{"id":"8daef80f-de71-4173-a7fa-24b542d14888","arxiv_id":"2507.18556","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors derive the large-deviation rate function for ionic current in a symmetric dilute electrolyte and show it is non-Gaussian away from equilibrium, with a boundary-layer mechanism at large voltage.","lead":"This paper derives the probability of rare, large fluctuations in the electric current flowing through a dilute electrolyte in a narrow channel. It shows those fluctuations are Gaussian near equilibrium but develop non-Gaussian tails and boundary-layer mechanisms under large voltages, and it gives formulas for those probabilities.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Most load-bearing gap: the contraction in Eq. 7 is over all space-time histories, but Sec. II B replaces it by time-independent optimizers without proof; if time-periodic profiles win, Eq. 22 and the boundary-layer mechanism are not the true rate function.","rationale":"The reader identified the time-independent optimizer assumption as the main structural risk. I agree. The paper itself flags that time-periodic traveling-wave solutions can emerge in periodic geometries, and it gives no proof or numerical stability analysis excluding them for the reservoir geometry. The finite-difference checks are all internal to the stationary Euler-Lagrange framework, so they cannot test the contraction in Eq. 7 against genuine time-dependent histories. This matters because the large-potential rate function Eq. 22, the boundary-layer mechanism, the Gallavotti-Cohen symmetry check, and the thermodynamic uncertainty relation are all derived from the stationary saddle. A time-dependent optimizer would invalidate the reported rate function rather than merely adjust its prefactor. The paper is otherwise coherent: the small-potential perturbative expression is consistent with the mean current and variance, and the symmetry reductions are plausible and partially checked numerically. The correct response is therefore to keep the reader's CONDITIONAL verdict, now explicitly conditioned on settling the time-dependent variational problem.","tokens_in":11490,"tokens_out":32752,"duration_ms":360381,"concrete_test":"Run a time-dependent MFT optimization for the same channel: discretize Eq. 5 on a space-time grid (e.g., Chebyshev in x, long time horizon with periodic-in-time relaxation), with reservoir boundary conditions, alpha=5 (beta z Delta_phi=10) and |j_q/<j_q>|=2. Compute the minimum action allowing arbitrary rho_i(x,t), then compare to the stationary value from Eq. 22. Alternatively, linearize the time-dependent MFT Hamilton equations about the stationary profile Eq. 21 and search for an unstable mode. If the time-dependent action is lower by more than numerical tolerance, or an unstable mode exists, the stationary ansatz is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central result is the rate function I(j_q) obtained from the contraction in Eq. 7. That contraction is a minimization over all space-time density fields. In Sec. II B the authors replace it by a minimization over time-independent profiles, stating that time dependence vanishes and noting that traveling-wave optimizers can occur under periodic boundary conditions. No argument is given that the reservoir geometry excludes such time-dependent optimizers. Every subsequent comparison—Eqs. 18 and 22 against finite-difference solutions—is a comparison against solutions of the same stationary Euler-Lagrange equations, so it cannot detect a time-dependent saddle with lower action. If the global minimizer is a time-periodic field, the derived EL equations, the boundary-layer mechanism, the GC symmetry check, and the TUR saturation are all statements about the wrong control problem, and I(j_q) in Eqs. 18/22 is not the large-deviation rate function. The authors' explicit admission makes this the load-bearing uncertainty.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper applies macroscopic fluctuation theory to a stochastic Poisson-Nernst-Planck description of a dilute, symmetric electrolyte in a one-dimensional channel. It derives Euler-Lagrange equations for the optimal density profiles conditioned on a prescribed ionic current, solves them analytically in the limits of small and large applied potentials, and obtains explicit rate functions for the long-time current distribution: a Gaussian near the mean with exponential tails at small voltage (Eqs. 16-18) and a strongly asymmetric, non-Gaussian rate function generated by boundary layers at large voltage (Eqs. 19-22). The paper also checks Gallavotti-Cohen symmetry and the thermodynamic uncertainty relation against these rate functions. The central claim is that ionic current fluctuations are generically