Continuously Parametrised Porous Media Model and Scaling Limits of Kinetically Constrained Models

G. S. Nahum

REVIEW 2 major objections 6 minor 16 references

Continuously Parametrised Porous Media Model and Scaling Limits of Kinetically Constrained Models

T0 review · 2 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper proves that a continuously parametrised family of kinetically constrained exclusion processes has a hydrodynamic limit given by the porous medium equation, covering every positive power-law exponent.

desk verdict New model, right strategy, but two concrete gaps in the proof mean Theorem 2.13 is not yet established as written. read the letter →

arxiv 2504.12524 v1 pith:T6TFQNMQ submitted 2025-04-16 math.PR

G. S. Nahum This is my paper · ORCID

classification math.PR MSC 60K3582C2235K65

keywords hydrodynamiclimitkineticallyconstrainedexclusionprocessesporousmediumequationBernsteinpolynomialbasisgradientconditionreplacementlemmanonlineardiffusionempiricalmeasure

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 aims to show that a wide class of one-dimensional symmetric exclusion processes with kinetic constraints still obeys a deterministic hydrodynamic equation after diffusive rescaling, provided the constraints satisfy a few explicit growth and algebraic conditions. To demonstrate the scope, it constructs a family of gradient models, continuously parametrised by $m\in[0,1]$ on top of an integer $n$, whose diffusion coefficient is $m\rho^{n+m-1}$ and whose hydrodynamic limit is the porous medium equation $\partial_t\rho=\partial_u(\rho^{n+m}\partial_u\rho)$. Previously such power-law diffusivities were available only for integer exponents, or for $m\in(0,2)$ with restrictions; the new Bernstein-basis construction removes the obstruction and yields all positive exponents. The hydrodynamic limit is proved for a macroscopically negligible perturbation of the dynamics by the symmetric simple exclusion process, which guarantees local mixing without changing the limiting equation.

What carries the argument

The load-bearing object is the interpolating constraint $c_N^m=p_n+\sum_{k=1}^{\ell_N}\binom{m}{k}(-1)^k(p_n-p_{n,k})$, a superposition of the integer PMM constraint $p_n$ with auxiliary constraints $p_{n,k}$; it is built so that the generator splits into Bernstein-model generators, which forces the gradient condition $j_{0,1}=\nabla H_N$. The proof's quantitative engine is the entropy method together with three replacement estimates: a one-block lemma and a two-block lemma control the difference between local density and particle occupation in a box, and an energy estimate gives the $H^1$ regularity of $\Phi(\rho)$ needed for uniqueness. In Regime I the mixing that makes the replacement lemmas work comes from mobile clusters: boxes of length at least $n+2$ containing at least $n+2$ particles and one vacancy can be reorganised by nearest-neighbour exchanges at rates bounded below independently of $N$.

What would settle it

Take a configuration on a large torus with $n+2$ consecutive particles followed by one vacancy in a box of length $n+2$, arranged so that no smaller aligned block of $n$ particles appears in the box; if the rate $c_N^m$ for a jump inside that box decays with $N$ or vanishes, the mobile-cluster lower bound fails and the Regime-I replacement lemmas cannot yield the hydrodynamic limit. A direct computation of $c_N^m$ on such a configuration settles the point.

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Extended reading notes

Core claim

The central statement is Theorem 2.13: for any sequence of initial measures in local equilibrium with a profile $\rho_{\mathrm{ini}}$, the empirical measure of the $N^2$-speed process converges in probability to the unique weak solution $\rho$ of $\partial_t\rho=\partial_u^2\Phi(\rho)$, whenever the constraints satisfy Assumption 2.6 and belong to either Regime I (mobile clusters with uniformly positive rates) or Regime II (uniformly positive rates everywhere). For the interpolating process of Definition 2.3 the diffusivity satisfies $\mathbb{E}_{\nu^N_\alpha}[c_N^m]\to m\alpha^{n+m-1}$ uniformly in $\alpha$, so the limiting equation is the porous medium equation $\partial_t\rho=\partial_u(\rho^{n+m}\partial_u\rho)$. This covers the integer Porous Media Models, the Bernstein models, and their superpositions, and the proof is designed to be independent of the choice of polynomial basis: it uses only the stated rate bounds, the gradient condition with its explicit primitive $H_N=h_N+g_N$, and the one-block and two-block replacement lemmas.

