{"id":"69f40c18-d985-48de-8f84-08a0d5a0d739","arxiv_id":"2411.17653","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a symmetric exclusion process with general non-reversible boundary reservoirs, the paper proves the hydrodynamic limit (heat equation with nonlinear Robin conditions) and a dynamical large deviations principle.","lead":"This paper proves two laws for a line of particles where each site holds at most one particle and the ends can create or destroy particles in a small window. The laws describe the average particle density in the large-system limit and the exact rate of rare fluctuations away from that average.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Concavity of B_k, D_-k is not a mild technicality: admissible l=2 non-reversible rates yield B_1(alpha)=c(1-alpha)^2, convex, so Theorem 2.6's scope is narrower than the advertised 'general' boundary rates.","rationale":"The reader's weakest-assumption analysis is on target. The concavity hypothesis is not cosmetic: it enters exactly where the LDP rate function must be a good rate function and where arbitrary finite-rate paths are approximated by smooth ones. The explicit l=2 example makes the restriction concrete and shows that the theorem's domain is not 'very general rates'. I would not move the verdict: the paper is a plausible CONDITIONAL because the missing concavity characterization and the deferred or omitted proof steps (Theorems 4.12, 4.14, Proposition C.6, and the suspicious (4.36) display) are fixable but leave the advertised LDP not fully proven. The hydrodynamic limit part is well supported by tightness, replacement lemmas, and the uniqueness appendix. Thus the correct disposition remains CONDITIONAL, with the same confidence as the reader.","tokens_in":49711,"tokens_out":16930,"duration_ms":159714,"concrete_test":"Fix l=2 and define R^-(eta,xi)=c(1-eta(2 epsilon_N)) if xi flips site epsilon_N from 0 to 1 and equals eta elsewhere, and R^-(eta,xi)=0 otherwise. Choose a similar right-boundary rate (e.g., destruction depending on a neighbor). First check irreducibility of the superposition of this boundary dynamics with the Kawasaki generator restricted to the two-site window. Then compute B_1(alpha)=c(1-alpha)^2 and B_1''(alpha)=2c>0, proving concavity fails. If the authors keep the theorem as is, this example establishes that the advertised generality is false; if they add the concavity hypothesis to the model section, the theorem becomes conditional on a class that has not been characterized.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central LDP (Theorem 2.6) is stated only under the concavity of B_k and D_-k in (2.18). This is load-bearing: Theorem 4.5 uses it for convexity and compact level sets, and Lemma 4.22 uses it for the I-density approximation. Concavity is not a consequence of the model's standing assumptions (irreducibility, R±>=0), and the paper verifies it only for the Appendix B example and claims it for [4] without a general characterization. A simple admissible family shows the restriction bites. Take l=2 and, for the left boundary, allow exactly the transition that flips site epsilon_N from 0 to 1, with rate c(1-eta(2 epsilon_N)); set all other boundary transition rates to 0. With the bulk Kawasaki exchanges inside the window, the combined boundary-plus-bulk dynamics is irreducible. Then B_1(alpha)=E_{nu_alpha}[R^- 1_{Sigma xi = Sigma eta + 1}]=c(1-alpha)^2, whose second derivative is 2c>0, so B_1 is strictly convex and the hypothesis fails. This rate is a local, non-reversible creation rule of exactly the kind the abstract calls 'very general', so Theorem 2.6 as written does not cover it. The paper needs either a proof that concavity follows from natural conditions on R±, a characterization of the admissible class, or an explicit statement that the LDP is restricted to concave boundary rates. There is also a suspicious inequality in Lemma 4.27: the bound (4.36) as printed would be infinite for a>0, and the later |a|^l domination is much weaker than the correct |a| log |a| form; a corrected domination is needed before the dominated-convergence step. Both are addressable, not reasons to reject the hydrodynamic limit, but they underscore that the LDP claim is not fully established as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a one-dimensional symmetric simple exclusion process in mild contact with boundary reservoirs through windows of fixed size l with general non-reversible rates R±. The main results are a hydrodynamic limit (Theorem 2.2) identifying the empirical density with the unique weak solution