non-Gaussian, with rare events controlled by delocalized density fluctuations at low voltage and by boundary-layer formation at high voltage.","tokens_in":11671,"tokens_out":4503,"duration_ms":49165,"significance":"If the results are correct, the paper provides a rare fully analytic example of current large deviations in a continuum electrolyte model, with parameter-free scaling forms and explicit symmetry checks. The comparison of the Gaussian small-voltage regime with the Nernst-Einstein conductivity and the identification of a boundary-layer mechanism at large voltage are conceptually useful for nanofluidic applications. The finite-difference solutions provide a numerical benchmark that supports the internal consistency of the stationary Euler-Lagrange framework. However, the unproven restriction to time-independent optimal profiles is a substantial caveat, because all figures compare against solutions of the same stationary equations and therefore cannot detect a competing time-periodic saddle.","major_comments":[{"comment":"The contraction in Eq. (7) is over all space-time density fields, but the paper restricts the minimization to time-independent profiles with the statement that time dependencies vanish and that traveling-wave optimizers occur only under periodic boundary conditions. No argument or numerical evidence is given that the reservoir geometry excludes time-periodic or other time-dependent saddles with lower action. Since Figs. 2-5 compare the analytical results only against finite-difference solutions of the same stationary Euler-Lagrange equations, they cannot validate this assumption. If a time-dependent field has lower action for some jq, then Eqs. (18) and (22), the boundary-layer mechanism, and the Gallavotti-Cohen and thermodynamic-uncertainty checks are statements about a constrained variational problem rather than about the true large-deviation rate function. I request either a proof of stationarity for this boundary condition (for example, by a convexity argument or a rigorous MFT result) or a direct numerical optimization over time-periodic fields for representative values of alpha and jq, and a corresponding adjustment of the claims if stationarity cannot be established.","section":"Sec. II B, Eq. (7)"},{"comment":"The central large-potential rate function is derived from the approximate optimal profile in Eq. (21), which is stated to be accurate for beta z |Delta phi| > 1. The manuscript does not provide an error estimate for the resulting rate function, particularly near jq = 0 where the non-analytic absolute-value terms dominate and for the threshold value beta z |Delta phi| = 1 where the crossover to the parabolic profile occurs. The agreement shown in Fig. 5 is qualitative ('indistinguishable to graphical accuracy'), and a quantitative relative-error plot as a function of alpha for several representative jq / <jq> would substantiate the claim that Eq. (22) is a controlled asymptotic result. This matters because the non-Gaussian tails and the subsequent symmetry checks rely on the accuracy of Eq. (22).","section":"Sec. III B, Eq. (22)"},{"comment":"The claim that both the perturbative and large-potential rate functions satisfy Gallavotti-Cohen symmetry is stated without derivation. For Eq. (18), the symmetry holds exactly because the bilinear term alone flips sign under jq -> -jq while the quadratic term is even. For Eq. (22), the presence of absolute values and inverse hyperbolic tangents makes the symmetry less transparent; I request an explicit algebraic verification that Eq. (22) satisfies I(jq) - I(-jq) = beta jq Delta phi for the entire domain of validity. Without this verification, the 'origin of the sharp linear feature' attributed to Gallavotti-Cohen symmetry is an assertion rather than a demonstrated property of the derived rate function.","section":"Sec. IV A, Eq. (26)"}],"minor_comments":[{"comment":"There are apparent typesetting errors: an unmatched opening bracket in the expression for I(jq) and the symbol 'q' before the square-root term, which should likely be a square-root sign. Please correct these and define all variables in the equation.","section":"Eq. (22)"},{"comment":"The summation notation \\sum_{i=\\pm} is unconventional; it should be \\sum_{i\\in\\{+,-\\}} or written explicitly. Also, the action's dependence on the noise amplitude appears only through D_i rho_i in the denominator; a brief comment on the Ito/Stratonovich convention for the noise term would help.","section":"Eq. (5)"},{"comment":"The assertion that P(j+,j-) takes the large-deviation form with rate function I2 is said to be 'a consequence of the expected finite correlation times'. Since this is the foundation for the entire contraction procedure, a reference to a proof or a more precise