{"id":"ec7409a3-9103-456d-abea-09ceddf107da","arxiv_id":"2502.00738","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"We prove hydrodynamic limits for facilitated exclusion processes with closed boundaries in symmetric, asymmetric and weakly asymmetric regimes via a coupling to simple exclusion processes.","lead":"This paper proves the hydrodynamic limit for facilitated exclusion processes confined by impenetrable walls, in three scaling regimes. The proof couples the constrained process to a simpler exclusion process, so previously known results transfer directly.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2's proof that ρ^ε→ρ relies on an estimate with ||∂_vω^ε||∞, which is not uniformly bounded in ε; the AFEPvv case of Theorem 2 is therefore not rigorously established as written.","rationale":"The central strategy is sound: the microscopic mapping is plausible and the macroscopic PDE computations in Lemma 1 check out. The reader's conditional verdict is appropriate because of unproven or imported ingredients. The most load-bearing additional flaw I find is not the high-density assumption, but a concrete invalid estimate in Lemma 2 that blocks the proof for the hyperbolic AFEPvv regime. The flaw is likely repairable, so it does not overturn the verdict, but it does mean the manuscript as written does not fully establish one of the three advertised regimes. I therefore keep the verdict CONDITIONAL/UNCHANGED rather than moving to reject or accept outright.","tokens_in":16342,"tokens_out":44957,"duration_ms":480621,"concrete_test":"Re-derive the convergence of ω^ε∘v^ε to ω∘v in L^1 using the standard composition estimate ∫|ω^ε(v^ε)−ω(v)| ≤ 2m||ω^ε−ω||_{L^1} + ∫|ω(v^ε)−ω(v)|, where the second term tends to 0 because v^ε→v uniformly and ω∈L^1 (approximate ω by continuous functions). If this succeeds, the gap is cosmetic and the AFEPvv case can be repaired; if it fails, construct a counterexample with ω^ε→ω in L^1, v^ε→v uniformly, but ω^ε∘v^ε↛ω∘v in L^1, which would invalidate Lemma 2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Lemma 2 (Section 5.2), the proof needs ρ^ε→ρ in L^1 to pass to the limit in the entropy inequality and to identify boundary traces. The estimate given is ||ω^ε_t∘v^ε_t − ω_t∘v_t||_{L^1} ≤ ||∂_vω^ε_t||_{L∞} ||v^ε_t−v_t||_{L∞} + (1/m)||ω^ε_t−ω_t||_{L^1}. The second term vanishes, but the first term is not shown to vanish: for the parabolic regularization (33), ||∂_vω^ε||_{L∞} typically blows up as ε→0 (viscous shock profiles have gradients of order 1/ε), while ||v^ε−v||_{L∞} is only controlled by ||ω^ε−ω||_{L^1}, which is at best O(ε). The product of these two quantities need not tend to zero. Since this L^1 convergence is what identifies the entropy solution and the boundary traces for the AFEPvv case, the proof of Theorem 2 for κ∈(1/2,1) is incomplete as written. This is a distinct, more internal weakness than the reader's focus on the high-density initial condition: even within the stated assumptions, the hyperbolic case is not fully proved by the manuscript's argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves hydrodynamic limits for the facilitated exclusion process on a finite segment with closed boundaries, in the symmetric (SFEP), weakly asymmetric (WAFEP), very weakly asymmetric (vWAFEP), and asymmetric with vanishing viscosity (AFEPvv) regimes. The method is a bijection φ from the ergodic component E_N of the FEP to the state space of a simple exclusion process, under which the FEP generator is mapped to the SEP generator. The known hydrodynamic limits of the SEP are then transferred back through the macroscopic version of φ, yielding the fast diffusion equation with Neumann boundary conditions (13), the convection-diffusion equation with Robin boundary conditions (14), and the scalar conservation law with Dirichlet boundary conditions (15). Section 5 contains explicit computations showing that the macroscopic mapping sends the SEP PDEs to the FEP PDEs.","tokens_in":16567,"tokens_out":9218,"duration_ms":97236,"significance":"If completed, the paper would provide a clean and reasonably general transfer principle: hydrodynamic limits for several FEP variants with closed boundaries are deduced from the corresponding SEP results. The explicit microscopic construction of φ, the macroscopic density transformation, and the careful treatment of entropy solutions are valuable and appear correct in the parabolic regimes. The paper does not rely on fitted parameters; the limiting PDEs