6), p. 873- 895, (2020)

    2020

  6. [14]

    Thierry Bodineau, Isabelle Gallagher, Laure Saint-Ra ymond, A microscopic view of the Fourier law , Comptes Rendus Physique Volume 20, Issue 5, July–August 20 19, Pages 402-418, https://doi.org/10.1016/j.crhy.2019.08.002

    2019 doi

  7. [15]

    Castorrini, Roberto; Liverani, Carlangelo Quantitative statistical properties of two- dimensional partially hyperbolic systems . Adv. Math. 409 (2022), part A, Paper No. 108625, 122 pp

    2022

  8. [16]

    Brownian Brownian motion

    Chernov, N.; Dolgopyat, D. Brownian Brownian motion . I. Mem. Amer. Math. Soc. 198 (2009), no. 927, viii+193 pp

    2009

  9. [17]

    Probability and analysis in in- teracting physical systems, 17–48, Springer Proc

    Chevyrev, Ilya; Friz, Peter K.; Korepanov, Alexey; Mel bourne, Ian; Zhang, Huilin Multiscale systems, homogenization, and rough paths . Probability and analysis in in- teracting physical systems, 17–48, Springer Proc. Math. St at., 283, Springer, Cham, 2019. HEAT EQUATION FROM...

    2019

  10. [18]

    Pellegrinotti, Err ico Presutti

    De Masi, Anna, Nicoletta Ianiro, A. Pellegrinotti, Err ico Presutti. A survey of the hydrodynamical behavior of many-particle systems , NASA STI/Recon Technical Re- port A 85 (1984): 123-294

    1984

  11. [19]

    De Masi, R

    A. De Masi, R. Esposito, J.L. Lebowitz, Incompressible Navier-Stokes and Eu- ler Limits of the Boltzmann Equation, Comm. Pure Appl. Math, Vol. XLII 1189- 1214(1989),

    1989

  12. [20]

    33 o Col´ oq

    Demers, Mark F.; Kiamari, Niloofar; Liverani, Carlang elo Transfer operators in hyperbolic dynamics—an introduction. 33 o Col´ oq. Bras. Mat. Instituto Nacional de Matem´ atica Pura e Aplicada (IMPA), Rio de Janeiro, 2021. 23 8 pp

    2021

  13. [21]

    De Simoi, Jacopo; Liverani, Carlangelo, Limit theorems for fast-slow partially hy- perbolic systems. Invent. Math. 213 (2018), no. 3, 811–1016

    2018

  14. [22]

    Hyperbolic dynamics, fluctuations and large deviations, 3 11–339, Proc

    De Simoi, Jacopo; Liverani, Carlangelo, The martingale approach after Varadhan and Dolgopyat . Hyperbolic dynamics, fluctuations and large deviations, 3 11–339, Proc. Sympos. Pure Math., 89, Amer. Math. Soc., Providence, RI, 2015

    2015

  15. [23]

    De Simoi, Jacopo; Liverani, Carlangelo Statistical properties of mostly contracting fast-slow partially hyperbolic systems . Invent. Math. 206 (2016), no. 1, 147–227

    2016

  16. [24]

    De Simoi, Jacopo; Liverani, Carlangelo; Poquet, Chris tophe; Volk, Denis Fast-slow partially hyperbolic systems versus Freidlin-Wentzell ra ndom systems . J. Stat. Phys. 166 (2017), no. 3-4, 650–679

    2017

  17. [25]

    Boldrighini, R

    C. Boldrighini, R. L. Dobrushin, and Yu. M. Suhov. One-dimensional hard rod car- icature of hydrodynamics . J. Statist. Phys., 31(3):577–616, 1983

    1983

  18. [26]

    One-dimensional hard-rod caricature of hydrodynamics: Navier-Stokes correction

    C Boldrighini, RL Dobrushin, and Y M Suhov. One-dimensional hard-rod caricature of hydrodynamics: Navier-Stokes correction . Technical report, Dublin Institute for Advances Studies, 1990. Preprint

    1990

  19. [27]

    Navier-Stokes correction

    C. Boldrighini and Y. M. Suhov. One-dimensional hard-rod caricature of hydrody- namics: “Navier-Stokes correction” for local equilibrium initial states. Comm. Math. Phys., 189(2):577–590, 1997