Load-bearing premise

The Regime-I verification rests on an unproved mobile-cluster assertion: every box of length at least $n+2$ containing at least $n+2$ particles and one vacancy is mobile, and jumps inside it occur at rates bounded below by a constant independent of the system size.

Editorial extensions

If this is right

The hydrodynamic limit provides a weak law of large numbers for the empirical measure, so local particle densities are well approximated by a deterministic PDE for large system sizes.
The interpolating process gives a constructive route to any power-law nonlinear diffusion $\partial_t\rho=\partial_u(\rho^p\partial_u\rho)$ for $p>0$, including fast diffusion for $p<1$ and slow diffusion for $p>1$, from exclusion-type microdynamics.
Because the proof is basis-independent, the same theorem covers any gradient constraint family that satisfies the uniform rate bounds, the uniform Cauchy condition on $h_N$, and one of the two mixing regimes.
The $H^1$ energy regularity of $\Phi(\rho)$ (or of $\rho$ itself in Regime II) yields uniqueness of the weak solution, so every subsequential limit of the empirical measures coincides with the same macroscopic profile.

Reading between the lines

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

The mobile-cluster condition could be tested numerically: simulate the interpolating model at large $N$ on a block with $n+2$ particles and one vacancy and measure the smallest jump rate; if it stays bounded below by a positive constant independent of $N$, the missing assertion in Section 2.1 is likely true for these models.
The same Assumption 2.6 could be checked for other finite-range gradient constraints, for example multi-species or higher-dimensional symmetric exclusion processes, potentially yielding hydrodynamic limits beyond the one-dimensional PMM setting.
The decomposition into Bernstein generators suggests a design principle: particle systems whose hydrodynamic equation is a prescribed polynomial or Bernstein diffusivity can be assembled as superpositions of elementary Bernstein models, and the proof here indicates which superpositions are tractable.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a promising paper with a genuinely new model, but the proof of the main theorem has holes that need filling before the result is fully established.

The interpolating process in Definition 2.3 is a real contribution. It uses the Bernstein basis to define a one-parameter family of exclusion processes with diffusivity ρ^{n+m} for integer n and m∈[0,1], extending the earlier monomial construction in [10] beyond exponent 2. The decomposition c^m_N = δ_N p_n + m p_{n,1} + Σ_{k≥2}|C(m,k)|p_{n,k} is clean, and the gradient property is inherited from the Bernstein models. The general conditions in Assumption 2.6 are also useful: they separate algebraic growth conditions from the dynamics, and the two regimes are a sensible organizing device.

The problems are in the proof. The verification that the interpolating model belongs to Regime I is essentially a sketch. The claim that boxes with at least n+2 particles and one vacancy are mobile clusters (which, for length exactly n+2, is internally inconsistent — presumably n+1 particles is meant) is asserted, but δ_N tends to 0 as ℓ_N→∞, and the p_{n,k} terms for large k also have coefficients going to 0. A uniform r⋆ needs a quantitative argument that, for a fixed cluster length κ⋆, some term with coefficient bounded away from 0 is always active. That is not supplied. Lemma 4.6 also has a clear typo: the auxiliary set Ω^{p,w,y} in (4.11) repeats the same condition twice, so the case split and the counting argument that follow do not work as written. The energy estimate in (5.2) has liminf where limsup is needed; that is likely a simple fix. Regime II is deferred with a 'proof is identical' remark, which is probably acceptable but not fully written out.