statement of the mixing condition would be appropriate.","section":"Sec. II B, Eq. (6)"},{"comment":"It would be helpful to state explicitly that Eq. (30) reduces to the variance quoted after Eq. (17) in the limit Delta phi -> 0, so that the reader can verify consistency between the small- and finite-voltage expressions.","section":"Sec. III A, Eq. (30)"}],"recommendation":"major_revision","confidential_remarks":"The paper is well written and the analytical results are potentially valuable, but the unproven stationary-saddle assumption is a genuine load-bearing gap. The authors should be asked to provide either a rigorous argument or a numerical search over time-dependent profiles before the claims about the true large-deviation rate function can be accepted. The derivation of Eq. (22) also needs a clearer error estimate and typesetting fixes. This is a case where the manuscript's scope can accommodate the requested additions, so revision rather than rejection is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core of this paper is a self-contained MFT calculation for a symmetric electrolyte in a 1D channel. The new thing is that the Poisson coupling makes the Euler-Lagrange equations nonlinear and coupled, and they still get explicit analytic rate functions: the parabolic profile at zero voltage, Eq. 15, the perturbative small-voltage expression Eq. 18, and the large-voltage boundary-layer mechanism with Eq. 22. They also show the rate function satisfies Gallavotti-Cohen symmetry and saturates the TUR in linear response. The finite-difference checks against their own EL equations are reassuring, and the derivation of Eq. 15 is exact. Credit where due: this is a useful addition to the MFT literature, and the boundary-layer picture for rare currents at high bias is a concrete and plausible mechanism.\n\nNow the soft spots. The important one, which the authors themselves flag in Sec. II B, is that the contraction in Eq. 7 is over all space-time histories, but they restrict to time-independent optimal profiles without proof. They mention traveling-wave instabilities under periodic boundary conditions but don't argue why reservoir boundary conditions exclude them. More to the point, every numerical check in Figs. 2-5 is against solutions of the same stationary EL equations, so those checks can't detect a time-dependent saddle with lower action. This is a real gap, though I don't see any concrete indication that such saddles exist here; it's an open technical assumption, not a demonstrated wrong answer.\n\nA secondary issue is that the large-potential formula Eq. 22 is presented as \"cumbersome\" to derive, and the derivation is compressed. They give the approximate profile and compare to numerics, which helps, but a referee will want more intermediate steps or a reproducible artifact. No code or data are provided, which limits how much the numerical claims can be independently checked.\n\nOverall, the paper is worth a serious referee. The central argument holds up as far as the stated model goes, and the time-independence assumption is a fair caveat rather than a fatal flaw. I'd send it to review and ask the authors to either prove or carefully bound the stationary-saddle assumption and to make the numerical finite-difference code available.","headline":"A self-contained MFT calculation that delivers explicit non-Gaussian rate functions for ionic currents in a 1D electrolyte channel; the main unresolved issue is the unproven restriction to stationary optimal profiles, which the authors do acknowledge.","tokens_in":12177,"tokens_out":2449,"would_cite":true,"duration_ms":25267,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"For a dilute electrolyte in a one-dimensional channel, the long-time distribution of ionic current is generically non-Gaussian, and the paper derives its rate function in both the small- and large-voltage limits.","keywords":["large deviations","ionic current","macroscopic fluctuation theory","Poisson-Nernst-Planck","electrolyte","non-Gaussian fluctuations","Gallavotti-Cohen symmetry","thermodynamic uncertainty relation"],"falsifier":"Simulate the stochastic Poisson-Nernst-Planck dynamics in a one-dimensional channel with $\\beta z\\Delta\\phi \\approx 4$ using a rare-event sampling method, and compare the measured rate function and conditional density profiles with Eqs. (21) and (22); observing traveling-wave optimal profiles, or rate functions that differ from Eq. (22), would falsify the stationary-profile result.","tokens_in":11247,"feed_emoji":"⚡","tokens_out":8346,"duration_ms":76563,"temperature":0.7,"pith_summary":"This paper determines how the time-averaged electric current through a dilute electrolyte channel fluctuates when it is observed for a long time. Starting from a