REVIEW 2 major objections 4 minor 86 references
Macroscopic fluctuation theory for the multi-time statistics of current in non-stationary diffusive systems
T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash
Pith's one-line read Multi-time current statistics from a density step are solved within macroscopic fluctuation theory: exact for non-interacting gases, explicit to second order for generic mobilities, and the step breaks the fractional Brownian motion of…
desk verdict Solid, well-verified extension of MFT to multi-time current statistics; the fBm breakdown for step initial conditions is the new result to remember, with the unproven quenched typicality assumption the main caveat. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the scaled cumulant generating functional $\chi(\Lambda(\tau))$ for the integrated current, defined through the least-action principle of macroscopic fluctuation theory (MFT), the coarse-grained hydrodynamic description of diffusive large deviations; the optimal density $q(x,\tau)$ and response field $p(x,\tau)$ satisfy the Euler-Lagrange equations with a time-dependent fugacity $\Lambda(\tau)$ entering as a source term $\Lambda(\tau)\Theta(x)$. The decisive simplification for the non-interacting case is a Cole-Hopf-type change of variables, $p=\ln P$ and $q=RP$, which decouples the coupled equations into a pair of linear diffusion equations with sources whose iterative solution yields all cumulants as products of Gaussian propagators, and hence the pair-sum formulas (5) and (6). For interacting systems the same equations are solved perturbatively in $\Lambda$; the two-time correlation then reduces to integrals over the spontaneous diffusion profile $q_0(x,\tau)=\bar\rho-(\Delta\rho/2)\,\mathrm{erf}(x/(2\sqrt{\tau}))$, with the mobility $\sigma$ evaluated at that profile.
What would settle it
Measure the quenched two-time correlation for a single fixed initial step profile rather than averaging over initial states; if the variance over initial profiles of the log-generating function grows like $\sqrt{T}$ rather than staying subleading, the assumption behind Eq. (29) fails and the predicted coefficients in (6) will shift. A numerical test at extreme density contrast such as $\rho_a=1$, $\rho_b=0$, comparing the large-$t_2$ decay predicted by (62) and (8), would settle this.
Extended reading notes
Core claim
On the paper's own terms, the central claim is that multi-time statistics of the integrated current in a diffusive system with a domain-wall initial condition are captured by the MFT action with a time-dependent source term, and that for $D(\rho)=1$, $\sigma(\rho)=2\rho$ this action can be evaluated in closed form. The resulting annealed cumulants have the form $(\rho_a+(-1)^n\rho_b)/\sqrt{\pi}$ times a pair sum over $0\le i<j\le n$ of coefficients $P_{ij}^{(n)}/\sqrt{t_j-t_i}$, where each $P_{ij}^{(n)}$ is a conditional probability for one Brownian motion to keep all sampled positions above the value it takes at times $i$ and $j$; the quenched cumulants contain additional pair terms $\sqrt{t_i+t_j}$ that arise from partitioning the time indices into independent Brownian blocks. For generic constant-diffusivity systems the same equations are solved perturbatively, giving the two-time correlations (7) and (8), whose leading piece is set by the mobility $\sigma(\bar\rho)$ and whose first non-trivial non-stationary correction is proportional to $(\Delta\rho)^2\sigma''(\bar\rho)$. The paper concludes that the two-time correlation for a step initial state is not that of fractional Brownian motion with Hurst exponent $H=1/4$, and it supports this conclusion by matching the hydrodynamic formulas to microscopic solutions of the SSEP.
Load-bearing premise
The paper's quenched results assume that the logarithm of the generating function for a fixed initial density profile is dominated by the typical step profile $r(x)$, so that atypical initial configurations contribute negligibly at the same order in $\sqrt{T}$; this concentration statement is stated in Eq. (29) but not proven for the step initial condition.
Editorial extensions
If this is right
- Because the scaled cumulant generating functional grows as $\sqrt{T}$ with coefficients depending only on time ratios, every multi-time current cumulant in the non-stationary domain-wall regime is fixed by the same MFT variational problem and is computable from formulas (5) and (6).
- For non-interacting gases and hard-core Brownian particles, explicit three- and four-time current correlations are obtained in closed form in terms of arcsin functions of time ratios, and they match the microscopic random-walk solution.
- Annealed and quenched two-time correlations differ qualitatively: as $t_2\to\infty$ at fixed $t_1$, the annealed correlation approaches half the equal-time variance while the quenched correlation decays, and the quenched cumulants carry distinct $\sqrt{t_1+t_2}$ pair terms.
