{"id":"ff4a68a9-f6be-4d34-bfe2-676caa0074ff","arxiv_id":"2508.19447","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For theta >= 1, the empirical density of the symmetric zero-range process with slow boundary reservoirs converges in probability to the unique weak solution of the nonlinear heat equation with Robin (theta = 1) or Neumann (theta > 1) boundary conditions.","lead":"The paper proves that a certain particle system with slow boundary reservoirs converges, after rescaling, to a nonlinear heat equation with boundary conditions that depend on how slow the reservoirs are. It is the first rigorous derivation of this hydrodynamic limit for the symmetric zero-range process with nearest-neighbor jumps.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Boundary replacement lemma (Lemma 3.14) is stated but not proved; the theorem's θ-dependent boundary conditions rest entirely on it.","rationale":"The reader's weakest assumption was (SG), but Remark 2.8 already observes that only a positive lower bound for each fixed ℓ,j is needed, and such a bound exists automatically for the finite irreducible zero-range chain on a box; the ℓ^{-2} rate is stronger than necessary for the fixed-ℓ Rayleigh estimates. The genuinely load-bearing unverified step is the boundary replacement lemma. The paper explicitly declines to prove Lemma 3.14, and the proof of the quoted boundary condition in §3.4.1 passes through R6, whose vanishing is exactly that lemma. Since the boundary condition is the paper's main new content, leaving this lemma as an exercise is a substantive gap. The energy estimate in Appendix A.1 is also skeletal, but it is secondary to the boundary structure. These are addressable gaps, not demonstrated contradictions, so the existing CONDITIONAL verdict remains appropriate; no change is needed. The reader's rationale did mention the omitted boundary replacement lemma, so there is partial agreement even though the reader's weakest_assumption focused on (SG).","tokens_in":42719,"tokens_out":21906,"duration_ms":255886,"concrete_test":"Write out the full proof of Lemma 3.14 by adapting Lemma 3.9 to the half-box Λ={1,...,εN} with directed averages. Concretely: (i) prove the analogue of the one-block estimate (3.39) for the directed average ⃗η_s^{εN}(1), using the spectral gap of the zero-range process on a reflecting half-box; (ii) verify that each boundary term in the Feynman–Kac computation (3.43)–(3.48) is bounded by C N^{1−θ}/M plus terms that vanish after N→∞ and ε→0; (iii) check that no contribution of order 1/N or larger survives from the site-1 generator. If such a surface term survives, R6 does not vanish and the boundary conditions in Definition 2.9 are unjustified. If the bound goes through, the gap is closed and the central claim is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main novelty of Theorem 2.12 is the boundary condition in (2.23): Robin-type terms appear exactly when θ=1, and flux terms survive for θ>1. In §3.4.1 these boundary conditions are obtained by sending the remainder R6_{N,ε}(G,t) to zero, and the vanishing of R6 is exactly the content of Lemma 3.14. That lemma is not proved: §3.6 says the proof is 'almost identical' to the bulk argument and omits it. This is not a routine transcription. The directed averages in (3.28) are one-sided, the boxes touch the boundary, and the one-block/two-block arguments in Lemmas 3.12–3.13 are formulated for symmetric bulk boxes under a translation-invariant reference measure. Boundary reservoirs can create extra surface terms in the Feynman–Kac and Rayleigh estimates (3.43)–(3.48) that are not obviously of order N^{1−θ}/M after summing. If any such surface term survives the limits N→∞, ε→0, the limiting equation would acquire additional boundary contributions and (2.24) would be wrong. The energy estimate in Appendix A.1 is also delegated to KL99 with little adaptation, but the boundary replacement gap is more load-bearing because it directly controls the boundary conditions that distinguish this paper from the torus case.