None of this looks fatal. The strategy is the standard entropy method, the structure is right, and the gaps appear fixable with work rather than new ideas. But as it stands, Theorem 2.13 is not completely proved. The paper deserves a serious referee: the model alone justifies referee time. I would send it out with a request for major revision and a careful check of the two-block estimate and the Regime I verification. For my reading group, I would wait for a revised version.

Referee Report

2 major / 6 minor

Summary. The paper introduces a new one-dimensional kinetically constrained symmetric exclusion process, the 'interpolating process' (Definition 2.3), which is constructed via a Bernstein-polynomial/generalized-binomial expansion and is intended to interpolate between the Porous Media Models PMM(n) and PMM(n+1). The associated hydrodynamic limit is claimed to be ∂tρ = ∂u(ρ^{n+m}∂uρ). The paper then states a general hydrodynamic-limit theorem (Theorem 2.13) for perturbed gradient exclusion KCMs satisfying Assumption 2.6 and either a mobile-cluster condition (Regime I) or a uniform positivity condition (Regime II). The proof follows the standard entropy method: tightness of the empirical measures, replacement lemmas (one-block and two-block), and an energy estimate for uniqueness of the limiting weak solution. The general framework is intended to be basis-independent and to cover the interpolating model as well as superpositions of Bernstein models.

Significance. The paper's central ambition is significant: it proposes a basis-independent set of sufficient conditions for hydrodynamic limits of gradient kinetically constrained exclusion processes, and it exhibits a genuinely new continuously parametrized model covering non-integer exponents. The construction is explicit and parameter-free in the sense that the target diffusivity emerges from the rate definition rather than being fitted; the diffusivity computation for the interpolating model is a clear strength. If the proof gaps are repaired, the framework would unify and extend the companion works [10] and [13]. However, the manuscript as written does not fully verify the mobile-cluster hypothesis for the new model, and the two-block estimate contains a defective auxiliary set; both points are load-bearing for Theorem 2.13. The paper is therefore not yet complete, but the issues appear local and fixable within the manuscript's scope.

major comments (2)

[Section 2.1, after Definition 2.3; Definition 2.7(ii)] The claim that every box of length at least n+2 containing at least n+2 particles and at least one vacancy is a mobile cluster with all required jump rates bounded below by a constant r⋆ independent of N is asserted but not proved. In the decomposition cm_N = δ_N p_n + m p_{n,1} + Σ_{k≥2} |C(m,k)| p_{n,k}, the coefficient δ_N = 1 − Σ_{k=1}^{ℓ_N} |C(m,k)| is a binomial tail that depends on ℓ_N and may vanish as ℓ_N → ∞; a uniform lower rate r⋆ must be derived from the remaining finite-sum terms (for example, from the coefficient m of p_{n,1} and from the structure of p_{n,k} for 2 ≤ k ≤ ℓ_N), but no such quantitative bound is computed. This is load-bearing because Lemma 4.6 invokes the uniform lower bound r⋆(Q_n η⋆) ≥ r0 after (4.17) and in the path-count argument; if r⋆ is not established independently of N, the two-block estimate in Regime I collapses and Theorem 2.13 is not proved for the interpolating model.
[Lemma 4.6, Eq. (4.11)] The auxiliary set Ω^{p,w,y} is garbled: its two clauses are identical, both reading (⟨τ_{p+w}η⟩_L ≥ (k⋆+1)/L or ⟨τ_yη⟩_L ≥ (k⋆+1)/L). Consequently, the assertion that outside Ω^{p,w,y} at most k⋆+1 values of r ∈ B_L give nonzero exchange rates is not justified (a block could be nearly full while the other is nearly empty without satisfying the displayed condition as written), and the subsequent case split into cases (1) and (2) is not exhaustive. The proof of the two-block estimate is therefore incomplete as written; the definition needs to be corrected (e.g., requiring one block to be high and the other low) or the counting argument must be revised accordingly.

minor comments (6)