- For any constant-diffusivity system, the first non-stationary correction to the two-time correlation is of order $(\Delta\rho)^2\sigma''(\bar\rho)$, so models with quadratic mobility have exact two-time correlations given by (7) and (8).
- The density-current correlations (9) and (10) describe the hydrodynamic density profile conditioned on an atypical current, coincide with the MFT optimal profile, and extend earlier equilibrium biased-ensemble results to the step and quenched settings.
Reading between the lines
- The paper leaves implicit that, if the concentration assumption in Eq. (29) holds, the partition-of-time-indices structure should survive for interacting integrable systems, so exact multi-time SSEP cumulants could be constructed from the same Brownian orthant probabilities.
- A testable extension beyond the paper's calculations is that the two-time correlation at intermediate density steps traces a one-parameter family of Gaussian processes connecting the $H=1/4$ fractional Brownian motion at $\Delta\rho=0$ to a genuinely aging process at strong steps.
- The annealed-quenched difference expressed in (89) depends only on the initial profile and the linear response field, suggesting it may generalize to observables other than the current and give a route to quenched multi-time statistics for arbitrary linear functionals.
- For Brownian hard rods, whose single-time MFT solution uses an interacting-to-non-interacting mapping, the same mapping could plausibly upgrade those results to multi-time current correlations; the paper lists this only as a forward direction.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript extends Macroscopic Fluctuation Theory (MFT) to multi-time statistics of the time-integrated current in one-dimensional diffusive systems on an infinite line with a domain-wall initial condition. It formulates an action principle for the annealed and quenched cumulant generating functionals, solves the resulting Euler-Lagrange equations exactly for the non-interacting case D=1, σ=2ρ, and obtains explicit multi-time cumulants (5) and (6). For constant diffusivity and arbitrary mobility it develops a perturbative expansion in the fugacity and in the density difference, obtaining the two-time correlations (7) and (8). The hydrodynamic predictions are checked by exact microscopic solutions for independent random walkers and for the SSEP, and by Monte Carlo simulations, including a microscopic check of the MFT optimal density profile through density-current correlations.
Significance. If the results hold, this is a substantial step forward: it provides the first explicit multi-time current large-deviation formulas for non-stationary diffusive systems and shows that the fractional-Brownian-motion covariance found for flat initial conditions does not persist for step initial conditions. A particular strength of the paper is that the central formulas are parameter-free and are not obtained by fitting; the non-interacting and SSEP results are independently derived from microscopic dynamics, and the optimal-profile calculation is verified against exact density-current correlations. The main unresolved issue is the quenched typicality assumption (29), which is load-bearing for the generic quenched results but is only verified in exactly solvable models.
major comments (2)
- [§2.2, Eq. (29)] The replacement of the quenched cgf by the generating function evaluated at the typical initial profile r(x) is asserted but not proven. The heuristic that the logarithm of the generating function is 'slowly varying' compared with the initial-state distribution is not sufficient: both the initial-state weight and the generating function are exponential in √T, so atypical initial profiles of cost exp(-√T F) could in principle be compensated by a change of order F in the scaled cumulant χ. The assumption is exact for the non-interacting gas, as shown by the microscopic factorization in Section 5, and it can be checked for the SSEP two-time correlation in Section 6, but for generic mobilities such as KMP or the inclusion process no proof or independent verification is given. Since Eq. (8) and the generic quenched multi-time claims rest on Eq. (29), this is a load-bearing gap. Please either provide a proof or a rigorous concentration argument, or explicitly restrict the generic quenched claims to the models in which the assumption is established.
- [§4.2, Eqs. (76)-(79)] The perturbative derivation leading to the quenched two-time correlation (8) proceeds by expanding q_0(x,τ) in powers of Δρ and exchanging the order of the time integrals and the expansion. The paper gives explicit evaluations in Appendix F but does not provide uniform control of the error terms in the long-time scaling limit; the integrands have singular kernels near τ=0 and τ=τ1, and the validity of the truncation at order (Δρ)^2 is not justified for generic σ(ρ). The independent SSEP check in Section 6 supports the final formula, so I do not regard this as a fatal flaw, but the manuscript should state the conditions under which the expansion is uniform and should either prove or clearly flag this point.
minor comments (4)
- [Figures 2-4] The simulation data are shown without error bars; please add error bars or state explicitly that they are smaller than the symbol size, and give the relevant finite-size parameters (number of lattice sites and boundary treatment).