of the heat equation with nonlinear Robin boundary conditions (2.10), and a dynamical large deviations principle (Theorem 2.6) for the empirical measure with speed N and a variational rate function I[0,T](·|γ), under a concavity assumption on the mean creation/destruction rates B_k, D_{-k} and a C^{2+β} initial profile. An example (Exclusion l3) shows multiple stationary profiles while the evolution equation is unique.","tokens_in":50080,"tokens_out":15599,"duration_ms":130783,"significance":"The hydrodynamic limit part is supported by a fairly detailed proof, including tightness, replacement lemmas, energy estimates, and a uniqueness theorem for nonlinear Robin problems in Appendix C; this is a genuine contribution. The LDP, if fully established, would be a significant step toward macroscopic fluctuation theory for non-reversible boundary-driven systems with nonlinear boundary conditions. However, the LDP proof has major omissions and at least one incorrect estimate, and the concavity hypothesis is not benign; the advertised 'very general rates' is narrower than what Theorem 2.6 actually covers. The paper is therefore not yet ready for publication, but the core ideas are promising.","major_comments":[{"comment":"The concavity of B_k and D_{-k} in (2.18) is a real restriction and does not follow from the standing assumptions. Consider l=2 with left-window rates that create a particle at epsilon_N from the empty state 00 and from the state 01 with rates 2c and c, respectively, and that also have positive destruction rates (e.g., 10->00, 01->00, 11->10, 11->01) so that the boundary-plus-bulk dynamics in the window is irreducible. Then B_1(alpha)=2c(1-alpha)^2 + c alpha(1-alpha) = c(2 - 3 alpha + alpha^2), whose second derivative is 2c>0, so B_1 is strictly convex. This shows that the admissible class is strictly larger than the concave class, and the paper does not characterize which natural rates satisfy concavity. The authors should either prove concavity under natural conditions, characterize the concave class, or explicitly state in the abstract and introduction that the LDP is restricted to the concave case.","section":"Section 2.4, Theorem 2.6 and Remark 2.7"},{"comment":"The energy estimate for the upper bound (Theorem 4.12) is stated with the remark 'The details are left to the reader,' and the hydrodynamic limit of the tilted process (Theorem 4.14) is dispatched with 'We leave the details to the reader.' Both results are essential: Theorem 4.12 controls the compactness of level sets and the upper bound, and Theorem 4.14 is the basis of the entropy identity (4.23) for the lower bound. The paper must provide full proofs or a precise reduction to [9] and [12] that accounts for the nonlinear Robin boundary terms introduced here.","section":"Section 4.3.2, Theorem 4.12; Section 4.4.1, Theorem 4.14"},{"comment":"The estimate for Phi^{-}_t(a) is incorrect as printed. The displayed bound sup_x {a x + C(e^{-|x|l} - 1)} is +infinity for a>0, since the linear term dominates as x tends to infinity. Consequently, the bound (4.37) is not justified, and the dominated convergence argument for I^(2) is not established. The correct growth of the Legendre transform of the boundary cost is of order |a| log |a| (or a log a), not |a|^l log(|a|^l). This is a load-bearing step in the proof that Pi_4 is I-dense (Theorem 4.18).","section":"Section 4.4.2, Lemma 4.27, inequality (4.36)"},{"comment":"The assertion after (3.10) that P_{nu_N}(M^H_N(t)=0 for all t and all H)=1 is false: M^H_N(t) defined in (3.5) is a martingale, not the zero process. The proof later uses this claim to conclude that the term in (3.18) vanishes. The correct argument would use the previously stated L^2 convergence of the martingale, lim_N E[M^H_N(t)^2]=0, together with Doob's inequality. As written, the hydrodynamic limit proof has a gap, though it appears fixable.","section":"Section 3.2, equations (3.5)-(3.10)"}],"minor_comments":[{"comment":"The definition D_{-k}(alpha)=B_k(alpha) cannot be correct, since it would identify destruction with creation; the formula for b± in (2.16)-(2.17) and the example in Appendix B use different coefficients for creation and destruction. It should be D_{-k}(alpha)=E_{nu_alpha}[sum_xi R±(eta,xi) 1_{sum xi = sum eta - k}]. Please correct this typo.","section":"Equation (2.18)"},{"comment":"In the definition of C^{n,m}_0(Omega_T), the phrase 'that below to C^{n,m}' should read 'that belong to C^{n,m}'.","section":"Section 2.2"},{"comment":"The sentence describing the attained supremum, 'x = (-1)/BD a>0 1/l log(|a|/(Cl))', is garbled and