{"id":"ba211e38-bfb4-4a2a-8cbc-c1cb6b366932","arxiv_id":"2608.03619","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Entropy production rate is bounded from below by a function of the asymmetry of two-time correlation functions of coarse-grained observables, for steady states at any lag and for time-dependent processes at vanishing lag.","lead":"This paper proves two new lower bounds on the entropy production rate of small stochastic systems, using only measurements of how two observable signals correlate over time. The bounds work even when most of the system's states are hidden, making them potentially useful for experiments on molecular machines and other driven nanoscale systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Optimized-bound formula Eq. (54) has a coefficient error (3/8 instead of 1/8) that can make the estimator exceed the true EPR; the main bounds (30), (34) are unaffected.","rationale":"The central theorems (30) and (34) are supported by a correct chain of inequalities, and the false-looking inequality in Eq. (23) is repaired by summing over unordered pairs rather than applied termwise. The stated Markov and local-detailed-balance assumptions are explicit scope conditions, not hidden flaws. The one unambiguous load-bearing error is the coefficient in Eq. (54), which affects the optimization contribution: it can produce an estimate larger than the true EPR if a user trusts the printed formula. Because the reader's verdict already conditions acceptance on fixing this coefficient and related issues, the conditional verdict should stand unchanged.","tokens_in":13233,"tokens_out":32106,"duration_ms":265034,"concrete_test":"Re-derive the minimum of the quadratic form (46) by substituting the optimal shift (53); confirm that the resulting denominator is D − ⅛ VᵀH⁻¹V. Then, for the four-state network of Sec. V.A with parameters from Fig. 1(e), compute both the printed bound (with −3/8) and the corrected bound (with −1/8) as functions of τ, and check whether the printed value exceeds the true EPR σ at any τ. If it does, the coefficient error is confirmed and must be fixed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most concrete load-bearing defect is in the optimization claim of Sec. IV.B. Starting from the quadratic form (46), D_{A',B'}(τ) = D_{A,B}(τ) + ½ sᵀH(τ)s + ½ sᵀV(τ), the optimal shift (53), s* = −H⁻¹V/2, yields D_min = D_{A,B} − ⅛ VᵀH⁻¹V. Substituting into the estimator (30) gives the optimized bound σ* = (2/τ)χ²/[D − ⅛ VᵀH⁻¹V]. The paper instead prints −3/8 in Eq. (54). Since VᵀH⁻¹V ≥ 0 for positive-definite H, the printed denominator is smaller than the correct one, so the printed optimized estimator is systematically too large and can violate the bound σ* ≤ σ. This is an unambiguous algebraic error, not a modeling assumption. It leaves the central inequalities (30) and (34) intact, but it makes the optimization result unreliable as stated and needs correction before the paper can be used as is.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript derives two lower bounds on the entropy production rate (EPR) of a partially observed Markovian system in terms of the time-asymmetry of two-time correlation functions of two arbitrary coarse-grained observables A and B. In a non-equilibrium steady state (NESS), Eq. (30) gives σ̂_{A,B}(τ)=(2/τ)χ_{A,B}(τ)^2/D_{A,B}(τ) ≤ σ for any lag τ; for time-dependent processes, the short-lag limit of the same expression bounds the instantaneous EPR, Eq. (34). The derivation combines Cauchy-Schwarz, a logarithmic inequality, the log-sum inequality, and the path-weight fluctuation theorem. The paper also analyzes tightness, proposes an analytic optimization of the NESS bound by constant shifts of the observables, and illustrates the bounds on a four-state Markov network and an overdamped Langevin particle on a ring.","tokens_in":13434,"tokens_out":17773,"duration_ms":152015,"significance":"If correct, the two inequalities are attractive additions to the thermodynamic-inference toolbox: they require only two-time correlation functions, allow coarse-grained observables, and need no fitting parameters. The NESS bound holding for arbitrary lag and the time-dependent bound being