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the symmetric nearest-neighbour zero-range process on the finite interval {1,...,N-1} in contact with reservoirs at the boundary sites 1 and N-1, with injection/removal rates scaled by N^{-theta}, theta >= 1. Under assumptions that the jump rate g is nondecreasing with bounded increments, the initial measures have relative entropy O(N) with respect to the NESS and are stochastically dominated by it, and that a spectral-gap lower bound (SG) holds, the authors claim that the empirical density converges in probability to the unique weak solution of the nonlinear heat equation partial_t rho = Delta Phi(rho) with Robin boundary conditions for theta=1 and Neumann conditions for theta>1 (Eqs. (2.23)-(2.24)). The proof follows the entropy method: Dynkin martingales, tightness, absolute continuity of limit points, bulk one-block and two-block replacement lemmas, a boundary replacement lemma, and an energy estimate for the regularity of Phi(rho).","tokens_in":43045,"tokens_out":15712,"duration_ms":178118,"significance":"If the proof is completed, this would be the first rigorous hydrodynamic limit for the boundary-driven zero-range process in the nearest-neighbour case with slow reservoirs, closing a gap left open in [FMN21] and extending the slow-boundary exclusion results to a model with unbounded occupation numbers. The theta=1 Robin-type transition is a genuinely new feature for zero-range. The paper has clear strengths: the explicit product NESS is used intelligently, the translation-invariant reference measure is a good choice, Lemma 2.7 on the coupled two-box spectral gap is a useful tool, and the uniqueness proof for the nonlinear Robin problem is careful. The bulk replacement estimates are detailed. However, the boundary treatment is incomplete in a load-bearing way: the boundary replacement lemma is stated without proof, and the remainder R5 in Section 3.4.1 is asserted to vanish by an argument that does not apply to it. The energy estimate is also delegated to [KL99] with little adaptation. These gaps must be addressed before the main theorem can be considered established.","major_comments":[{"comment":"The proof asserts that E_mu[|R5|] tends to 0 'by the same argument given for R3'. This is not justified and is load-bearing. R3 is a bulk remainder with an explicit 1/N factor and a smooth factor Delta G_s; after summation by parts it is controlled by the continuity of Delta G. R5 has no such factor: it contains g(eta_s(1)) - (1/epsilon N) sum_{y=1}^{epsilon N} g(eta_s(y)) (and the analogous right-boundary term), multiplied by partial_u G_s(0), plus for theta=1 additional terms with G_s(0). A single-site rate does not converge in L1 to its spatial block average under product-type measures with slowly varying profile; the L1 difference is of order one. Vanishing of R5 is a boundary one-block replacement, not a corollary of the R3 estimate. Without a proof of this step, the boundary terms in (3.27) cannot be replaced by Phi of directed density averages, and the boundary conditions in (2.24","section":"Section 3.4.1, Eq. (3.27), R5"},{"comment":"The boundary replacement lemma is stated without proof, with only the remark that the proof is 'almost identical' to the bulk argument. This lemma is exactly what sends R6 to zero in Section 3.4.1 and hence produces the Robin (theta=1) and flux (theta>1) boundary terms in (2.24). The directed averages in (3.28) are one-sided, the boxes touch the boundary, and the reference measure is not translation-invariant near the boundary; the bulk one-block and two-block estimates (Lemmas 3.12-3.13) are formulated for symmetric bulk boxes. Surface terms in the Feynman-Kac/Rayleigh estimates (3.43)-(3.48) could survive after summation. A complete proof, or a precise reduction to Lemmas 3.12-3.13, is required.","section":"Section 3.6, Lemma 3.14"},{"comment":"The proof of the energy estimate stops at (A.3) with 'the exact same argument given in the proof of [KL99, Lemma 7.3]'. This estimate is used to prove Proposition 3.8, i.e., item ii) of Definition 2.9, namely Phi(rho) in L^2([0,T],H^1). The adaptation is not routine: the process has boundary reservoirs, the Dirichlet form D_N^0 in (3.44) is only the bulk part, and the auxiliary Lemmas B.1-B.3 involve boundary corrections. The reader cannot verify that the expression W_N is controlled without seeing the argument. Please include the details or a precise statement of the modified estimate.","section":"Appendix A.1, Lemma A.1 / Eq. (A.3)"}],"minor_comments":[{"comment":"The text says 'By the bulk replacement lemma, namely Lemma 3.14'; Lemma 3.14 is the boundary replacement lemma. The reference should be to Lemma 3.9.","section":"Appendix A.1, first paragraph"},{"comment":"In the last term there is a misplaced parenthesis: 'delta { g(eta_s(N-1) - 1/(epsilon N) ...' should read 'delta ( g(eta_s(N-1)) - 1/(epsilon N) ...'