stochastic Poisson-Nernst-Planck description, the authors derive the Euler-Lagrange equations for the most likely ion-density profile conditioned on a given current, and solve them analytically for a symmetric $z{:}z$ electrolyte. The result is that the current distribution is Gaussian only in a small window around its mean; rare fluctuations have exponential tails. At small applied voltage the rare events are delocalized density fluctuations across the channel, while at large voltage they are boundary layers near the channel ends. The paper also shows that the rate functions obey Gallavotti-Cohen symmetry and satisfy the thermodynamic uncertainty relation, which fixes the asymmetry of the distribution and the relation between noise and dissipation.","feed_headline":"Rare ionic currents go non-Gaussian in dilute electrolytes","feed_subtitle":"Rate functions from stochastic Poisson-Nernst-Planck theory show exponential tails and boundary-layer rare events.","key_machinery":"The central object is the macroscopic fluctuation theory action for the stochastic Poisson-Nernst-Planck equations, Eq. (5), a Gaussian action over joint fluctuations of ionic densities and currents. The rate function is obtained by contracting this action over optimal density profiles, which satisfy the coupled Euler-Lagrange equations Eq. (9) together with Poisson's equation Eq. (3). In the symmetric case the optimization separates into charge and mass sectors; the mass-current minimization sets the optimal charge density to zero, and the remaining equation is solvable analytically in the small- and large-potential limits, yielding the parabolic profile of Eq. (15) and the boundary-layer profile of Eq. (21). The boundary-layer length $L/\\beta z|\\Delta\\phi|$ is the field penetration depth that controls the large-voltage mechanism.","core_discovery":"For a symmetric $z{:}z$ electrolyte between two reservoirs in a one-dimensional channel held at a potential drop $\\Delta\\phi$, the long-time rate function $I(j_q)$ for the ionic current is given by Eq. (18) at small applied potentials and Eq. (22) at large applied potentials. In both regimes the rate function is non-quadratic: the current is Gaussian only near its mean, with exponential tails at large deviations. The optimal density profile that generates a rare current is a delocalized parabolic profile at small voltage, Eq. (15), and becomes a boundary-layer profile of width $L/\\beta z|\\Delta\\phi|$ at large voltage, Eq. (21). Under large driving the rate function is asymmetric, and the paper shows that this asymmetry is exactly the Gallavotti-Cohen symmetry $I(j_q)-I(-j_q)=\\beta j_q\\Delta\\phi$, with the strong thermodynamic uncertainty relation saturated in linear response and loosened beyond it.","pith_inferences":["If the boundary-layer mechanism persists in two- and three-dimensional nanopores, rare ionic-current fluctuations should be spatially localized near the pore entrances; the one-dimensional prediction gives a concrete signature for imaging or coarse-grained simulations to test.","The exponential tails imply that device metrics such as memristor switching probabilities or ionic-diode noise, which are often estimated from Gaussian linear-response assumptions, may be dominated by rare events at large driving.","A direct numerical search over time-periodic density profiles would test whether the stationary-profile assumption hides a dynamical phase transition; the authors note that traveling-wave optimizers occur in related periodic-boundary problems.","Applying the same contraction to concentration-gradient-driven mass currents would produce coupled charge and mass profiles rather than the simple $\\bar{q}=0$ reduction, potentially exposing a transition between delocalized and boundary-layer rare-event mechanisms."],"forward_implications":["Measured current fluctuations in dilute electrolyte channels will be Gaussian only close to the mean; rare events decay exponentially rather than as a Gaussian, so Gaussian-noise models underestimate the probability of large deviations.","At high applied voltage, the most probable way to produce a rare current is to develop steep density layers at the channel ends, a spatial signature that could be searched for in simulations or experiments.","The asymmetry of the current distribution is not an adjustable feature: Gallavotti-Cohen symmetry forces $I(j_q)-I(-j_q)=\\beta j_q\\Delta\\phi$, tying the rare-event statistics directly to the potential drop.","The thermodynamic uncertainty relation is saturated for $|\\beta z\\Delta\\phi|<1$ and becomes loose at larger driving, indicating that high-voltage operation is thermodynamically inefficient for producing low-noise currents.","The finite-difference solution of the Euler-Lagrange equations