- [§1, Eqs. (1a)-(1b)] The two displayed definitions of µ_A and µ_Q appear identical in the typeset version; the overline notation that distinguishes the annealed and quenched averages over initial conditions should be restored and defined at first use.
- [§5 and §6] The notation for the average over initial configurations, ⟨...⟩ for the evolution average, and the overline for the initial-state average is introduced in Section 5, but it is not used consistently in Sections 2 and 4; please standardize it.
- [Appendix J, Eq. (J.6)] The expression '4TOwen' should read '4 T_Owen' or '4 T_Owen(...)'; the missing multiplication symbol makes the formula hard to parse.
Circularity Check
No circularity: the multi-time MFT results are derived from the hydrodynamic action and independently verified by microscopic solutions.
full rationale
The paper's central derivation is self-contained. Section 2 constructs the multi-time MFT action from the fluctuating-hydrodynamics path integral (15) and the Euler-Lagrange equations (26), with no fitted parameters. Section 3 solves the D=1, sigma=2rho case explicitly by a Cole-Hopf transformation and obtains the scgfs (48); Section 5 independently re-derives the same scgfs from the microscopic random-walk dynamics (95,97), so the non-interacting results are not assumed. For generic diffusivity D=1 and arbitrary mobility, Section 4 solves the Euler-Lagrange equations perturbatively to obtain (7) and (8), and Section 6 plus Appendices H-K verify those formulas for the SSEP from the microscopic dynamics and the known equal-time correlations of [21]. The flat-initial-condition limits reproduce previously known fractional-Brownian-motion correlations as checks, not as inputs. The only load-bearing uncontrolled point is the quenched concentration statement (29), which assumes atypical initial profiles do not contribute; this is an unproven approximation for generic mobilities and therefore a correctness risk, but it is not a circular identification or a fitted result, and it is independently checked in the non-interacting and SSEP cases. Self-citations to [27] and [34] supply prior framework and notation, but the relevant equations are re-derived here, so no central claim reduces by construction to a self-citation.
Assumptions & free parameters
assumptions (5)
- domain assumption The fluctuating hydrodynamics equation (11) with noise covariance (12) is the correct large-scale description of the diffusive systems considered.
- domain assumption The free energy functional F(ρ) in (19) gives the large-deviation probability of initial density profiles.
- domain assumption For the quenched ensemble, the log of the cgf is dominated by the typical initial profile r(x) in the large-time limit (Eq. 29).
- domain assumption The series expansion of the MFT equations in the fugacity Λ and in Δρ is valid to the order used.
- standard math Known exact results for the SSEP equal-time correlations from [21] are correct.
Cite this review
Pith. "Pith review of Macroscopic fluctuation theory for the multi-time statistics of current in non-stationary diffusive systems." pith.science (2026). https://pith.science/paper/OPB36JDU
@misc{pith2026260812119,
author = {Pith},
title = {Pith review of: Macroscopic fluctuation theory for the multi-time statistics of current in non-stationary diffusive systems},
year = {2026},
howpublished = {\url{https://pith.science/paper/OPB36JDU}},
note = {Machine review of arXiv:2608.12119}
}
read the original abstract
The statistics of current fluctuations has long been a central object of study in non-equilibrium physics. Most existing work has focused on one-time statistics, while their multi-time generalisation remains comparatively less explored. We address this gap by extending the fluctuating hydrodynamics framework of Macroscopic Fluctuation Theory (MFT) to study multi-time statistics in the non-stationary state of a diffusive system on an infinite line. For the simplest cases of a non-interacting lattice gas and hard-core Brownian point particles, we present explicit solution of the MFT leading to multi-time large-deviation statistics. For generic systems, the MFT is solved perturbatively, yielding explicit results for two-time correlations. These reveal that the connection between current fluctuations and fractional Brownian motion, previously observed for flat initial conditions, does not persist for step initial conditions. We independently verify these hydrodynamic results by solving the corresponding microscopic dynamics for the non-interacting gas and for the symmetric simple exclusion process. Additional confirmation comes from numerical simulations.