.","section":"Eq. (3.27)"},{"comment":"The statement uses 'sup_{|y|<=epsilon N}' inside the expectation while y subsequently appears in the sums; clarify whether y is a fixed index over which the sup is taken. Also the order of limits in the lemma statement should match the order used at the end of the proof (N -> infinity, then epsilon -> 0, then ell -> infinity).","section":"Lemma 3.13"},{"comment":"The notation 'Q_N' with 'N >= 0' appears to be a typo for 'N >= 1'.","section":"Remark 3.2"}],"recommendation":"major_revision","confidential_remarks":"The central result is plausible and the overall strategy is standard, but the omitted boundary replacement lemma and the unjustified vanishing of R5 are genuine proof gaps, not mere presentation issues. The energy estimate delegation is also a concern. I would not reject: these gaps are likely fixable with known entropy-method tools, but they must appear in the manuscript. The paper should be checked carefully for the R5 issue before acceptance, since it concerns the boundary conditions that constitute the main novelty."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves the hydrodynamic limit for the symmetric nearest-neighbor zero-range process in contact with slow reservoirs for θ ≥ 1. That is a real result: [FMN21] left it conditional on replacement lemmas, and here the limiting PDE—nonlinear heat equation with Robin boundary terms at θ=1 and Neumann at θ>1—is derived rather than assumed. The entropy method is adapted honestly: non-invariant reference measure, spectral gap for a coupled two-box process, one- and two-block estimates. The hydrostatic limit follows as a corollary. The central derivation is coherent and the theorem is believable.\n\nThe soft spots are the boundary replacement lemma and the energy estimate. Lemma 3.14 is stated, then omitted with the remark that the proof is 'almost identical' to the bulk argument. That sentence is doing too much work. The boundary conditions in (2.23) depend on exactly this lemma: the directed averages in (3.28) are one-sided, the boxes are anchored at the boundary, and the bulk proof uses translation invariance and symmetric boxes. Surface terms in the Rayleigh estimates (3.43)–(3.48) are not obviously of order N^{1−θ}/M after summing. I would not desk-reject over this, but I would not accept it either without seeing those estimates written out. The boundary replacement lemma is load-bearing, not a routine transcription.\n\nThe energy estimate in Appendix A.1 also ends with 'the exact same argument' from KL99 Lemma 7.3. That is less concerning because the boundary conditions there are classical and the cited lemma is standard, but still a referee should ask for the adaptation.\n\nAssumption (SG) is imported: spectral gap order ℓ^{-2} for the zero-range process on a box. The paper gives three classes of rates for which this is known. For the theorem as stated, this assumption is an explicit hypothesis, so that is acceptable; a reader should just note the theorem is conditional on (SG). The paper does state it as an assumption.\n\nThe citation pattern is reasonable. The paper leans on several works coauthored by the present authors, but in this area those are the standard tools, and I do not see evidence of self-citation inflation.\n\nVerdict: deserves a serious referee. The right report is 'accept after major revision' where the missing boundary replacement proof and the energy estimate are supplied. It would be a useful reading-group paper as well, especially to see how the entropy method handles non-invariant reference measures.","headline":"Rigorous hydrodynamic limit for a standard boundary-driven zero-range model, but the proof leans on an unproved boundary replacement lemma that carries the θ-dependent boundary conditions.","tokens_in":43517,"tokens_out":2234,"would_cite":true,"duration_ms":24041,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82C22","35K55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The zero-range process with slow boundary reservoirs has a rigorous hydrodynamic limit: its empirical density converges to the unique weak solution of a nonlinear heat equation, with boundary conditions that switch from Robin (at the critic","keywords":["hydrodynamic limit","zero-range