extends directly to asymmetric electrolytes, complex geometries, and mass-current boundary conditions, where no closed-form rate function is available."],"supporting_citations":[{"why":"Supplies the macroscopic fluctuation theory framework that converts the stochastic hydrodynamic action into a rate-function optimization.","marker":"[15]"},{"why":"Provides the contraction principle used to obtain the current rate function from the joint density-current action.","marker":"[46]"},{"why":"Particle-based Brownian dynamics from which the stochastic Poisson-Nernst-Planck equations are obtained.","marker":"[56]"},{"why":"Stochastic density-functional equations for interacting particles that underlie the fluctuating hydrodynamics and the Gaussian action.","marker":"[57]"},{"why":"Optimal density profiles for current fluctuations in diffusive systems, which provide the zero-voltage analogue of the parabolic profile.","marker":"[41]"},{"why":"Gallavotti-Cohen fluctuation theorem that fixes the asymmetry of the rate function through entropy production.","marker":"[39]"},{"why":"Thermodynamic uncertainty relation bounding the certainty of a current by entropy production, checked in Sec. IV B.","marker":"[69]"},{"why":"Earlier version of the thermodynamic uncertainty relation used to interpret the efficiency of large-voltage driving.","marker":"[70]"}],"fun_headline_variants":["Ionic currents show exponential tails at high voltage","Rare ion currents: from Gaussian to exponential tails","Large deviations give non-Gaussian ionic currents","Boundary layers shape rare ionic current tails","Dilute electrolytes: rare currents break Gaussian statistics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculation assumes that the long-time minimization over density profiles is achieved by stationary, time-independent profiles; if a time-periodic or traveling-wave profile is the true optimum, the derived rate functions are not the actual large-deviation answers.","fun_headline_variants_meta":{"raw":{"variants":["Ionic currents show exponential tails at high voltage","Rare ion currents: from Gaussian to exponential tails","Large deviations give non-Gaussian ionic currents","Boundary layers shape rare ionic current tails","Dilute electrolytes: rare currents break Gaussian statistics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000635,"raw_usage":{"total_tokens":2887,"prompt_tokens":863,"completion_tokens":2024,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":1963}},"tokens_in":479,"tokens_out":2024,"duration_ms":15389,"temperature":1.0,"reasoning_tokens":1963,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:11:22.006184+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the stochastic Poisson-Nernst-Planck dynamics in a one-dimensional channel with $\\beta z\\Delta\\phi \\approx 4$ using a rare-event sampling method, and compare the measured rate function and conditional density profiles with Eqs. (21) and (22); observing traveling-wave optimal profiles, or rate functions that differ from Eq. (22), would falsify the stationary-profile result.","supporting_citations":[{"cited_title":"Bertini , author A","cited_arxiv_id":null,"evidence_quote":"Supplies the macroscopic fluctuation theory framework that converts the stochastic hydrodynamic action into a rate-function optimization."},{"cited_title":"Touchette ,\\ @noop journal journal Physics Reports \\ volume 478 ,\\ pages 1 ( year 2009 ) NoStop","cited_arxiv_id":null,"evidence_quote":"Provides the contraction principle used to obtain the current rate function from the joint density-current action."},{"cited_title":"Donev , author A","cited_arxiv_id":null,"evidence_quote":"Particle-based Brownian dynamics from which the stochastic Poisson-Nernst-Planck equations are obtained."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Stochastic density-functional equations for interacting particles that underlie the fluctuating hydrodynamics and the Gaussian action."},{"cited_title":"Bodineau \\ and\\ author B","cited_arxiv_id":null,"evidence_quote":"Optimal density profiles for current fluctuations in diffusive systems, which provide the zero-voltage analogue of the parabolic profile."},{"cited_title":"Gallavotti \\ and\\ author E","cited_arxiv_id":null,"evidence_quote":"Gallavotti-Cohen fluctuation theorem that fixes the asymmetry of the rate function through entropy production."},{"cited_title":"Hasegawa \\ and\\ author T","cited_arxiv_id":null,"evidence_quote":"Thermodynamic uncertainty relation bounding the certainty of a current by entropy production, checked in Sec. IV B."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier version of the thermodynamic uncertainty relation used to interpret the efficiency of large-voltage driving."}],"review_version":2}