pith. sign in

arxiv: 2606.01937 · v1 · pith:VNQSP4R5new · submitted 2026-06-01 · 🧮 math.PR · math-ph· math.MP

Stationary fluctuations for an exclusion process with mass and energy conservation

Hugo Da Cunha , Makiko Sasada This is my paper

Pith reviewed 2026-06-28 12:50 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords exclusion processfluctuation fieldsstochastic Burgers equationconservation lawsBoltzmann-Gibbs principlenonlinear fluctuating hydrodynamicsspectral gapuniversality classes
0
0 comments X

The pith

For suitable parameters, fluctuation fields in a mass-and-energy conserving exclusion process converge to uncoupled stochastic Burgers equations.

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

The paper introduces an exclusion process that conserves both the number of particles and their total energy, unlike standard multi-species models. Applying nonlinear fluctuating hydrodynamics shows that different parameter choices lead to various universality classes for the fluctuations. The main result rigorously establishes that, when parameters are chosen appropriately, the stationary fluctuation fields of the conserved quantities converge to uncoupled stochastic Burgers equations in the appropriate scaling limit. This is proven using a second-order Boltzmann-Gibbs principle, spectral gap estimates, and equivalence of ensembles. Additionally, the work provides a general proof that the Jacobian matrix of the macroscopic current is diagonalizable with distinct real eigenvalues for multi-component systems.

Core claim

The authors construct an exclusion process with mass and energy conservation that mimics interacting oscillators. For suitable parameter choices where the Jacobian of the macroscopic current has distinct real eigenvalues, the fluctuation fields converge in the scaling limit to a pair of uncoupled stochastic Burgers equations. The proof relies on establishing the second-order Boltzmann-Gibbs principle for this model, combined with spectral gap and equivalence of ensembles results. They also prove in general that the Jacobian matrix for such multi-component systems is diagonalizable with distinct real eigenvalues.

What carries the argument

The second-order Boltzmann-Gibbs principle established for this model, which controls fluctuations of the conserved quantities to enable convergence to the stochastic Burgers equations.

Load-bearing premise

The model parameters must be chosen so that the second-order Boltzmann-Gibbs principle holds and the Jacobian matrix of the macroscopic current has distinct real eigenvalues.

What would settle it

A direct numerical simulation of the particle system for the chosen parameters showing that the fluctuation fields do not converge to the uncoupled stochastic Burgers equations would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.01937 by Hugo Da Cunha, Makiko Sasada.

Figure 1
Figure 1. Figure 1: Schematic illustration of the asymmetric dynamics of the EPE: par￾ticle jump rates (top) and energy transfer rates (bottom). We now introduce an asymmetric version of the dynamics by considering the infinitesimal generator L := LS + N −γLA, where γ > 0 and the antisymmetric part LA acts on local func￾tions f : Ω −→ R via LAf(η) = 1 2 X x,y∈Z |x−y|=1 (y − x)(αp + αeηx)c p x→y (η) [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 2
Figure 2. Figure 2: Illustation of the path of transformations used in the proof of the moving energy Theorem 27. The initial configuration η is represented in ➀ where there is a particle with energy m at x and a particle with energy ℓ at y, with potentially some other particles in between. In step ➁, we bring the particle at x close to the particle at y by performing nearest-neighbour exchanges. In step ➂, we transfer an ene… view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the path of transformation used in the moving particle Lemma 28 to exchange the values of ηx and ηx+1 when both sites are occupied by a particle. so we focus on the second term. When both sites x and x + 1 are occupied by a particle, it is possible to exchange the values of ηx and ηx+1 by using the energy exchange dynamics. More precisely, we define a sequence (η (k) )0≤k≤n recursively by s… view at source ↗
read the original abstract

We introduce a novel exclusion process with two conservation laws, mass and energy, designed to mimic the essential features of continuous systems like interacting oscillators within the framework of interacting particle systems. This distinguishes our model from conventional multi-species processes where only particle numbers are conserved. As a basis for our fluctuation analysis, we first show that applying nonlinear fluctuating hydrodynamics (NFH) to this model reveals a wide variety of universality classes depending on the parameter choices. The main objective of this work is to study the stationary fluctuations of these conserved quantities. For a suitable choice of parameters, we rigorously show that the fluctuation fields converge to uncoupled stochastic Burgers equations (SBE) in the scaling limit. The proof relies on the second-order Boltzmann-Gibbs principle that we establish for this model, along with the spectral gap estimate and the equivalence of ensembles. Of independent interest is our general proof of the diagonalizability of the Jacobian matrix for the macroscopic current with distinct real eigenvalues. While this property is often taken as given in the physics literature, we establish it rigorously for multi-component systems even when the eigenvectors cannot be explicitly computed, offering a firm mathematical foundation for a broad class of models.