available for relaxation and driven processes are genuinely useful. The derivations of Eqs. (30) and (34) are transparent and the short-time expansion leading to Eq. (38) is clean. The discussion of tightness, in particular the observation that saturation of the component inequalities forces equilibrium, is a valuable caveat. However, the optimization section contains algebraic errors that must be corrected before the results there can be used.","major_comments":[{"comment":"Eq. (54) is not a valid consequence of the preceding minimization. From Eq. (46), D_{A',B'}(τ)=D_{A,B}(τ)+(1/2)s^T H(τ)s+(1/2)s^T V(τ). Minimizing with respect to s gives s* = −H^{-1}V/2 and D_min = D_{A,B} − (1/8) V^T H^{-1}V, not the denominator with the coefficient 3/8 printed in Eq. (54). Since V^T H^{-1}V ≥ 0 for positive definite H, the printed denominator is too small; the printed optimized estimator can therefore exceed the true EPR and violates the very bound it is supposed to optimize. The coefficient must be 1/8.","section":"Sec. IV.B, Eq. (54)"},{"comment":"Expanding (d'_{αβ})^2 directly gives the cross term −2 s_A s_B ΔA_{αβ}ΔB_{αβ}. The matrix h_{αβ} in Eq. (45) has off-diagonal entry −2ΔB_{αβ}ΔA_{αβ}, so s^T h s contains −4 s_A s_B ΔAΔB, overcounting this term by a factor of two. Consequently the Hessian entry in Eq. (49) should be H_2(τ)=2(C_{A,B}+C_{B,A}−2⟨AB⟩), not 4(C_{A,B}+C_{B,A}−2⟨AB⟩). The quadratic form (46), the optimal shift (53), and the optimized bound (54) are all built on this incorrect cross term. Both this factor and the coefficient error in Eq. (54) must be corrected, and the illustrative optimization results in Fig. 1(e,f) should be re-examined.","section":"Sec. IV.B, Eqs. (45) and (49)"}],"minor_comments":[{"comment":"The statement that the NESS bound holds 'for any lag τ≥0' should be qualified to τ>0 (or to the limiting interpretation at τ=0), since at τ=0 both numerator and denominator vanish and the estimator is indeterminate.","section":"Sec. III.B, Eq. (30)"},{"comment":"The claim that the Hessian H(τ) is positive definite for τ>0 is not always guaranteed; for example, if one of the observables is constant on the observed meso-states, H has a zero eigenvalue. Positive semidefiniteness suffices for the minimization argument, but the uniqueness statement needs qualification.","section":"Sec. IV.B"},{"comment":"The heading 'upper bound on the correlation asymmetry' is somewhat misleading because the intermediate result (Eq. (25)) is immediately used as a lower bound on the EPR; retitling the subsection would improve readability.","section":"Sec. III.A"},{"comment":"There are several typographical issues, including 'indepedent' in Sec. III.B, 'intervalls' in the caption of Fig. 3, and inconsistent ordering of C_{A^2,B^2} and C_{B^2,A^2} in Eq. (38) compared with Eq. (22).","section":"Various"}],"recommendation":"major_revision","confidential_remarks":"The central inequalities (30) and (34) withstand scrutiny; the defects are local to Sec. IV.B and, in my view, correctable by a focused revision rather than a reworking of the paper's main argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: the two central bounds are correct and genuinely useful; the optimization section has a clear algebraic error that must be fixed.\n\nWhat's new: I'm not aware of an existing NESS lower bound on EPR from arbitrary coarse-grained observables valid for any lag, nor a time-dependent bound in the zero-lag limit. The estimator \\hatσ = (2/τ)χ²/D is simple and operational. The derivation is clean: Cauchy–Schwarz, the log-sum inequality, and the fluctuation theorem are all used appropriately. I particularly like the handling of coarse-graining — instead of requiring full state resolution, you work with lumped meso-states and the log-sum inequality takes care of the hidden transitions. The shift optimization is a nice practical touch.\n\nSoft spots: Eq. (54) has an incorrect coefficient — the denominator should be D − (1/8)VᵀH⁻¹V, not D − (3/8)VᵀH⁻¹V. I recomputed the minimum of the quadratic form (46); the factor 1/8 is what you get. As printed, the optimized estimator can exceed the true EPR, which defeats the purpose. This is a fixable typo, but it has to be corrected.