should be replaced by a clean formula such as x = (1/l) log(a/(Cl)) for a>0, with the corresponding expression for a<0.","section":"Section 4.4.2, proof of Lemma 4.27"}],"recommendation":"major_revision","confidential_remarks":"The paper has a promising structure and the hydrodynamic part is largely sound, but the LDP proof is not self-contained: several key results are delegated to 'left to the reader' and the incorrect estimate (4.36) breaks the lower-bound argument as written. The concavity assumption also narrows the advertised generality. I believe the issues are likely fixable, but they require substantial revision and re-verification, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the hydrodynamic limit (Theorem 2.2) looks right, and a lot of the proof is actually in the paper: tightness, replacement lemmas, energy bounds, and a real uniqueness theorem (Theorem C.1) for nonlinear Robin boundary conditions. That part deserves a careful read and will be useful. Second, the LDP (Theorem 2.6), which the abstract advertises, is not fully established as written. The concavity assumption on B_k and D_-k is an assumption, not a consequence of the standing hypotheses, and several critical upper-bound and approximation steps are deferred to [9,12] or simply left to the reader. The paper is honest about some of this, but the gap between \"very general rates\" and \"concave rates\" is real and should be stated plainly in the abstract.\n\nWhat is genuinely new: the l-window boundary with arbitrary rates, the resulting nonlinear Robin hydrodynamic equation, the uniqueness theorem for that PDE, and an example with multiple stationary profiles but a unique evolution. That is a clear step beyond [4,11,22] and beyond the linear Robin case in [12]. No parameter fitting or normalization forces the results, and the self-citation pattern is not a problem when the cited results are the ones actually doing the work.\n\nThe main soft spot is exactly what the stress-test flag says. The l=2 example works: allow one left-boundary creation at rate c(1-eta(2 epsilon_N)), set the other boundary rates to zero, keep Kawasaki exchange inside the window. The dynamics is irreducible, and B_1(alpha) = c(1-alpha)^2, which is strictly convex. So Theorem 2.6 does not cover a natural non-reversible boundary family of exactly the kind the paper claims to treat. The paper should either characterize the admissible class, prove concavity from structural conditions on R±, or explicitly restrict the LDP statement to concave rates.\n\nThere is also a smaller technical issue in Lemma 4.27: the bound (4.36) with |a|^l is not the right growth for a Legendre transform of e^{-|x|}; the correct form should involve |a| log|a|. The domination step needs repair before the dominated convergence argument goes through. This is likely fixable, but as printed it is a gap.\n\nWho is this for? Researchers working on boundary-driven exclusion processes and macroscopic fluctuation theory. The hydrodynamic limit alone is worth a serious referee. The LDP needs substantial revision: clean up the concavity hypothesis, fill or explicitly import the deferred estimates, and correct the Lemma 4.27 bound. I would send it to peer review, but I would expect major revisions rather than acceptance in the present form.","headline":"The hydrodynamic limit is real and mostly well proved; the advertised dynamical LDP is narrower than claimed, since the concavity assumption is load-bearing and the proof leans on deferred results.","tokens_in":50639,"tokens_out":1907,"would_cite":true,"duration_ms":20590,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60F10","82C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a dynamical large deviations principle for exclusion processes whose boundary reservoirs inject and remove particles through general, possibly non-reversible rates.","keywords":["hydrodynamic limit","dynamical large deviations","exclusion process","non-reversible boundary","Robin boundary conditions","mild contact","boundary reservoirs"],"falsifier":"Compute B_1''(α) for a three-site boundary window with parameters a0,a1,a2 outside the Appendix B relation, for example with a2 much smaller than a+2b; the second derivative becomes positive somewhere, so concavity fails. To decide the theorem itself, one would look for such non-concave rates together with a smooth initial profile for which the rate function I_{[0,T]} stops being lower semicontinuous or has non-compact level sets, making the large-deviations upper and lower bounds