    1997

  20. [28]

    A.; Sina ˘ ı, Ya

    Bunimovich, L. A.; Sina ˘ ı, Ya. G. Statistical properties of Lorentz gas with periodic configuration of scatterers. Comm. Math. Phys. 78 (1980/81), no. 4, 479–497

    1980

  21. [29]

    Averaging and invariant measures

    Dolgopyat, Dmitry. Averaging and invariant measures . Mosc. Math. J. 5 (2005), no. 3, 537–576, 742

    2005

  22. [30]

    Dolgopyat, Dmitry; Liverani, Carlangelo Energy transfer in a fast-slow Hamiltonian system. Comm. Math. Phys. 308 (2011), no. 1, 201–225

    2011

  23. [31]

    Annalen der Physik

    Einstein, Albert, ¨Uber die von der molekularkinetischen Theorie der W¨ arme geforderte Bewegung von in ruhenden Fl¨ ussigkeiten suspendierten Teilchen. Annalen der Physik. 17 (8): 549–560 (1905)

    1905

  24. [32]

    Evans, R.F

    L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions , CRC Press, Boca Ratan, {1992}

    1992

  25. [33]

    Ferrari, S.Olla, Macroscopic diffusive fluctuations for generalized hard rods dy- namics, 2023, https://doi.org/10.48550/arXiv.2305.13037

    P. Ferrari, S.Olla, Macroscopic diffusive fluctuations for generalized hard rods dy- namics, 2023, https://doi.org/10.48550/arXiv.2305.13037

    2023 doi

  26. [34]

    Fritz, J. (1987). On the hydrodynamic limit of a one-dimensional Ginzburg-La ndau lattice model. The a priori bounds . Journal of Statistical Physics, 47, 551-572

    1987

  27. [35]

    Fritz, T

    J. Fritz, T. Funaki, and J. L. Lebowitz. Stationary states of random hamiltonian systems. Probability Theory and Related Fields, 99(2):211–236, 199 4

  28. [36]

    Fritz, J´ ozsef; Liverani, Carlangelo; Olla, StefanoReversibility in infinite Hamiltonian systems with conservative noise . Comm. Math. Phys. 189 (1997), no. 2, 481–496

    1997

  29. [37]

    Cristian Giardin` a and Jorge Kurchan, The Fourier law in a momentum-conserving chain, J. Stat. Mech. (2005) P05009, DOI 10.1088/1742-5468/2005 /05/P05009

    2005 doi

  30. [38]

    Nonlinear diffusion limit for a system with nearest neighbor interactions , Commun.Math

    Guo, M.Z., Papanicolaou, G.C., Varadhan, S.R.S. Nonlinear diffusion limit for a system with nearest neighbor interactions , Commun.Math. Phys. 118, 31–59 (1988). https://doi.org/10.1007/BF01218476

    1988 doi

  31. [39]

    Hennion Sur un th´ eor` eme spectral et son application aux noyaux lip chitziens

    H. Hennion Sur un th´ eor` eme spectral et son application aux noyaux lip chitziens. Proc. Amer. Math. Soc. 118.2, pp. 627–634 (1993). 58 GIOV ANNI CANESTRARI, CARLANGELO LIVERANI, AND STEF ANO O LLA

    1993

  32. [40]

    Bulletin of the American Mathematical So- ciety

    Hilbert, David, Mathematical Problems. Bulletin of the American Mathematical So- ciety. 8 (10): 437–479 (1902)

    1902

  33. [41]

    Kelly, David; Melbourne, Ian Deterministic homogenization for fast-slow systems with chaotic noise . J. Funct. Anal. 272 (2017), no. 10, 4063–4102

    2017

  34. [42]

    General topology

    Kelley, John L. General topology. Reprint of the 1955 ed ition [Van Nostrand, Toronto, Ont.]. Graduate Texts in Mathematics, No. 27. Spri nger-Verlag, New York- Berlin, 1975

    1955

  35. [43]

    and Landim, C

    Kipnis, C. and Landim, C. (1999). Scaling Limits of Inte racting Particle Systems, Grundlehren Math. Wiss. 320. Berlin: Springer. MR1707314