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Reference graph
Works this paper leans on
-
[1]
Derrida B 2007J. Stat. Mech.P07023
-
[2]
Derrida B 2025SciPost Phys. Lect. Notes106
-
[3]
Mallick K 2015Physica A41817
-
[4]
Pr¨ ahofer M and Spohn H 2002Current Fluctuations for the Totally Asymmetric Simple Exclusion Process(Boston, MA: Birkh¨ auser Boston) p 185
-
[5]
Bertini L, De Sole A, Gabrielli D, Jona-Lasinio G and Landim C 2005Phys. Rev. Lett.94030601
-
[6]
Bertini L, Sole A D, Gabrielli D, Jona-Lasinio G and Landim C 2006J. Stat. Phys.123237
-
[7]
Lazarescu A 2015J. Phys. A: Math. Theor.48503001
-
[8]
Dandekar R, Krapivsky P L and Mallick K 2023Phys. Rev. E107044129
Show all 86 references
-
[9]
Banerjee T, Majumdar S N, Rosso A and Schehr G 2020Phys. Rev. E101052101
-
[10]
Di Bello C, Hartmann A K, Majumdar S N, Mori F, Rosso A and Schehr G 2023Phys. Rev. E 108014112
-
[11]
Lee H, Levitov L S and Yakovets A Y 1995Phys. Rev. B514079
-
[12]
McCulloch E, De Nardis J, Gopalakrishnan S and Vasseur R 2023Phys. Rev. Lett.131210402
-
[13]
Rep.4781
Touchette H 2009Phys. Rep.4781
-
[14]
Lecomte V, Imparato A and van Wijland F 2010Prog. Theo. Phys. Suppl.184276
-
[15]
Akkermans E, Bodineau T, Derrida B and Shpielberg O 2013EPL (Europhysics Letters)103 20001
-
[16]
Bodineau T and Derrida B 2004Phys. Rev. Lett.92180601
-
[17]
Harris R J, R´ akos A and Sch¨ utz G M 2005J. Stat. Mech.2005P08003
-
[18]
Bodineau T and Derrida B 2006J. Stat. Phys.123277
-
[19]
Bodineau T and Derrida B 2005Phys. Rev. E72066110
-
[20]
Appert-Rolland C, Derrida B, Lecomte V and van Wijland F 2008Phys. Rev. E78021122
-
[21]
Derrida B and Gerschenfeld A 2009J. Stat. Phys.1361
-
[22]
Derrida B and Gerschenfeld A 2009J. Stat. Phys.137978
-
[23]
Mallick K, Moriya H and Sasamoto T 2022Phys. Rev. Lett.129040601
-
[24]
Sharma K, Saha S, Jangid S and Sadhu T 2026Phys. Rev. E113L052101
-
[25]
Grabsch A, Moriya H, Mallick K, Sasamoto T and B´ enichou O 2024Phys. Rev. Lett.133117102
-
[26]
Sadhu T and Derrida B 2016J. Stat. Mech.2016113202
-
[27]
Krapivsky P L, Mallick K and Sadhu T 2015J. Stat. Mech.2015P09007
-
[28]
Sadhu T and Derrida B 2015J. Stat. Mech.2015P09008
-
[29]
Poncet A, Grabsch A and B´ enichou O 2025arXiv2505.21446
-
[30]
Bertini L, De Sole A, Gabrielli D, Jona-Lasinio G and Landim C 2015Rev. Mod. Phys.87593
-
[31]
Bertini L, De Sole A, Gabrielli D, Jona-Lasinio G and Landim C 2002J. Stat. Phys.107635
-
[32]
Bertini L, De Sole A, Gabrielli D, Jona-Lasinio G and Landim C 2001Phys. Rev. Lett.87040601
-
[33]
Saha S, Jangid S, Sharma K and Sadhu T 2026arXiv2605.26069
-
[34]
Saha S, Jangid S, de Pirey T A, Klamser J U and Sadhu T 2026arXiv2601.02319
-
[35]
Spohn H 1991Large Scale Dynamics of Interacting Particles(New York: Springer-Verlag)
-
[36]
Derrida B 2011J. Stat. Mech.2011P01030
-
[37]
Bodineau T and Derrida B 2026arXiv2605.01900
-
[38]
Bettelheim E, Smith N R and Meerson B 2022Phys. Rev. Lett.128130602
-
[39]
Krajenbrink A and Le Doussal P 2021Phys. Rev. Lett.127064101
-
[40]
Leibovich N and Barkai E 2013Phys. Rev. E88032107
-
[41]
Krapivsky P L, Mallick K and Sadhu T 2015J. Stat. Phys.160885
-
[42]
Lizana L, Lomholt M A and Ambj¨ ornsson T 2014Physica A395148
-
[43]
Krapivsky P L and Meerson B 2012Phys. Rev. E86031106
-
[44]
Krapivsky P L, Mallick K and Sadhu T 2014Phys. Rev. Lett.113078101