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

0 major / 3 minor

Summary. The manuscript introduces a novel exclusion process on the lattice that conserves both mass and energy (distinct from standard multi-species models), derives its nonlinear fluctuating hydrodynamics to exhibit multiple universality classes depending on parameters, and proves that, for suitable parameter choices, the stationary fluctuation fields of the conserved quantities converge to a pair of uncoupled stochastic Burgers equations. The argument rests on establishing the second-order Boltzmann-Gibbs principle for the model, together with spectral-gap and equivalence-of-ensembles estimates, and includes an auxiliary general result proving that the Jacobian matrix of the macroscopic current is diagonalizable with distinct real eigenvalues even when eigenvectors cannot be written explicitly.

Significance. If the central convergence statement holds, the work supplies a rigorous, parameter-tuned example of the stochastic-Burgers universality class arising from a two-conservation-law particle system that mimics features of continuous oscillator chains. The general diagonalizability theorem is of independent interest because it removes an assumption frequently invoked without proof in the physics literature on multi-component fluctuating hydrodynamics.

minor comments (3)
  1. The abstract and introduction repeatedly refer to “a suitable choice of parameters” without an explicit, self-contained statement of the admissible range (e.g., inequalities on the interaction rates or on the conserved densities). Adding a short, numbered list of the required conditions early in §2 would improve readability.
  2. Notation for the two conserved fields (mass and energy) is introduced in §1 but then reused with slightly varying symbols in the fluctuation-field definitions of §4; a single, consistently indexed notation table would eliminate minor confusion.
  3. The statement of the second-order Boltzmann-Gibbs principle (Theorem 3.2) is given in a form that mixes microscopic and macroscopic variables; separating the microscopic error bound from the macroscopic replacement step would make the subsequent application in the fluctuation proof easier to follow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the significance of the diagonalizability result, and recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring detailed rebuttal. We will make any minor editorial changes in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via independent proofs

full rationale

The central claim is convergence of fluctuation fields to uncoupled SBEs, obtained by establishing the second-order Boltzmann-Gibbs principle, spectral gap, and equivalence of ensembles for the model, plus a general proof that the macroscopic current Jacobian is diagonalizable with distinct real eigenvalues. These are presented as results proven within the paper rather than fitted parameters renamed as predictions or reduced via self-citation chains. No quoted step equates a claimed output to an input by construction (e.g., no self-definitional ratios or ansatzes smuggled via prior work). The derivation therefore remains independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, ad-hoc axioms, or new invented entities are stated; the work rests on standard background results in interacting particle systems.

pith-pipeline@v0.9.1-grok · 5735 in / 1100 out tokens · 18242 ms · 2026-06-28T12:50:23.534179+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

34 extracted references · 18 canonical work pages

  1. [1]

    Bernardin and G

    C. Bernardin and G. Stoltz. Anomalous diffusion for a class of systems with two conserved quantities.Non- linearity, 25(4):1099, March 2012.https://doi.org/10.1088/0951-7715/25/4/1099

    work page doi:10.1088/0951-7715/25/4/1099 2012

  2. [2]

    Bertini and G

    L. Bertini and G. Giacomin. Stochastic burgers and kpz equations from particle systems.Communications in Mathematical Physics, 183(3):571–607, 1997.https://doi.org/10.1007/s002200050044

    work page doi:10.1007/s002200050044 1997

  3. [3]

    Cannizzaro, P

    G. Cannizzaro, P. Cardoso, L. Gr¨ afner, and A. Occelli. Equilibrium fluctuations for a multi-species particle system with long jumps, April 2026. arXiv.2604.03777

    Pith/arXiv arXiv 2026

  4. [4]

    Cannizzaro, P

    G. Cannizzaro, P. Gon¸ calves, R. Misturini, and A. Occelli. From ABC to KPZ.Probability Theory and Related Fields, 191(1):361–420, February 2025.https://doi.org/10.1007/s00440-024-01314-z

    work page doi:10.1007/s00440-024-01314-z 2025

  5. [5]

    P. Caputo. On the spectral gap of the Kac walk and other binary collision processes.ALEA Lat. Am. J. Probab. Math. Stat. 4, pages 205–222, 2008.https://alea.impa.br/articles/v4/04-10.pdf

    2008

  6. [6]

    E. A. Carlen, M. C. Carvalho, and M. Loss. Determination of the spectral gap for Kac’s master equation and related stochastic evolution.Acta Mathematica, 191(1):1–54, March 2003.https://doi.org/10.1007/ BF02392695

    2003

  7. [7]

