{"id":"137bcbc4-f9ed-46d2-aaea-f7455d2bba4c","arxiv_id":"2608.12119","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For diffusive systems on a line with a stepped initial density, the paper derives multi-time current statistics from macroscopic fluctuation theory, giving exact results for non-interacting particles and a perturbative two-time correlation for generic systems.","lead":"This paper extends macroscopic fluctuation theory to describe how current flowing across a point at different times is correlated, for systems that start from a sharp step in density. It derives exact formulas for simple gases and a general approximation for interacting systems, verified by microscopic calculations and simulations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Quenched typicality assumption (Eq. 29) is the load-bearing step for the generic quenched results (6, 8); it is verified microscopically only for the non-interacting gas and SSEP two-time correlations.","rationale":"The reader identifies the quenched typicality assumption, Eq. (29), as the weakest assumption. I agree: this is the only major step in the MFT derivation that is asserted rather than proven, and it is necessary for the quenched generic results. However, the concern is limited in scope. The annealed results, including the headline breakdown of fractional Brownian motion for step initial conditions, do not rely on Eq. (29). The non-interacting quenched formulas (3-6) are derived independently from the microscopic random-walk solution, where the factorization of the generating function makes the typicality assumption exact. The SSEP two-time quenched correlation is obtained by an exact microscopic computation of the annealed-quenched difference, Eq. (126)-(127), so it also does not depend on Eq. (29). What remains unverified is the extension to generic mobilities beyond these solvable cases. This justifies a conditional acceptance rather than unconditional acceptance, but it is not a reason to reject: the assumption is physically standard and is consistent with all available exact checks. The proposed one-loop computation would directly test the assumption and could upgrade the verdict to unconditional acceptance if the correction is subextensive. No self-evident internal inconsistency, algebraic error, or unsupported claim in the annealed construction was found; the Monte Carlo verification, while lacking error bars, is secondary to the exact microscopic derivations. Therefore the reader's CONDITIONAL verdict is appropriate and should be unchanged.","tokens_in":37427,"tokens_out":21201,"duration_ms":189727,"concrete_test":"Perform the one-loop Gaussian calculation around the quenched saddle point: expand the action (21), including the initial-condition free energy F(rho(x,0)), to quadratic order in delta rho(x,0) around the typical step profile r(x), with Lambda(t) fixed, and integrate out the Gaussian fluctuations. Compare the resulting correction to the quenched cgf mu_Q with the leading sqrt(T) term as T goes to infinity. If the correction is O(1) or smaller, Eq. (29) is consistent; if it is O(sqrt(T)), atypical initial configurations contribute at the same order and the quenched formulas (6, 8) are invalid. For the non-interacting gas this calculation should reproduce the exact microscopic concentration of log Z and can be used to validate the method before applying it to generic sigma(rho).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim includes explicit quenched multi-time formulas for generic diffusive systems: Eq. (6) for non-interacting gas and Eq. (8) for constant-diffusivity systems with arbitrary mobility. The derivation of these quenched results rests on Eq. (29): the quenched cgf is replaced by the generating function evaluated at the typical initial profile r(x), i.e., atypical initial fluctuations are assumed not to contribute at leading order in the long-time scaling. This is a Laplace/self-averaging assumption, not proven in the manuscript. It is exact for the non-interacting gas (Section 5, because the quenched generating function factorizes and only the mean occupation enters) and for the SSEP two-time current correlation (Section 6, where the annealed-quenched difference is computed exactly via Eq. (126)). However, for generic mobilities (e.g., KMP or inclusion process) there is no independent check that rare initial profiles with cost exp(-sqrt(T) F) cannot compensate their rarity by producing atypically large log-generating functions. If Eq. (29) fails, the quenched generic two-time correlation (8) and the quenched multi-time formulas (6) would be incorrect, even though the annealed result (7) and the fractional-Brownian-motion breakdown would survive. The concern is therefore load-bearing specifically for the quenched generic claims, not for the annealed claim or for the exactly solvable models.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":37674,"tokens_out":7542,"duration_ms":79358,"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":[{"comment":"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.","section":"§2.2, Eq. (29)"},{"comment":"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.","section":"§4.2, Eqs. (76)-(79)"}],"minor_comments":[{"comment":"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).","section":"Figures 2-4"},{"comment":"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.","section":"§1, Eqs. (1a)-(1b)"},{"comment":"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.","section":"§5 and §6"},{"comment":"The expression '4TOwen' should read '4 T_Owen' or '4 T_Owen(...)'; the missing multiplication symbol makes the formula hard to parse.","section":"Appendix J, Eq. (J.6)"}],"recommendation":"major_revision","confidential_remarks":"This is a strong and well-written paper with substantial results and unusually careful cross-checks. The main issue is the unproven quenched typicality assumption (29), which affects only the generic quenched claims and is fixable either by a proof or by a clearly stated restriction of the claims. I would be comfortable with acceptance once this point is addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good paper, worth a careful read. Two things to know up front. The genuinely new result is the multi-time current statistics for a step initial condition, and the one people will remember is the breakdown of fractional Brownian motion: for flat initial conditions the two-time current correlation is fBm with Hurst 1/4, for a domain wall it is not, starting at second order in the density difference. The other thing to know is that the paper verifies itself unusually well: exact microscopic solutions for independent random walkers and for SSEP, plus Monte Carlo, all consistent with the hydrodynamic formulas.