Large deviations of ionic currents in dilute electrolytes

David T. Limmer; Jafar Farhadi

REVIEW 3 major objections 4 minor 70 references

Large deviations of ionic currents in dilute electrolytes

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2507.18556 v1 pith:JHLQMKK4 submitted 2025-07-24 cond-mat.stat-mech cond-mat.mes-hallphysics.chem-ph

Jafar Farhadi , David T. Limmer This is my paper · ORCID

classification cond-mat.stat-mechcond-mat.mes-hallphysics.chem-ph

keywords largedeviationsioniccurrentmacroscopicfluctuationtheoryPoisson-Nernst-Planckelectrolytenon-GaussianfluctuationsGallavotti-Cohensymmetrythermodynamicuncertaintyrelation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

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.

Now 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.

A 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.

Overall, 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.

Referee Report

3 major / 4 minor

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.

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 (3)

[Sec. II B, Eq. (7)] 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.
[Sec. III B, Eq. (22)] 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).
[Sec. IV A, Eq. (26)] 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.

minor comments (4)

[Eq. (22)] 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.
[Eq. (5)] 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.
[Sec. II B, Eq. (6)] 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.
[Sec. III A, Eq. (30)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ionic-current rate functions are derived from the stated stochastic PNP action with physical inputs only, and no fitted quantity or self-citation chain determines the predicted form.

full rationale

I find no circularity. The derivation is self-contained: Sec. II A defines the stochastic action from the specified fluctuating Poisson-Nernst-Planck equations (Eqs. 2–5); Eq. 7 defines the rate function through the contraction/minimization principle; Eqs. 8–9 are obtained by functional differentiation; and the small- and large-voltage rate functions, Eqs. 18 and 22, are evaluated by inserting the analytically derived optimal profiles (Eqs. 15 and 21) into the action and integrating. No parameter is fitted to the current distribution being predicted; the only inputs are physical model parameters (D, z, epsilon, L, beta, rho_hat, Delta_phi). The Gallavotti-Cohen and thermodynamic-uncertainty checks are post hoc constraints applied to the derived rate function, not ingredients used to construct it. Self-citations appear only in background or standard-technique contexts and are not load-bearing for the rate-function form. The time-independent optimization assumption in Sec. II B is a substantive correctness assumption about the saddle-point class, but it is not circular: assuming stationarity does not define the rate function or make Eqs. 18 and 22 true by construction. The finite-difference comparisons validate the analytical approximations against numerical solutions of the same Euler-Lagrange equations; they are not fits to an external target.

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

No parameters are fitted to data. All model inputs (D, z, epsilon, L, beta, rho_hat, Delta_phi) are physical inputs fixed by the model; the integration constants c1 and c2 in Eq. 20 are fixed by boundary conditions. The theory is parameter-free conditional on the stochastic PNP model. No new particles, forces, dimensions, or conserved quantities are introduced; the boundary-layer profiles are derived structures, not postulated entities.

assumptions (4)

domain assumption The stochastic Poisson-Nernst-Planck equations (Eqs. 1-3) capture the long-time fluctuating dynamics of a dilute strong electrolyte.
The derivation starts from this model; neglected hydrodynamic coupling, short-range ion correlations, and concentration-dependent friction are outside the model's validity.
domain assumption Optimal density and current fields can be chosen time-independent in the long-time limit.
Stated in Sec. II B; the authors note time-periodic traveling-wave optimizers can arise under periodic boundary conditions, so reservoir boundary conditions are assumed to select stationary profiles.
standard math The contraction principle and Laplace method apply to the Gaussian action, so the rate function is the minimum action over density fields.
Standard large-deviation result invoked in Eq. 7; requires finite correlation times and exponential scaling in observation time.
ad hoc to paper For the symmetric electrolyte, symmetry makes I2(jq,jrho)=I2(jq,-jrho), and convexity places the optimum at jrho=0 with qbar=0.
Used in Sec. III to reduce the Euler-Lagrange equations; the implication qbar=0 follows from the stated symmetry and convexity but is not fully demonstrated.

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Cite this review

Pith. "Pith review of Large deviations of ionic currents in dilute electrolytes." pith.science (2026). https://pith.science/paper/JHLQMKK4

@misc{pith2026250718556,
  author       = {Pith},
  title        = {Pith review of: Large deviations of ionic currents in dilute electrolytes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JHLQMKK4}},
  note         = {Machine review of arXiv:2507.18556}
}

read the original abstract

We evaluate the exponentially rare fluctuations of the ionic current for a dilute electrolyte by means of macroscopic fluctuation theory. We consider the fluctuating hydrodynamics of a fluid electrolyte described by a stochastic Poisson-Nernst-Planck equation. We derive the Euler-Lagrange equations that dictate the optimal concentration profiles of ions conditioned on exhibiting a given current, whose form determines the likelihood of that current in the long-time limit. For a symmetric electrolyte under small applied voltages, number density fluctuations are small, and ionic current fluctuations are Gaussian with a variance determined by the Nernst-Einstein conductivity. Under large applied potentials, where number densities vary, the ionic current distribution is generically non-Gaussian. Its structure is constrained thermodynamically by Gallavotti-Cohen symmetry and the thermodynamic uncertainty principle.

Figures

Figures reproduced from arXiv: 2507.18556 by the authors.