are derived from the dynamics. However, the hyperbolic case is not rigorously established as written because of a gap in the convergence argument in Lemma 2, and the central microscopic conjugacy Theorem 1 is stated without proof.","major_comments":[{"comment":"The proof of L1 convergence uses the estimate ||ω^ε_t ∘ v^ε_t − ω_t ∘ v_t||_{L1([0,1])} ≤ ||∂_v ω^ε_t||_{L∞} ||v^ε_t − v_t||_{L∞} + (1/m)||ω^ε_t − ω_t||_{L1([0,1])}. The paper asserts that the first term vanishes because ω^ε is smooth and v^ε converges uniformly to v. This is not justified: smoothness alone provides no uniform bound on ||∂_v ω^ε||_{L∞}, and for the parabolic regularization (33) the spatial gradients of ω^ε are expected to blow up as ε→0, while ||v^ε−v||_{L∞} is only controlled by ||ω^ε−ω||_{L1}. No estimate is given showing that the product tends to zero. Since this L1 convergence is the step that identifies ρ as an entropy solution and supplies the boundary traces, the proof of Theorem 2 for AFEPvv with κ∈(1/2,1) is incomplete as written.","section":"§5.2, Lemma 2"},{"comment":"The microscopic mapping theorem is stated without proof. This is a load-bearing step: it is the only mechanism by which the FEP dynamics is transferred to the SEP dynamics, and all subsequent hydrodynamic statements depend on it. The construction with fictive boundary particles, the numbering convention, and the treatment of the rate factor N^{−κ} need a full verification, or at least a precise reference that covers the finite-interval closed-boundary setting. The sentence 'proved by straightforwardly checking all possible transitions' and the citation to [2] are not sufficient for a central conjugacy result of this kind.","section":"§3.1, Theorem 1"},{"comment":"The hydrodynamic limits for the SEP are imported via the slow-reservoir equivalence of [13] and the results of [3,7,17]. For the asymmetric case (26) in particular, the paper should verify that the hypotheses of [13] hold for the closed system, namely attractiveness and the precise slow-reservoir scaling, and should state exactly which theorem of [17] applies to the closed-boundary ASEPvv process with the acceleration Θ_N = N^{1+κ}. As written, this transfer input is only cited, not checked, and it is a substantial part of the proof of Theorem 2.","section":"§4.2, Proposition 2"}],"minor_comments":[{"comment":"The upper limit of the sum in the definition of L^TA_M is written as N−2; it should be M−2.","section":"§3.1, Theorem 1, Eq. (11)"},{"comment":"Reference [8] contains a typographical error: 'arXiv:2401:16535' should be 'arXiv:2401.16535'.","section":"References"},{"comment":"The notation ⟨·,·⟩ is introduced as the scalar product in L²([0,1]), but the test functions in (35) also depend on time; the text should clarify that the time integral is taken separately.","section":"§5.2, after Eq. (35)"},{"comment":"The passage from a deterministic sequence of configurations to a random initial law via Skorokhod and dominated convergence is correct in spirit, but the random lattice size M = M(η(0)) should be handled explicitly, since Proposition 2 is stated for a deterministic sequence M_N.","section":"§4, proof of Theorem 2"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the gap in Lemma 2 for the AFEPvv case; this is a load-bearing error that needs a genuine repair, not merely a reference. The parabolic regimes appear sound subject to a proof of Theorem 1. The paper relies heavily on prior works of the authors ([8] and [17]), which is acceptable if the cited statements are precisely matched to this setting; the editor may wish to have the relevance of [17, Theorem 2.8] and [13] independently checked."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a genuine step forward. It proves the hydrodynamic limit for the facilitated exclusion process on a finite interval with impermeable walls, in the symmetric, weakly asymmetric, and (attempted) hyperbolic regimes, by coupling to the simple exclusion process. The weak-solution cases look solid; the hyperbolic case has a proof gap in Lemma 2 that must be addressed.\n\nThe microscopic bijection φ is explicit and well-motivated, and the induced macroscopic map Φ is handled with care. Proposition 1 (the macroscopic mapping) is clearly argued, and the computations in Section 5.1 for weak solutions check out: the heat equation maps to the fast diffusion equation, and viscous Burgers maps to the convection-diffusion equation with the correct boundary terms. The paper is also honest about the initial condition being restricted to the high-density ergodic component, and it cites prior work properly, including the coupling ideas from [2] and [10].