    Chang, C

    C. Chang, C. Landim, and S. Olla. Equilibrium fluctuations of asymmetric simple exclusion processes in dimensiond≥3.Probability Theory and Related Fields, 119(3):381–409, 2001.https://doi.org/10.1007/ PL00008764

    2001

  8. [8]

    Diaconis and L

    P. Diaconis and L. Saloff-Coste. Comparison Theorems for Reversible Markov Chains.The Annals of Applied Probability, 3(3):696 – 730, 1993.https://doi.org/10.1214/aoap/1177005359

    work page doi:10.1214/aoap/1177005359 1993

  9. [9]

    Franceschini, P

    C. Franceschini, P. Gon¸ calves, M. Jara, and B. Salvador. Non-equilibrium fluctuations for SEP(α) with open boundary.Stochastic Processes and their Applications, 178:104463, December 2024.https://doi.org/10. 1016/j.spa.2024.104463

    arXiv 2024

  10. [10]

    Gon¸ calves and K

    P. Gon¸ calves and K. Hayashi. Derivation of Anomalous Behavior from Interacting Oscillators in the High- Temperature Regime.Communications in Mathematical Physics, 403(3):1193–1243, November 2023.https: //doi.org/10.1007/s00220-023-04818-2

    work page doi:10.1007/s00220-023-04818-2 2023

  11. [11]

    Gon¸ calves, K

    P. Gon¸ calves, K. Hayashi, and J. Mangi. Stochastic oscillators out of equilibrium: scaling limits and corre- lation estimates, May 2025. arXiv:2505.10256

    arXiv 2025

  12. [12]

    Gon¸ calves, K

    P. Gon¸ calves, K. Hayashi, and M. Sasada. Characterization of Gradient Condition for Asymmetric Partial Exclusion Processes and Their Scaling Limits.Latin American Journal of Probability and Mathematical Statistics, 22(1):331, 2025.https://doi.org/10.30757/ALEA.v22-12

    work page doi:10.30757/alea.v22-12 2025

  13. [13]

    Gon¸ calves and M

    P. Gon¸ calves and M. Jara. Scaling Limits of Additive Functionals of Interacting Particle Systems.Commu- nications on Pure and Applied Mathematics, 66(5):649–677, 2013.https://doi.org/10.1002/cpa.21441

    work page doi:10.1002/cpa.21441 2013

  14. [14]

    Gon¸ calves and M

    P. Gon¸ calves and M. Jara. Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems. Archive for Rational Mechanics and Analysis, 212(2):597–644, May 2014.https://doi.org/10.1007/ s00205-013-0693-x

    2014

  15. [15]

    Gon¸ calves, M

    P. Gon¸ calves, M. Jara, and S. Sethuraman. A stochastic burgers equation from a class of microscopic inter- actions.Annals of Probability, 43(1):286–338, 2015.https://doi.org/10.1214/13-AOP878

    work page doi:10.1214/13-aop878 2015

  16. [16]

    Gubinelli and N

    M. Gubinelli and N. Perkowski. Energy solutions of KPZ are unique.Journal of the American Mathematical Society, 31(2):427–471, April 2018.https://www.ams.org/jams/2018-31-02/S0894-0347-2017-00889-0/

    2018

  17. [17]

    Gubinelli and N

    M. Gubinelli and N. Perkowski. The infinitesimal generator of the stochastic Burgers equation. Probability Theory and Related Fields, 178(3):1067–1124, December 2020.https://doi.org/10.1007/ s00440-020-00996-5

    2020

  18. [18]

    M. Z. Guo, G. C. Papanicolaou, and S. R. S. Varadhan. Nonlinear diffusion limit for a system with nearest neighbor interactions.Communications in Mathematical Physics, 118(1):31–59, 1988.https://doi.org/10. 1007/BF01218476. STATIONARY FLUCTUATIONS FOR AN EXCLUSION PROCESS WITH TWO CONSERVATION LAWS 41

    1988

  19. [19]

    Hayashi and S

    K. Hayashi and S. Olla. Nonlinear fluctuations for a chain of weakly anharmonic oscillators with stochastic perturbation, October 2025. arXiv:2510.12922

    Pith/arXiv arXiv 2025

  20. [20]

    Holley and D

    R. Holley and D. Stroock. Generalized Ornstein-Uhlenbeck Processes and Infinite Particle Branching Brow- nian Motions.Publications of the Research Institute for Mathematical Sciences, 14(3):741–788, December 1978.https://doi.org/10.2977/prims/1195188837

    work page doi:10.2977/prims/1195188837 1978

  21. [21]