impossible.","tokens_in":49503,"feed_emoji":"🧮","tokens_out":5177,"duration_ms":50338,"temperature":0.7,"pith_summary":"This paper treats a one-dimensional symmetric exclusion process in weak contact with reservoirs at both ends, where the boundary dynamics can create and destroy several particles at a time through arbitrary rates. It establishes two scaling results: the empirical density obeys a hydrodynamic limit given by the heat equation with nonlinear Robin boundary conditions, and, under a concavity condition on the boundary rates, the full trajectory law satisfies a large deviations principle with speed N and an explicit rate function. The significance is that the boundary and bulk stationary states need not agree, so the usual Bernoulli reference measures are not stationary for the full process; the paper shows the hydrodynamics and large deviations still hold, including cases where the hydrodynamic equation has several stationary profiles. This opens the way to studying metastability and transition times between stable density profiles.","feed_headline":"Large deviations proven for exclusion process with general reservoirs","feed_subtitle":"Hydrodynamic limit and large-deviation rate function now cover nonlinear Robin boundaries and multiple stationary profiles.","key_machinery":"The load-bearing objects are the expected boundary creation and destruction rates B_k(α) and D_{-k}(α), defined as expectations under Bernoulli product measures of density α of the rates at which the boundary adds or removes exactly k particles. Their difference gives the boundary flux F_±, and their exponential generating function b_±(α,M) supplies the boundary cost in the rate function. The argument is carried by a variational decomposition of the rate function into a bulk part, expressed as a weighted Sobolev norm of a current, plus a boundary part given by a Legendre transform of the b_± terms, together with an I-density approximation showing that any finite-cost path can be smoothed. Concavity of B_k and D_{-k} is used to make the rate function convex with compact level sets, and a uniqueness theorem for the nonlinear Robin initial-value problem, including its tilted analogue, closes the hydrodynamic and large-deviation proofs.","core_discovery":"On the paper's own terms, the central discovery is that the non-reversible boundary dynamics contribute to the dynamical large-deviation rate function through the boundary functionals b±(α,M), and that the whole rate functional is a good rate function whenever the expected k-particle creation and destruction rates B_k and D_{-k} are concave. In that regime the tilted dynamics have a hydrodynamic limit, and the lower bound is obtained by showing that every finite-cost trajectory can be approximated by smooth trajectories in a class of well-behaved paths. The companion hydrodynamic limit theorem identifies the macroscopic density as the unique weak solution of the heat equation with nonlinear Robin boundary conditions, with boundary fluxes given by the expectations F_-(ρ(t,0)) and F_+(ρ(t,1)). A concrete model, called the Exclusion l3 model, is exhibited for which the hydrodynamic equation has a unique time-dependent solution but several distinct stationary solutions.","pith_inferences":["One could test whether the concavity of B_k and D_{-k} is equivalent to convexity of a boundary free energy; if so, the rate function's good properties would follow from a more physical structural condition rather than a case-by-case check.","The multiple stationary solutions of the Exclusion l3 model suggest that the dynamical LDP is a natural first ingredient for an Eyring–Kramers type estimate of transition times between stable profiles, a question the paper does not address.","The authors' stated program of first taking the diffusive limit and then letting the reservoir intensity grow would turn this weak-contact LDP into a Γ-convergence route toward the rate function of strongly coupled reservoirs; the present theorem supplies the inner limit of that program.","For rates that violate concavity, the theorem as stated does not apply, but the rate functional may still be a valid LDP rate function in a non-convex form; the variational proof here would need a different convexity argument to cover such cases."],"forward_implications":["The density of the process converges to the unique weak solution of ∂tρ=Δρ with nonlinear Robin boundary conditions ∇ρ_t(0)=-F_-(ρ_t(0)) and ∇ρ_t(1)=F_+(ρ_t(1)).","Typical fluctuations away from this hydrodynamic path are exponentially rare, with speed N and cost given by the explicit rate function I_{[0,T]}(π|γ).","When the boundary rates