\n\nThe soft spots are these. Theorem 1, the exact mapping of generators, is stated without proof; \"straightforward checking\" is acceptable for a lemma but a referee will want the transitions spelled out. The SEP hydrodynamic limits are imported from [13] via the slow-reservoir equivalence; that is a legitimate shortcut, but it makes the proof depend on a substantial external result. The main issue is in Lemma 2, the hyperbolic case. The proof that ρ^ε converges to ρ in L1 uses the bound ||ω^ε∘v^ε - ω^ε∘v||_{L1} ≤ ||∂_vω^ε||∞ ||v^ε-v||∞. The derivative ||∂_vω^ε||∞ is not uniformly bounded for the viscous approximation (33); for a smeared shock it blows up like 1/ε, and ||v^ε-v||∞ is only controlled by the L1 error of ω^ε, which is O(ε) at best. The product need not vanish. The sentence \"as ω^ε is smooth and v^ε converges uniformly\" does not justify the limit. This leaves the AFEPvv case of Theorem 2 unproved as written. The gap is likely fixable—BV estimates or a different composition lemma would probably close it—but it is real.\n\nNo signs of fitting or invented entities; the assumptions are explicit and the PDEs are derived from the dynamics. Who is this for? Researchers in hydrodynamic limits for kinetically constrained particle systems. It deserves a serious referee, but the hyperbolic case needs a rigorous fix before acceptance.","headline":"Weak-solution cases are in good shape, but the hyperbolic AFEPvv case has a real gap in Lemma 2 that needs fixing before the paper is complete.","tokens_in":17133,"tokens_out":5137,"would_cite":true,"duration_ms":53083,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82C22","35K55","35L65"],"pacs":[],"model":"deepseek-v4-flash","headline":"For the facilitated exclusion process with closed boundaries, the hydrodynamic limit in the symmetric, weakly asymmetric, and asymmetric vanishing-viscosity regimes is obtained by coupling the process to a simple exclusion process through…","keywords":["Facilitated exclusion process","Hydrodynamic limit","Simple exclusion process","Closed boundaries","Fast diffusion equation","Conservation law","Kinetically constrained model","Entropy solution"],"falsifier":"Simulate the closed symmetric FEP from a smooth ergodic initial profile strictly above $1/2$, with $\\sigma=1$, and compare the empirical measure at a fixed time with the unique weak solution of (13) started from the same profile. The theorem predicts convergence in probability for every test function; observing a systematic loss of mass near the walls, which the Neumann condition forbids, or a limiting profile that solves a different PDE would falsify the claim.","tokens_in":16086,"feed_emoji":"🧮","tokens_out":8348,"duration_ms":78216,"temperature":0.7,"pith_summary":"The paper proves the hydrodynamic limit of the facilitated exclusion process (FEP) on a finite one-dimensional segment with impermeable walls, in the symmetric, weakly asymmetric, and asymmetric vanishing-viscosity regimes, when the process starts in its high-density ergodic phase. The limit is expressed by a deterministic density profile that solves a specific PDE dictated by the regime: a fast diffusion equation with Neumann boundary conditions, a convection-diffusion equation with Robin boundary conditions, or a scalar conservation law with Dirichlet boundary conditions. The proof does not analyze the kinetically constrained dynamics directly. Instead, it constructs an exact one-to-one map from the ergodic FEP configurations to configurations of the simple exclusion process (SEP), so the FEP becomes a SEP under a simple change of coordinates. Known hydrodynamic limits for SEP with closed boundaries then transfer through the same map to the FEP equations.","feed_headline":"One exact map gives the hydrodynamic limits of facilitated exclusion","feed_subtitle":"A bijection to the simple exclusion process turns known closed-boundary results into PDE limits for the kinetically constrained model.","key_machinery":"The load-bearing object is the bijection $\\varphi$ from ergodic FEP configurations, meaning configurations with no two adjacent empty sites, to SEP configurations. Starting from a configuration, one adds a fictive particle at each wall, numbers the particles, and marks a SEP site occupied exactly when the corresponding FEP particle has a particle to its