process","slow boundary","Robin boundary condition","nonlinear heat equation","entropy method","replacement lemma","spectral gap"],"falsifier":"Run microscopic simulations for a specific jump rate that satisfies (ND) and (2.1) but violates (SG) — for example, a rate with strong blocking that makes the spectral gap decay faster than ℓ^{-2} — and test whether the empirical density still converges to the nonlinear heat equation with the stated boundary conditions; a persistent mismatch as N grows would refute the sufficiency of the theorem's hypotheses, while convergence would indicate that (SG) is not necessary.","tokens_in":42648,"feed_emoji":"","tokens_out":3334,"duration_ms":38620,"temperature":0.7,"pith_summary":"This paper proves that the large-scale density of a symmetric zero-range process on a finite interval, in contact with boundary reservoirs that inject and remove particles at rates slowed by a factor N^{-θ} with θ ≥ 1, converges in probability to the unique weak solution of a nonlinear heat equation. The limiting equation is ∂_t ρ = ΔΦ(ρ), where Φ is the flux function determined by the jump rate g, and the boundary conditions depend sharply on θ: for θ = 1 they are Robin-type, while for θ > 1 they degenerate to zero-flux Neumann conditions as the reservoirs become asymptotically invisible. This is the first rigorous derivation in this boundary-driven setting, closing a gap left open by an earlier heuristic result. The proof follows the entropy method but requires substantial new ingredients because the invariant measure with reservoirs is not translation-invariant; the authors replace it by the invariant measure of the dynamics restricted to the bulk and control the discrepancy via a spectral gap estimate.","feed_headline":"Zero-range density converges to nonlinear heat equation","feed_subtitle":"First rigorous hydrodynamic limit for boundary-driven zero-range process: slow reservoirs produce Robin conditions at the critical rate, Neu","key_machinery":"The central objects are the flux function Φ = R^{-1}, where R(φ) is the expected occupation under the product invariant measure with fugacity φ (so that Φ(α) is the expected jump rate at density α), and the entropy method with a translation-invariant reference measure: the invariant measure of the dynamics restricted to the bulk, denoted ¯ν_φ. The one-block estimate uses a spectral gap lower bound (Assumption (SG)) together with the Rayleigh estimate, while the two-block estimate requires a new spectral gap bound for a coupled two-box zero-range process. The indicator κ̃ = κ1_{θ=1} encodes the critical slowdown: only at θ = 1 do the reservoirs survive in the macroscopic boundary conditions.","core_discovery":"Theorem 2.12: for θ ≥ 1, the sequence of empirical measures π_t^N converges in probability to the absolutely continuous measure ρ(t,u)du, where ρ is the unique weak solution of the nonlinear heat equation ∂_t ρ = ΔΦ(ρ) with boundary conditions ∂_uΦ(ρ(0)) = κ̃(λΦ(ρ(0)) − α) and ∂_uΦ(ρ(1)) = κ̃(β − δΦ(ρ(1))), with κ̃ = κ1_{θ=1}. The proof is built on the entropy method of Guo, Papanicolaou and Varadhan, adapted to the non-conservative, non-translation-invariant setting by using the bulk-restricted invariant measure as a reference, establishing one-block and two-block replacement lemmas via spectral gap estimates, and proving a spectral gap bound for a coupled two-box process. The restriction θ","pith_inferences":["The method likely extends to long-range jumps, where the same entropy approach with a translation-invariant reference measure and replacement lemmas would produce convergence in probability to a fractional nonlinear heat equation—this is suggested by the paper's discussion of the exclusion analogue, but is not proved here.","For θ < 1 the paper's strategy would require the stationary measure as reference, at the cost of losing translation invariance; a plausible outcome is that Dirichlet boundary conditions emerge, but the necessary equivalence-of-ensembles and spectral gap estimates are not yet available.","The stochastic domination hypothesis µ_N ≤ ¯ν_N may be replaceable by a weaker uniform moment bound combined with a truncation argument, since its only role is to bound high-density configurations and moments of g; a testable variation is to verify the hydrodynamic limit for product initial measures with slowly varying parameter without imposing domination.","One can