[Section 2.1, diffusivity display] The text states that EνNα[cm_N] → mα^{m−1} uniformly, but the computation immediately below it yields mα^n Σ_{k=0}^{ℓ_N} (−1)^k C(m,k)(1−α)^k, which converges to mα^{n+m}; this is also the diffusivity needed for the stated equation ∂tρ = ∂u(ρ^{n+m}∂uρ). The displayed limit should be corrected.
[Lemma 2.9] The statement lists '(2.11), (2.7),(2.8), (2.10) and (2.11)' with (2.11) repeated; presumably one of these should be the verification of (2.9) or a renumbering is needed.
[Definition 2.3] The auxiliary constraints p_{n,k} and p_j_{n,k} use a notation that is hard to follow; in particular, the superscript j appears in 'pj n,k' but the definition of p_j_{n,k} is not explicitly separated from p_{n,k}. Please clarify the indexing.
[Section 3.3, Eqs. (3.13)–(3.16)] In the decomposition of the product difference, the maps φ^{εN,L}_m, φ^L_m, and φ^L_m are introduced; two of them are printed with the same symbol φ^L_m. If they are different objects, they should be renamed; if they are identical, the decomposition is redundant.
[Lemma 4.6 proof] The letter r is used both for an index in B_L and for the rate lower bound r⋆ (and r0), which makes the displays around (4.17) hard to read. Please use a different symbol for the spatial index.
[Throughout] There are several typos and inconsistencies: 'interacing' and 'condtition' in the Introduction, 'Riez' for Riesz in Section 5, 'arrodingly' in Section 3.3, and the varying use of 'Porous Media Model' versus 'Porous Medium Equation'. A thorough proofreading is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the model's diffusivity is explicitly built into the rate definition, and the hydrodynamic limit is a derived theorem rather than a fitted prediction.

full rationale

The paper does not exhibit a circular reduction. Definition 2.3 specifies the microscopic rates c^m_N directly as a basis expansion involving p_n and p_{n,k}; the diffusion coefficient is then computed from that definition via the generalized binomial theorem, E_{\nu^N_\alpha}[c^m_N] \to m\alpha^{n+m-1} uniformly in \alpha, and the hydrodynamic equation is derived from the generator by the entropy method. Thus the PDE is an output of the construction, not a separately fitted quantity. The proof imports technical results from the same author's companion works [10] and [13], but those are parameter-free mathematical lemmas (gradient property, generalized binomial estimates, uniqueness arguments) with stated assumptions that do not include the hydrodynamic limit of the interpolating process; citing them is not circular, even though the papers share authors. The skeptical concerns about the Regime-I verification---namely the unproved mobile-cluster lower bound r_\star and the garbled auxiliary set \Omega^{p,w,y} in (4.11)---are correctness or completeness gaps in the proof, not examples of conclusions being equivalent to their inputs by definition. A gap is not a circular step. No fitted parameter is called a prediction, no target result is assumed in the definition of the model, and no self-citation chain forces the central claim.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard entropy-method machinery, the reversibility of Bernoulli product measures, the gradient condition inherited from the Bernstein models, and an unproved mobile-cluster assertion for the new interpolating process. The model parameters n, m, and ell_N are chosen by hand to realize the target diffusivity; no data fitting occurs. The entropy estimate also imports uniqueness results from companion papers.

free parameters (5)

n = positive integer
Base integer parameter of the porous media model being interpolated; chosen by hand to define the family.
m = in [0,1]
Continuation exponent added to n; chosen to realize the target diffusivity rho^{n+m}.
ell_N = integer with 2 <= ell_N <= N/2 - 2 and ell_N -> infinity
Truncation length of the binomial expansion; controls the range of the rates and enters the rate bounds in Assumption 2.6.
p_N = sequence with p_N -> 0 and p_N N^2 -> infinity
Strength of the SSEP perturbation used to guarantee local mixing in Regime I while remaining macroscopically negligible.
kappa_star and r_star = not computed
Regime I assumes existence of mobile clusters of length kappa_star and a uniform lower jump rate r_star independent of N; these are structural assumptions rather than fitted numbers.