right; the SEP lattice has one site per FEP particle. The map is one-to-one and turns the FEP generator with rates $\\sigma$ and $\\sigma+pN^{-\\kappa}$ into the SEP generator with the same rates, so closed boundaries become closed boundaries automatically. At the macroscopic level $\\varphi$ lifts to the profile map $\\omega = \\Phi(\\rho) = (2\\rho-1)/\\rho$ composed with the cumulative-mass reparametrization, and this map sends each known SEP hydrodynamic equation to the FEP equation claimed in Theorem 2.","core_discovery":"The central discovery is Theorem 2: for initial laws supported on the ergodic component $E_N$ and associated with a continuous density profile $\\rho_{\\mathrm{ini}}$ with $\\inf \\rho_{\\mathrm{ini}} > 1/2$, the empirical measure of the closed-boundary FEP converges in probability to $\\rho_t(u)\\,du$, where $\\rho_t$ is the unique weak solution of the fast diffusion equation with Neumann boundary conditions for the symmetric and very weakly asymmetric cases, the unique weak solution of the convection-diffusion equation with Robin boundary conditions in the weakly asymmetric case, and the unique entropy solution of the scalar conservation law with Dirichlet boundary conditions for the asymmetric vanishing-viscosity case with $\\kappa \\in (1/2,1)$. The densities are related to those of the auxiliary SEP through the macroscopic map $\\omega = (2\\rho-1)/\\rho$ after a mass-preserving change of coordinates; this is what turns the heat equation into fast diffusion, viscous conservation laws into convection-diffusion, and the first-order conservation law into the FEP conservation law.","pith_inferences":["The same coupling suggests an extension to FEP in contact with reservoirs: any SEP hydrodynamic result with slow or fast boundaries should transfer to a FEP result whenever the reservoir interaction can be expressed in the $\\varphi$-coordinates, giving a route to boundary-driven phase diagrams beyond the closed case.","The strict condition $\\inf \\rho_{\\mathrm{ini}} > 1/2$ and the restriction $\\kappa>1/2$ look technical rather than structural; a natural next step is to approximate profiles that touch $1/2$, in which case the limiting equation would be expected to develop a free boundary of Stefan type rather than stay in the same PDE class.","One could test numerically whether the entropy-solution boundary values $1/2$ and $1$ in the AFEPvv regime are selected by the vanishing-viscosity limit of the microscopic asymmetric jumps, or whether the finite-rate walls produce a boundary layer that the current leading-order limit does not resolve."],"forward_implications":["For the closed symmetric FEP, and also the very weakly asymmetric case $\\kappa>1$, the macroscopic density obeys the fast diffusion equation $\\partial_t \\rho = \\sigma \\partial_u^2((2\\rho-1)/\\rho)$ with zero flux of $a(\\rho)=(2\\rho-1)/\\rho$ at both walls.","In the weakly asymmetric regime $\\kappa=1$, the limit is the convection-diffusion equation $\\partial_t \\rho = \\sigma \\partial_u^2 a(\\rho) - p \\partial_u h(\\rho)$ with the Robin-type boundary balance $\\sigma\\partial_u a(\\rho) - p h(\\rho) = 0$ at each boundary.","In the asymmetric vanishing-viscosity regime $\\kappa\\in(1/2,1)$, the limit is the entropy solution of $\\partial_t \\rho + p \\partial_u h(\\rho)=0$ with boundary values fixed at $1/2$ on the left and $1$ on the right.","Because the same bijection works for every regime, the boundary conditions at the walls are not imposed by hand; they are forced by the mapping from the closed SEP, whose hydrodynamic limits are already known.","The convergence holds for every time $t\\ge 0$ once the process starts inside the ergodic component, so no waiting time for transience enters the statement."],"supporting_citations":[{"why":"Supplies the original FEP-to-SEP mapping that the paper generalizes to finite closed boundaries.","marker":"[2]"},{"why":"Provides the construction of initial measures concentrated on the ergodic component and associated to continuous profiles bounded away from 1/2.","marker":"[8]"},{"why":"Gives the hydrodynamic limit of the symmetric exclusion process with slow boundaries, from which the closed SSEP limit is deduced.","marker":"[3]"},{"why":"Gives the hydrodynamic limits for weakly and very weakly asymmetric exclusion with slow boundaries, used for the WAFEP and vWAFEP cases.","marker":"[7]"},{"why":"Supplies the entropy-solution hydrodynamic limit of one-dimensional ASEP with reservoirs, used for