\n\nWhat is actually new: the explicit multi-time correlations (5,6) for the non-interacting gas with step initial condition, the perturbative two-time formula (7) with its explicit delta-rho-squared correction, and the density-current correlation (9,10) which independently checks the optimal MFT profile. The flat-initial-condition limits recover earlier results, so those work as checks. The formalism leans on the authors' earlier tracer paper, and they say so; that is honest, not a problem.\n\nThe soft spots. The quenched typicality assumption, Eq. (29), is the load-bearing step for the quenched generic formulas, and the stress-test note is right: it is not proven for arbitrary mobility. The paper shows it is exact for the non-interacting gas and verifies it for SSEP through the exact annealed-quenched difference (126). For KMP or the inclusion process, which share the same quadratic mobility, there is no independent microscopic check. So the quenched generic claim (8) depends on a physically reasonable but unproven concentration argument. The annealed result (7) and the fBm breakdown do not rest on it. I call this a real caveat, in proportion: it does not sink the paper, but a referee should ask whether the quenched generic step can be sharpened. The other issue is minor: the simulation plots lack error bars, though the sample sizes are 10^7 to 10^8 and the agreement is visually clean.\n\nFor whom: MFT and 1D transport people, and anyone tracking the current-fluctuation / fBm connection. It deserves a serious referee. My recommendation is to send it to review; the referee should press on Eq. (29) and on error bars, but the central results look solid.","headline":"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.","tokens_in":38202,"tokens_out":3400,"would_cite":true,"duration_ms":28146,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82C22","60F10","82C31"],"pacs":["05.40.-a","05.60.-k"],"model":"deepseek-v4-flash","headline":"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…","keywords":["multi-time statistics","current fluctuations","macroscopic fluctuation theory","large deviations","domain-wall initial condition","symmetric simple exclusion process","fractional Brownian motion","annealed and quenched ensembles"],"falsifier":"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.","tokens_in":37234,"feed_emoji":"📈","tokens_out":15345,"duration_ms":137195,"temperature":0.7,"pith_summary":"This paper asks how the fluctuating current crossing a point in an infinite diffusive system is correlated at several different times when the system starts from a sharp density step, a domain-wall initial state. It shows that macroscopic fluctuation theory can be extended to this multi-time, non-stationary setting and that, for non-interacting particles, the variational problem can be solved exactly, yielding explicit formulas for every $n$-time cumulant of the integrated current in both annealed and quenched averages over initial conditions. For generic systems with constant diffusivity and arbitrary mobility, a perturbative solution gives explicit two-time current correlations up to second order in the density jump $\\Delta\\rho=\\rho_a-\\rho_b$. The central physical result is that for a step initial condition the two-time current correlation is no longer the covariance of fractional Brownian motion, the behaviour previously found for flat initial states; the step leaves a lasting memory of the initial profile. These hydrodynamic predictions are verified by exact microscopic solutions for independent random walkers and for the symmetric simple exclusion process, and by Monte Carlo simulations.","feed_headline":"Domain-wall currents escape fractional Brownian motion","feed_subtitle":"Exact MFT formulas for annealed and quenched current correlations reveal lasting memory of the initial step.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the single-time MFT formalism for current statistics on the infinite line, including the Euler-Lagrange equations and the Cole-Hopf transformation that this paper extends to multi-time.","marker":"[22]"},{"why":"Develops the multi-time MFT framework for tracer particles with annealed and quenched ensembles; the present action and perturbative scheme build directly on it.","marker":"[27]"},{"why":"Provides the microscopic multi-time tracer correlations for hard-core Brownian particles whose structure is matched by the formulas (3)-(6).","marker":"[28]"},{"why":"Gives multi-time tracer correlations in the high-density SSEP with uniform initial condition; the time dependencies found here coincide up to numerical prefactors with these results.","marker":"[29]"},{"why":"Supplies the exact equal-time density correlations of the SSEP used to verify the two-time current correlations microscopically.","marker":"[21]"},{"why":"Establishes the two-time current correlation method for the finite-size SSEP that is adapted here to the infinite line with a domain wall.","marker":"[26]"},{"why":"Defines the macroscopic fluctuation theory action and large-deviation framework on which the paper's variational calculation rests.","marker":"[30]"}],"fun_headline_variants":["Step initial states strip currents of fBm memory","Multi-time currents break from fBm in step-initial diffusion","Two-time correlations expose fBm failure for step profiles","Exact multi-time statistics defy fBm for step initials"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Step initial states strip currents of fBm memory","Multi-time currents break from fBm in step-initial diffusion","Two-time correlations expose fBm failure for step profiles","Exact multi-time statistics defy fBm for step initials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001063,"raw_usage":{"total_tokens":4499,"prompt_tokens":1028,"completion_tokens":3471,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":644,"completion_tokens_details":{"reasoning_tokens":3402}},"tokens_in":644,"tokens_out":3471,"duration_ms":24278,"temperature":1.0,"reasoning_tokens":3402,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:15:19.956555+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the single-time MFT formalism for current statistics on the infinite line, including the Euler-Lagrange equations and the Cole-Hopf transformation that this paper extends to multi-time."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Develops the multi-time MFT framework for tracer particles with annealed and quenched ensembles; the present action and perturbative scheme build directly on it."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the microscopic multi-time tracer correlations for hard-core Brownian particles whose structure is matched by the formulas (3)-(6)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the exact equal-time density correlations of the SSEP used to verify the two-time current correlations microscopically."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the two-time current correlation method for the finite-size SSEP that is adapted here to the infinite line with a domain wall."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the macroscopic fluctuation theory action and large-deviation framework on which the paper's variational calculation rests."}],"review_version":1}