    Jacod and A

    J. Jacod and A. N. Shiryaev.Limit Theorems for Stochastic Processes, volume 288 ofGrundlehren der mathematischen Wissenschaften. Springer, Berlin, Heidelberg, 2003.http://link.springer.com/10.1007/ 978-3-662-05265-5

    2003

  22. [22]

    Kanegae and H

    K. Kanegae and H. Wachi. The analogue of Aldous’ spectral gap conjecture for the generalized exclusion process.Communications in Algebra, 54(3):1256–1272, 2026.https://doi.org/10.1080/00927872.2025. 2550819

    work page doi:10.1080/00927872.2025 2026

  23. [23]

    Landa, P

    M. Kardar, G. Parisi, and Y.-C. Zhang. Dynamic Scaling of Growing Interfaces.Physical Review Letters, 56(9):889–892, March 1986. Publisher: American Physical Society,https://doi.org/10.1103/PhysRevLett. 56.889

    work page doi:10.1103/physrevlett 1986

  24. [24]

    Kipnis and C

    C. Kipnis and C. Landim.Scaling Limits of Interacting Particle Systems, volume 320 ofGrundlehren der mathematischen Wissenschaften. Springer, Berlin, Heidelberg, 1999.http://link.springer.com/10.1007/ 978-3-662-03752-2

    1999

  25. [25]

    Kipnis and S

    C. Kipnis and S. R. S. Varadhan. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions.Communications in Mathematical Physics, 104(1):1–19, 1986.https: //doi.org/10.1007/BF01210789

    work page doi:10.1007/bf01210789 1986

  26. [26]

    I. Mitoma. Tightness of Probabilities OnC([0,1];Y ′) andD([0,1];Y ′).The Annals of Probability, 11(4):989– 999, November 1983. Publisher: Institute of Mathematical Statistics,https://doi.org/10.1214/aop/ 1176993447

    work page doi:10.1214/aop/ 1983

  27. [27]

    Nagahata

    Y. Nagahata. Fluctuation Dissipation Equation for Lattice Gas with Energy.Journal of Statistical Physics, 110(1):219–246, January 2003.https://doi.org/10.1023/A:1021022829221

    work page doi:10.1023/a:1021022829221 2003

  28. [28]

    S. Olla. Role of conserved quantities in Fourier’s law for diffusive mechanical systems.Comptes Rendus Physique, 20(5):429–441, 2019. Fourier and the science of today / Fourier et la science d’aujourd’hui,https: //doi.org/10.1016/j.crhy.2019.08.001

    work page doi:10.1016/j.crhy.2019.08.001 2019

  29. [29]

    J. Quastel. Diffusion of color in the simple exclusion process.Communications on Pure and Applied Mathe- matics, 45(6):623–679, 1992.https://doi.org/10.1002/cpa.3160450602

    work page doi:10.1002/cpa.3160450602 1992

  30. [30]

    Saito, M

    K. Saito, M. Sasada, and H. Suda. 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field.Communications in Mathematical Physics, 372(1):151–182, 2019.https://doi.org/10. 1007/s00220-019-03506-4

    2019

  31. [31]

    M. Sasada. On the Spectral Gap of the Kac Walk and Other Binary Collision Processes on d- Dimensional Lattice. In Kenji Iohara, Sophie Morier-Genoud, and Bertrand R´ emy, editors,Symmetries, Integrable Systems and Representations, pages 543–560, London, 2013. Springer.https://doi.org/10.1007/ 978-1-4471-4863-0_23

    2013

  32. [32]

    Sch¨ utz and S

    G. Sch¨ utz and S. Sandow. Non-Abelian symmetries of stochastic processes: Derivation of correlation functions for random-vertex models and disordered-interacting-particle systems.Physical Review E, 49(4):2726–2741, April 1994.https://doi.org/10.1103/PhysRevE.49.2726

    work page doi:10.1103/physreve.49.2726 1994

  33. [33]

    H. Spohn. Nonlinear Fluctuating Hydrodynamics for Anharmonic Chains.Journal of Statistical Physics, 154(5):1191–1227, 2014.https://doi.org/10.1007/s10955-014-0933-y

    work page doi:10.1007/s10955-014-0933-y 2014

  34. [34]

    Spohn and G

    H. Spohn and G. Stoltz. Nonlinear Fluctuating Hydrodynamics in One Dimension: The Case of Two Conserved Fields.Journal of Statistical Physics, 160(4):861–884, August 2015.https://doi.org/10.1007/ s10955-015-1214-0. Hugo Da Cunha – Universit´e Lyon 1, Centrale Lyon, INSA Lyon, Universit´e Jean Monnet, CNRS, ICJ UMR5208, 69622 Villeurbanne, France and JSPS ...

    2015