the AFEPvv case.","marker":"[17]"},{"why":"Establishes that closed attractive dynamics are exponentially close to dynamics with slow reservoirs, transferring the boundary results to closed SEP.","marker":"[13]"},{"why":"Introduces the Otto-type boundary conditions used to define entropy solutions on the bounded interval.","marker":"[14]"},{"why":"Provides the existence and uniqueness theory for entropy solutions of scalar conservation laws, invoked for the AFEPvv limit.","marker":"[15]"},{"why":"Supplies the representation theorem used to extend the hydrodynamic limit from deterministic initial configurations to random initial laws.","marker":"[4]"}],"fun_headline_variants":["One map gives FEP hydrodynamics from SEP limits","Bijection turns SEP results into FEP PDE limits","Facilitated exclusion hydrodynamics via a single map","Closed-boundary FEP limits follow from a mapping"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument runs entirely through the bijection $\\varphi$, which is defined only on configurations with no two adjacent empty sites, so the initial measure must live on that set and the initial density profile must stay strictly above $1/2$; at the critical density the map collapses and the claimed PDEs are not the right description.","fun_headline_variants_meta":{"raw":{"variants":["One map gives FEP hydrodynamics from SEP limits","Bijection turns SEP results into FEP PDE limits","Facilitated exclusion hydrodynamics via a single map","Closed-boundary FEP limits follow from a mapping"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000131,"raw_usage":{"total_tokens":1066,"prompt_tokens":823,"completion_tokens":243,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":439,"completion_tokens_details":{"reasoning_tokens":180}},"tokens_in":439,"tokens_out":243,"duration_ms":3343,"temperature":1.0,"reasoning_tokens":180,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T17:52:30.418469+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the closed symmetric FEP from a smooth ergodic initial profile strictly above $1/2$, with $\\sigma=1$, and compare the empirical measure at a fixed time with the unique weak solution of (13) started from the same profile. The theorem predicts convergence in probability for every test function; observing a systematic loss of mass near the walls, which the Neumann condition forbids, or a limiting profile that solves a different PDE would falsify the claim.","supporting_citations":[{"cited_title":"Ayyer, S","cited_arxiv_id":null,"evidence_quote":"Supplies the original FEP-to-SEP mapping that the paper generalizes to finite closed boundaries."},{"cited_title":"Da Cunha, C","cited_arxiv_id":null,"evidence_quote":"Provides the construction of initial measures concentrated on the ergodic component and associated to continuous profiles bounded away from 1/2."},{"cited_title":"Baldasso, O","cited_arxiv_id":null,"evidence_quote":"Gives the hydrodynamic limit of the symmetric exclusion process with slow boundaries, from which the closed SSEP limit is deduced."},{"cited_title":"Capit ˜ao and P","cited_arxiv_id":null,"evidence_quote":"Gives the hydrodynamic limits for weakly and very weakly asymmetric exclusion with slow boundaries, used for the WAFEP and vWAFEP cases."},{"cited_title":"Xu, Hydrodynamics for One-Dimensional ASEP in Contact with a Class of Reservoirs","cited_arxiv_id":null,"evidence_quote":"Supplies the entropy-solution hydrodynamic limit of one-dimensional ASEP with reservoirs, used for the AFEPvv case."},{"cited_title":"Weak reservoirs are superexponentially irrelevant for misanthrope processes","cited_arxiv_id":"2310.17038","evidence_quote":"Establishes that closed attractive dynamics are exponentially close to dynamics with slow reservoirs, transferring the boundary results to closed SEP."},{"cited_title":"Otto, Initial-boundary value problem for a scalar conservation law","cited_arxiv_id":null,"evidence_quote":"Introduces the Otto-type boundary conditions used to define entropy solutions on the bounded interval."},{"cited_title":"M ´alek, J","cited_arxiv_id":null,"evidence_quote":"Provides the existence and uniqueness theory for entropy solutions of scalar conservation laws, invoked for the AFEPvv limit."},{"cited_title":"Billingsley, Convergence of Probability Measures","cited_arxiv_id":null,"evidence_quote":"Supplies the representation theorem used to extend the hydrodynamic limit from deterministic initial configurations to random initial laws."}],"review_version":1}