assumptions (4)

domain assumption Bernoulli product measures nu^N_alpha are reversible for the symmetric exclusion generators in (2.1).
Used throughout the paper to define expectations, Dirichlet forms, and reference measures for the entropy method.
domain assumption The gradient condition c_N(e0,1 - e1,0) = grad H_N holds for the models considered.
Central to writing the generator action on the empirical measure and to deriving the hydrodynamic equation; for the new model it is inherited from [13, Proposition 2.7].
ad hoc to paper Boxes of length at least n+2 containing at least n+2 particles and one vacancy are mobile clusters with rate lower bound independent of N.
Asserted in Section 2.1 and used to place the interpolating model in Regime I; not fully proved and needed for the replacement lemmas.
domain assumption Oleinik's method gives uniqueness of weak solutions in Regime I, and the argument in [10, Appendix B] gives uniqueness in Regime II.
Uniqueness is imported from cited works rather than proved in this manuscript, and the Regime II case is not written out.

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Cite this review

Pith. "Pith review of Continuously Parametrised Porous Media Model and Scaling Limits of Kinetically Constrained Models." pith.science (2026). https://pith.science/paper/T6TFQNMQ

@misc{pith2026250412524,
  author       = {Pith},
  title        = {Pith review of: Continuously Parametrised Porous Media Model and Scaling Limits of Kinetically Constrained Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T6TFQNMQ}},
  note         = {Machine review of arXiv:2504.12524}
}

read the original abstract

We investigate the emergence of non-linear diffusivity in kinetically constrained, one-dimensional symmetric exclusion processes satisfying the gradient condition. Recent developments introduced new gradient dynamics based on the Bernstein polynomial basis, enabling richer diffusive behaviours but requiring adaptations of existing techniques. In this work, we exploit these models to generalise the Porous Media Model to non-integer parameters and establish simple conditions on general kinetic constraints under which the empirical measure of a perturbed version of the process converges. This provides a robust framework for modelling non-linear diffusion from kinetically constrained systems.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [10]

Nahum, and Marielle Si mon

Patricia Gon¸ calves, Gabriel S. Nahum, and Marielle Si mon. From exclusion to slow and fast diﬀusion. arXiv preprint , January 2023

work page 2023

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G. S. Nahum. A gradient model for the bernstein polynomi al basis. arXiv preprint arXiv:2411.15954v2 , 2024

work page Pith review arXiv 2024

[3] [1]

Baldasso, O

R. Baldasso, O. Menezes, A. Neumann, and R.R. Souza. Excl usion process with slow boundary. Journal of Statistical Physics , 167(5):1112–1142, 2017

work page 2017

[4] [2]

Bonorino, R

L. Bonorino, R. De Paula, P. Gon¸ calves, and A. Neumann. H ydrodynamics for the porous medium model with slow reservoirs. Journal of Statistical Physics , 179(3):0022–4715, 2020

work page 2020

[5] [3]

P. Cardoso. Hydrodynamic limit of symmetric exclusion process with lon g jumps and a slow barrier . PhD thesis, Instituto Superior T´ ecnico, Universidade de Lisb oa, 2022

work page 2022

[6] [4]

Cardoso, P

P. Cardoso, P. Gon¸ calves, and G. S. Nahum. Derivation of non-polynomial fractional diﬀusions from the generalized exclusion with a slow barrier. arXiv preprint arXiv:2412.12414 , 2024. 81 pages, 4 ﬁgures

arXiv 2024

[7] [5]

Linear and nonl inear fractional pdes from interacting particle systems

Pedro Cardoso and Patr ´ ıcia Gon¸ calves. Linear and nonl inear fractional pdes from interacting particle systems. arXiv preprint arXiv:2407.02246 , pages 1–27, 2024

arXiv 2024

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Renato De Paula, Patricia Gon¸ calves, and Adriana Neuma nn. Energy estimates and convergence of weak solutions of the porous medium equation. Nonlinearity, 34(11):7872, October 2021

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