admit more than one stationary density profile, the stationary large-deviations rate function has at least two critical points, signaling metastable behavior of the particle system.","The rate function splits into separate bulk and boundary costs, so the macroscopic cost of creating a density anomaly through the reservoirs can be compared directly with the cost of transporting density through the interior.","If the concavity assumption holds, the same proof covers the full class of mild-contact boundary dynamics, bringing the general non-reversible setting in line with the previously understood reversible and Dirichlet-style cases."],"supporting_citations":[{"why":"Supplies the decomposition of the rate function into bulk and boundary variational problems and the I-density approximation strategy used for the lower bound.","marker":"[12]"},{"why":"Introduces the predecessor model with specific non-reversible boundary rates; the present paper extends it to general rates and verifies concavity for those rates.","marker":"[4]"},{"why":"Provides the general entropy, relative-entropy, and Feynman–Kac machinery used for tightness, super-exponential estimates, and the standard large-deviations proof steps.","marker":"[1]"},{"why":"Gives the dynamical large deviations proof for boundary-driven exclusion whose energy estimates and super-exponential estimates are adapted here.","marker":"[9]"},{"why":"Underpins existence, uniqueness, and regularity for the nonlinear Robin boundary-value problem used in the hydrodynamic limit and in the tilted process.","marker":"[7]"}],"fun_headline_variants":["Exclusion rate function now covers non-reversible boundaries","Nonlinear Robin conditions: exclusion hydrodynamics and LDP","Multiple stationary states in exclusion with general reservoirs","Boundary non-reversibility shapes exclusion large deviation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the expected rates at which the boundary adds or removes k particles, B_k(α) and D_{-k}(α), being concave functions of the local density α on [0,1]; if a natural family of rates violates this assumption, the large deviations principle as stated may fail.","fun_headline_variants_meta":{"raw":{"variants":["Exclusion rate function now covers non-reversible boundaries","Nonlinear Robin conditions: exclusion hydrodynamics and LDP","Multiple stationary states in exclusion with general reservoirs","Boundary non-reversibility shapes exclusion large deviation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000318,"raw_usage":{"total_tokens":1704,"prompt_tokens":758,"completion_tokens":946,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":374,"completion_tokens_details":{"reasoning_tokens":884}},"tokens_in":374,"tokens_out":946,"duration_ms":60030,"temperature":1.0,"reasoning_tokens":884,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:53:21.149982+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute B_1''(α) for a three-site boundary window with parameters a0,a1,a2 outside the Appendix B relation, for example with a2 much smaller than a+2b; the second derivative becomes positive somewhere, so concavity fails. To decide the theorem itself, one would look for such non-concave rates together with a smooth initial profile for which the rate function I_{[0,T]} stops being lower semicontinuous or has non-compact level sets, making the large-deviations upper and lower bounds impossible.","supporting_citations":[{"cited_title":"and Neumann, A","cited_arxiv_id":null,"evidence_quote":"Supplies the decomposition of the rate function into bulk and boundary variational problems and the I-density approximation strategy used for the lower bound."},{"cited_title":"and Gon¸ calves, P","cited_arxiv_id":null,"evidence_quote":"Introduces the predecessor model with specific non-reversible boundary rates; the present paper extends it to general rates and verifies concavity for those rates."},{"cited_title":"and Landim, C","cited_arxiv_id":null,"evidence_quote":"Provides the general entropy, relative-entropy, and Feynman–Kac machinery used for tightness, super-exponential estimates, and the standard large-deviations proof steps."},{"cited_title":"Landim, C","cited_arxiv_id":null,"evidence_quote":"Gives the dynamical large deviations proof for boundary-driven exclusion whose energy estimates and super-exponential estimates are adapted here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Underpins existence, uniqueness, and regularity for the nonlinear Robin boundary-value problem used in the hydrodynamic limit and in the tilted process."}],"review_version":1}