quantitatively test the critical slowdown: for θ = 1, the boundary flux should equal λΦ(ρ(0)) − α in the limit, while for θ > 1 it should vanish; finite-size simulations measuring the boundary flux for increasing N would directly probe the sharp transition."],"forward_implications":["The hydrostatic limit follows as a corollary: the stationary measure ¯ν_N converges to the stationary profile ¯ρ_θ, the solution of the elliptic problem with the corresponding Robin or Neumann boundary conditions.","For θ > 1 the reservoirs have no macroscopic influence: the limiting equation is the nonlinear heat equation with zero-flux Neumann boundary conditions, so the system behaves as isolated in the limit.","The uniqueness of the weak solution, proved in Lemma 2.10, ensures that the identification of the limit is complete: any limit point must equal the unique solution.","The replacement lemmas are stated for the jump rate g but extend to any Lipschitz cylinder function Ψ, yielding a Boltzmann-Gibbs-type principle for a whole class of observables, not just the jump rate.","The spectral gap assumption (SG) is only used in the replacement lemmas; the paper notes that a positive lower bound depending on g, ℓ and j would suffice."],"supporting_citations":[{"why":"Supplies the entropy method framework, the one-block/two-block replacement lemmas, the Rayleigh estimate, and the equivalence of ensembles used throughout the proof.","marker":"[KL99]"},{"why":"Original source of the entropy method for hydrodynamic limits, which the paper follows and adapts.","marker":"[GPV88]"},{"why":"Identified heuristically the limiting PDE and boundary conditions for this model, conditional on replacement lemmas; the present paper closes the gap rigorously.","marker":"[FMN21]"},{"why":"Provides the explicit product-form invariant measure with reservoirs (the NESS), which is essential for the entropy bounds and the identification of the flux function.","marker":"[LMS05]"},{"why":"Shows how to use a spectral gap bound and the Rayleigh estimate to obtain one-block estimates, which the paper adapts and extends to the two-block setting.","marker":"[FVST23]"},{"why":"One of the cited references proving the spectral gap bound (Assumption (SG)) for a large class of jump rates, which underpins the replacement lemmas.","marker":"[LSV96]"},{"why":"Provides the PDE tools for uniqueness of weak solutions with Robin boundary conditions, used in Lemma 2.10.","marker":"[BDGN20]"}],"fun_headline_variants":["Slow reservoir boundary drives zero-range to nonlinear heat equation","Robin boundary emerges from slow reservoirs in zero-range hydrodynamic limit","Slow reservoirs induce Robin boundary in zero-range limit","Zero-range density obeys nonlinear heat equation under slow boundary"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"Assumption (SG): the spectral gap of the zero-range process on a box of size ℓ with j particles must be bounded below by C/ℓ², uniformly in j; if this rate fails, the denominators in the Rayleigh estimates inside the replacement lemmas can lose positivity and the proof collapses.","fun_headline_variants_meta":{"raw":{"variants":["Slow reservoir boundary drives zero-range to nonlinear heat equation","Robin boundary emerges from slow reservoirs in zero-range hydrodynamic limit","Slow reservoirs induce Robin boundary in zero-range limit","Zero-range density obeys nonlinear heat equation under slow boundary"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000672,"raw_usage":{"total_tokens":2855,"prompt_tokens":662,"completion_tokens":2193,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":406,"completion_tokens_details":{"reasoning_tokens":2129}},"tokens_in":406,"tokens_out":2193,"duration_ms":17666,"temperature":1.0,"reasoning_tokens":2129,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T15:47:31.454863+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run microscopic simulations for a specific jump rate that satisfies (ND) and (2.1) but violates (SG) — for example, a rate with strong blocking that makes the spectral gap decay faster than ℓ^{-2} — and test whether the empirical density still converges to the nonlinear heat equation with the stated boundary conditions; a persistent mismatch as N grows would refute the sufficiency of the theorem's hypotheses, while convergence would indicate that (SG) is not necessary.","supporting_citations":[],"review_version":1}