-
[45]
Grabsch A and B´ enichou O 2024Phys. Rev. Lett.132217101
-
[46]
Berlioz T, B´ enichou O and Grabsch A 2025Phys. Rev. Lett.134247101 Macroscopic fluctuation theory for multi-time statistics50
-
[47]
Saha S and Sadhu T 2023J. Stat. Mech.2023073207
-
[48]
Berlioz T, Venturelli D, Grabsch A and B´ enichou O 2024J. Stat. Mech.2024113208
-
[49]
Rana J and Sadhu T 2023Phys. Rev. E107L012101
-
[50]
Krapivsky P L, Mallick K and Sadhu T 2015J Phys A: Math Theo48015005
-
[51]
Kipnis C, Marchioro C and Presutti E 1982J. Stat. Phys.2765
-
[52]
Giardin` a C, Kurchan J and Redig F 2007J. Math. Phys.48033301
-
[53]
Saha S and Sadhu T 2026arXiv2501.03164v2
-
[54]
Mandelbrot B B and Van Ness J W 1968SIAM Review10422
-
[55]
Sadhu T, Delorme M and Wiese K J 2018Phys. Rev. Lett.120040603
-
[56]
Sadhu T and Wiese K J 2021Phys. Rev. E104054112
-
[57]
Henkel M and Pleimling M 2008Local Scale-Invariance in Disordered Systems(Berlin, Heidelberg: Springer) p 107
-
[58]
Berthier L and Biroli G 2011Rev. Mod. Phys.83587
-
[59]
Grabsch A, Poncet A, Rizkallah P, Illien P and B´ enichou O 2022Science Advances8eabm5043
-
[60]
Poncet A, Grabsch A, Illien P and B´ enichou O 2021Phys. Rev. Lett.127220601
-
[61]
Tailleur J, Kurchan J and Lecomte V 2007Phys. Rev. Lett.99150602
-
[62]
Tailleur J, Kurchan J and Lecomte V 2008J. Phys. A41505001
-
[63]
Martin P, Siggia E and Rose H 1973Phys. Rev. A8423
-
[64]
De Dominicis C 1975Lettere al Nuovo Cimento (1971-1985)12567
1971
-
[65]
Janssen H K 1976Zeitschrift f¨ ur Physik B Condensed Matter23377
-
[66]
Jangid S, Saha S, Sharma K, Kethepalli J, Guiselin B, De Nardis J and Sadhu T 2026arXiv 2607.14073
-
[67]
Saha S, Kethepalli J, Guiselin B, Nardis J D and Sadhu T 2026arXiv2604.24741
-
[68]
Kethepalli J, Urilyon A, Sadhu T and De Nardis J 2026SciPost Phys.20105
-
[69]
Harris T E 1965J. Appl. Probab.2323
-
[70]
Percus J K 1974Phys. Rev. A9557
-
[71]
Alexander S and Pincus P 1978Phys. Rev. B182011
-
[72]
Kollmann M 2003Phys. Rev. Lett.90180602
-
[73]
Bacon R H 1963The Annals of Mathematical Statistics34191
-
[74]
Derrida B and Sadhu T 2019J. Stat. Phys.176773
-
[75]
Derrida B and Sadhu T 2019J. Stat. Phys.177151
-
[76]
B´ enichou O, Illien P, Oshanin G, Sarracino A and Voituriez R 2015Phys. Rev. Lett.115220601
-
[77]
Iomin A, M´ endez V and Horsthemke W 2018Fractional dynamics in comb-like structures(World Scientific)
-
[78]
Venturelli D, Illien P, Grabsch A and B´ enichou O 2025J Phys A: Math Theo58215001
-
[79]
Venturelli D, Illien P, Grabsch A and B´ enichou O 2025Phys. Rev. Lett.135127101
-
[80]
Cantini L 2026arXiv2606.27186
-
[81]
Bodineau T and Derrida B 2011J. Stat. Phys.145745
-
[82]
Doyon B, Perfetto G, Sasamoto T and Yoshimura T 2023SciPost Phys.15136
-
[83]
Owen D B 1980Communications in Statistics - Simulation and Computation9389
-
[84]
Brychkov Y A and Savischenko N V 2016Integral Transforms and Special Functions27163
-
[85]
Korotkov N E and Korotkov A N 2020Integrals related to the error function(Chapman and Hall/CRC)
-
[86]
Prudnikov A P, Brychkov Yu A and Marichev O I 1986Integrals and Series: Special Functions vol 2 (New York: Gordon & Breach Science Publishers)
Reviewed August 16, 2026 · model on record in the stance chip above.
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