**Figure 2.** FIG. 2. Optimal mass density profile for ∆ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Rate function for ionic currents in the limit of low [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Optimal mass density profile for large [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Rate function for ionic currents for large applied [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. (top) Illustration of Gallavotti-Cohen symmetry for [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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author author R. L. \ Jack , author T. Nemoto ,\ and\ author V. Lecomte ,\ @noop journal journal Journal of Statistical Mechanics: Theory and Experiment \ volume 2020 ,\ pages 053204 ( year 2020 ) NoStop

2020
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Donev , author A

author author A. Donev , author A. J. \ Nonaka , author C. Kim , author A. L. \ Garcia ,\ and\ author J. B. \ Bell ,\ @noop journal journal Physical Review Fluids \ volume 4 ,\ pages 043701 ( year 2019 ) NoStop

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author author D. S. \ Dean ,\ @noop journal journal Journal of Physics A: Mathematical and General \ volume 29 ,\ pages L613 ( year 1996 ) NoStop

1996
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Bonneau , author V

author author H. Bonneau , author V. D \'e mery ,\ and\ author E. Rapha \"e l ,\ @noop journal journal Journal of Statistical Mechanics: Theory and Experiment \ volume 2025 ,\ pages 033201 ( year 2025 ) NoStop

2025
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Babick ,\ in\ @noop booktitle Suspensions of colloidal particles and aggregates \ ( publisher Springer ,\ year 2016 )\ pp.\ pages 75--118 NoStop

author author F. Babick ,\ in\ @noop booktitle Suspensions of colloidal particles and aggregates \ ( publisher Springer ,\ year 2016 )\ pp.\ pages 75--118 NoStop

2016
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Farhadi , author J

author author J. Farhadi , author J. B. \ Fernandes , author K. Shekhar ,\ and\ author K. K. \ Mandadapu ,\ @noop journal journal Physical Review E \ volume 111 ,\ pages 064412 ( year 2025 ) NoStop

2025
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author author J. W. \ Thomas ,\ @noop title Numerical partial differential equations: finite difference methods ,\ Vol. volume 22 \ ( publisher Springer Science & Business Media ,\ year 2013 ) NoStop

2013
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author author R. J. \ LeVeque ,\ @noop title Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems \ ( publisher SIAM ,\ year 2007 ) NoStop

2007
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Onsager \ and\ author S

author author L. Onsager \ and\ author S. Machlup ,\ @noop journal journal Physical Review \ volume 91 ,\ pages 1505 ( year 1953 ) NoStop

1953
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Palaia , author A

author author I. Palaia , author A. J. \ Asta , author P. B. \ Warren , author B. Rotenberg ,\ and\ author E. Trizac ,\ @noop journal journal arXiv:2303.07859 \ ( year 2023 ) NoStop

2023
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author author S. R. \ De Groot \ and\ author P. Mazur ,\ @noop title Non-equilibrium thermodynamics \ ( publisher Courier Corporation ,\ year 2013 ) NoStop

2013
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author author J. L. \ Lebowitz \ and\ author H. Spohn ,\ @noop journal journal Journal of Statistical Physics \ volume 95 ,\ pages 333 ( year 1999 ) NoStop

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author author J. M. \ Horowitz \ and\ author T. R. \ Gingrich ,\ @noop journal journal Nature Physics \ volume 16 ,\ pages 15 ( year 2020 ) NoStop

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Seifert ,\ @noop journal journal Annual Review of Condensed Matter Physics \ volume 10 ,\ pages 171 ( year 2019 ) NoStop

author author U. Seifert ,\ @noop journal journal Annual Review of Condensed Matter Physics \ volume 10 ,\ pages 171 ( year 2019 ) NoStop

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Hasegawa \ and\ author T

author author Y. Hasegawa \ and\ author T. Van Vu ,\ @noop journal journal Physical review letters \ volume 123 ,\ pages 110602 ( year 2019 ) NoStop

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author author A. C. \ Barato \ and\ author U. Seifert ,\ @noop journal journal Physical review letters \ volume 114 ,\ pages 158101 ( year 2015 ) NoStop

2015

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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author author D. T. \ Limmer , author C. Y. \ Gao ,\ and\ author A. R. \ Poggioli ,\ @noop journal journal The European Physical Journal B \ volume 94 ,\ pages 1 ( year 2021 ) NoStop

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Onsager \ and\ author S

author author L. Onsager \ and\ author S. K. \ Kim ,\ @noop journal journal The Journal of Physical Chemistry \ volume 61 ,\ pages 198 ( year 1957 ) NoStop

1957

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Toquer , author L

author author D. Toquer , author L. Bocquet ,\ and\ author P. Robin ,\ @noop journal journal The Journal of Chemical Physics \ volume 162 ( year 2025 ) NoStop

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author author C. Y. \ Gao \ and\ author D. T. \ Limmer ,\ @noop journal journal The Journal of chemical physics \ volume 151 ( year 2019 ) NoStop

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Lepri \ and\ author A

author author S. Lepri \ and\ author A. Politi ,\ @noop journal journal Physical Review E—Statistical, Nonlinear, and Soft Matter Physics \ volume 83 ,\ pages 030107 ( year 2011 ) NoStop

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Ray \ and\ author D

author author U. Ray \ and\ author D. T. \ Limmer ,\ @noop journal journal Physical Review B \ volume 100 ,\ pages 241409 ( year 2019 ) NoStop

work page 2019

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Chen , author G

author author X. Chen , author G. Cao , author A. Han , author V. K. \ Punyamurtula , author L. Liu , author P. J. \ Culligan , author T. Kim ,\ and\ author Y. Qiao ,\ @noop journal journal Nano letters \ volume 8 ,\ pages 2988 ( year 2008 ) NoStop

work page 2008

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Camalet , author J

author author S. Camalet , author J. Lehmann , author S. Kohler ,\ and\ author P. H \"a nggi ,\ @noop journal journal Physical review letters \ volume 90 ,\ pages 210602 ( year 2003 ) NoStop

work page 2003