    1999

  36. [44]

    Komorowski, J.L

    T. Komorowski, J.L. Lebowitz, S. Olla, Heat flow in a periodically forced, thermostat- ted chain II , J.Stat.Phys., 2023, 190 (4), pp.87. https://doi.org/10. 1007/s10955-023- 03103-9

    2023

  37. [45]

    Dynamical syste ms, theory and applications (Rencontres, Battelle Res

    Lanford, Oscar E., III Time evolution of large classical systems. Dynamical syste ms, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), p p. 1–111, Lecture Notes in Phys., Vol. 38, Springer, Berlin-Ne w York, 1975

    1974

  38. [46]

    Lepri, R

    S. Lepri, R. Livi, A. Politi, Thermal Conduction in classical low-dimensional lattices , Phys. Rep. 377, 1-80 (2003)

    2003

  39. [47]

    Proceedings of the International Congress of Mathematicians—Rio de Jan eiro 2018

    Liverani, Carlangelo Transport in partially hyperbolic fast-slow systems . Proceedings of the International Congress of Mathematicians—Rio de Jan eiro 2018. Vol. III. Invited lectures, 2643–2667, World Sci. Publ., Hackensack , NJ, 2018

    2018

  40. [48]

    Liverani, Carlangelo; Olla, Stefano, Ergodicity in infinite Hamiltonian systems with conservative noise . Probab. Theory Related Fields 106 (1996), no. 3, 401–445

    1996

  41. [49]

    Liverani, Carlangelo; Olla, Stefano Toward the Fourier law for a weakly interacting anharmonic crystal . J. Amer. Math. Soc. 25 (2012), no. 2, 555–583

    2012

  42. [50]

    (1955): On the derivation of the equation of hydrodynamics from sta- tistical mechanics

    Morrey, C.B. (1955): On the derivation of the equation of hydrodynamics from sta- tistical mechanics. Comm. Pure Appl. Math. 8, 279–326

    1955

  43. [51]

    S. Olla, S. Varadhan, H. Yau, Hydrodynamical limit for a Hamiltonian system with weak noise , Commun. Math. Phys. 155 (1993), 523-560

    1993

  44. [52]

    S. Olla, M. Sasada, Energy Diffusion in anharmonic chain with conservative noise , Prob.Th.Rel.Fields, 157, 721–775 (2013), https://doi.or g/10.1007/s00440-012-0469- 5

    2013 doi

  45. [53]

    Ya. B. Pesin and Ya. G. Sinai, Gibbs measures for partially hyperbolic attractors , Ergodic Theory Dynam. Systems 2 (1982), no. 3–4, 417–438 (19 83)

    1982

  46. [54]

    Koushik Ray, Green’s function on lattices, available a t https://arxiv.org/abs/1409.7806

    arXiv

  47. [55]

    Saint-Raymond, Hydrodynamic limits of the Boltzmann equation , Lecture Notes in Mathematics 1971, Springer-Verlag (2009)

    L. Saint-Raymond, Hydrodynamic limits of the Boltzmann equation , Lecture Notes in Mathematics 1971, Springer-Verlag (2009)

    1971

  48. [56]

    Saito, K., Sasada, M., Thermal conductivity for coupled charged harmonic oscilla tors with noise in a magnetic field , Commun. Math. Phys. 361(3), 951–995 (2018)

    2018

  49. [57]

    Spohn, Large Scale Dynamics of Interacting Particles , Texts and monographs in Physics, Springer (1991)

    H. Spohn, Large Scale Dynamics of Interacting Particles , Texts and monographs in Physics, Springer (1991)

    1991

  50. [58]

    Spohn, Hydrodynamic scales of integrable many-particle systems , arXiv:2301.08504v1, 2023

    H. Spohn, Hydrodynamic scales of integrable many-particle systems , arXiv:2301.08504v1, 2023

    2023

  51. [59]

    S.R.S. Varadhan, Nonlinear diffusion limit for a system with nearest neighbor interactions-II, Asymptotic problems in probability theory: stochastic mo dels and diffusions on fractals (Sanda/Kyoto, 1990), 75–128, Pitman Res. Notes Math. Ser., 283, Longman Sci. Tech., Harlow, 199...

    1990

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