\n\nThe relation to Refs [9,10,42] is stated a little loosely; the authors claim to be the first to use arbitrary coarse-grained observables, but they don't give a precise comparison to Ohga–Ito–Kolchinsky or Cheng et al. That's worth tightening in the final version.\n\nThe extension to overdamped Langevin dynamics is plausible via the continuous-space limit, but a direct derivation for a Fokker–Planck system would be stronger. Minor point.\n\nAlso, as the authors note, the bound is not tight out of equilibrium because the log inequality (a−b)/(a+b) ≤ ln(a/b)/2 is strict for a≠b. That's not a flaw; it's an honest limitation, but it means the method gives a lower bound, not an estimator that converges to EPR in general.\n\nWho it's for: experimentalists and theorists working on thermodynamic inference from correlation measurements. Worth a serious referee — the error in Eq. (54) is exactly the kind of thing a careful referee should catch before publication.\n\nRecommendation: accept after a minor revision that fixes Eq. (54) and sharpens the comparison with the closest prior work. Send it to peer review rather than desk reject.","headline":"Two solid new correlation-based lower bounds on entropy production, with a fixable algebra error in the optimization formula (Eq. 54) that must be corrected.","tokens_in":13985,"tokens_out":4523,"would_cite":true,"duration_ms":37752,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The entropy production rate is bounded from below by the time-asymmetry of two-time correlation functions.","keywords":["entropy production","thermodynamic inference","coarse-graining","correlation functions","non-equilibrium steady state","stochastic thermodynamics","Markov processes","overdamped Langevin dynamics"],"falsifier":"Simulate a three-state Markov network with known transition rates and a coarse-grained pair of observables; compute the two-time correlations exactly or with high statistics, evaluate $\\hat\\sigma_{A,B}(\\tau)$, and compare it with the true entropy production rate. Any case with $\\hat\\sigma_{A,B}(\\tau)>\\sigma$ (or $\\hat\\sigma_{A,B}(t)>\\sigma(t)$ in the time-dependent limit) would refute the bound.","tokens_in":13013,"feed_emoji":"📊","tokens_out":7262,"duration_ms":57656,"temperature":0.7,"pith_summary":"The paper proves that the entropy production rate of any system with Markovian underlying dynamics, including overdamped Langevin dynamics, is bounded from below by a simple ratio built from the time-asymmetry of two-time correlation functions of two arbitrary coarse-grained observables. In a nonequilibrium steady state the bound holds for every correlation lag; for time-dependent processes it holds in the vanishing-lag limit. Because correlation functions are routinely accessible in experiments, this gives a model-free, operational estimate of irreversibility in partially observed systems. The bound can be tightened by applying constant shifts to the observables, and its quality is illustrated on a coarse-grained Markov network and an overdamped Brownian particle on a ring.","feed_headline":"Correlation asymmetry sets a lower bound on entropy production","feed_subtitle":"Two coarse-grained observables suffice to estimate dissipation, in steady or time-dependent systems.","key_machinery":"The argument rewrites the correlation asymmetry $\\chi_{A,B}(t,\\tau)$ as a sum over current-like quantities $J_{\\alpha\\beta}$ between observed mesostates, then applies a chain of inequalities: Cauchy-Schwarz to separate the asymmetry from a correlation-derived denominator $D_{A,B}$, the elementary inequality $(a-b)/(a+b)\\le \\ln(a/b)/2$ to connect these currents to a Kullback-Leibler divergence, and the log-sum inequality to coarse-grain this divergence down to the observable level. In the steady state, the final divergence equals $\\tau\\sigma$ by the path-weight fluctuation theorem; for time-dependent processes, the short-time expansion of the propagator makes the same divergence equal to $\\sigma(t)$ in the limit $\\tau\\to0$.","core_discovery":"The paper's central claim is that for any Markov process satisfying local detailed balance, the estimator $\\hat\\sigma_{A,B}(\\tau) = \\frac{2}{\\tau}\\frac{\\chi_{A,B}(\\tau)^2}{D_{A,B}(\\tau)}$ — where $\\chi_{A,B}$ is the antisymmetric part of the two-time correlation function $\\langle A(0)B(\\tau)\\rangle$ and $D_{A,B}$ is an accessible combination of correlation functions — is a lower bound on the mean entropy production rate $\\sigma$ in a non-equilibrium steady state for arbitrary lag $\\tau$. For time-dependent processes, the same estimator in the limit $\\tau\\to0$ bounds the instantaneous entropy production rate $\\sigma(t)$. This result holds whether the observed states are fully resolved or coarse-grained into mesostates, and it requires no knowledge of the hidden dynamics.","pith_inferences":["The vanishing-lag estimator is essentially a differential measurement of time-reversal asymmetry; this suggests that experiments with fast sampling, such as fluorescence correlation spectroscopy or single-molecule tracking, could extract dissipation from the slope of correlation functions without resolving hidden states.","The optimization over constant shifts hints that the tightest bound for given observables may not come from zero-mean observables, and that a data-driven scan over shifts could be a practical protocol for tightening thermodynamic estimates.","Because the proof only needs the path-weight structure at the underlying level, analogous bounds might hold for other divergence-based irreversibility measures or for higher-order correlation functions, though the paper does not claim this."],"forward_implications":["In a nonequilibrium steady state, a lower bound on entropy production follows from correlation functions at any single lag, so experimental data with limited sampling rates still yield a valid bound.","For time-dependent relaxation or periodic driving, the bound holds instantaneously in the vanishing-lag limit, requiring only the short-time slope of correlation functions.","Constant shifts of the observables leave the correlation asymmetry invariant while the denominator has a unique global minimum, so the bound can be optimized analytically by a computed shift (Eq. 53).","The bound applies to discrete Markov chains and to overdamped Langevin dynamics, including partially observed or coarse-grained versions, because the latter arise as limits of the former."],"supporting_citations":[{"why":"Supplies the entropy production formula and the path-weight fluctuation theorem used to identify the Kullback-Leibler divergence with $\\tau\\sigma$.","marker":"[5]"},{"why":"Provides the Markov-network setup and local detailed balance condition underlying the proof.","marker":"[4]"},{"why":"Defines the entropy production rate in terms of currents and affinities in stochastic thermodynamics.","marker":"[3]"},{"why":"Presents the prior correlation-based bound for normalized state observables that this paper generalizes to arbitrary coarse-grained observables.","marker":"[42]"}],"fun_headline_variants":["Correlation asymmetry lower-bounds entropy production","Two-time correlations infer dissipation lower bound","Entropy rate bound from coarse-grained correlation asymmetry","Dissipation inferred from correlation functions only","Markovian systems: correlation asymmetry sets entropy bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The hidden microscopic dynamics must be a Markov process obeying local detailed balance, so that the entropy production formula and the path-weight fluctuation theorem apply.","fun_headline_variants_meta":{"raw":{"variants":["Correlation asymmetry lower-bounds entropy production","Two-time correlations infer dissipation lower bound","Entropy rate bound from coarse-grained correlation asymmetry","Dissipation inferred from correlation functions only","Markovian systems: correlation asymmetry sets entropy bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000369,"raw_usage":{"total_tokens":1916,"prompt_tokens":817,"completion_tokens":1099,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":433,"completion_tokens_details":{"reasoning_tokens":1032}},"tokens_in":433,"tokens_out":1099,"duration_ms":8716,"temperature":1.0,"reasoning_tokens":1032,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:49:39.970792+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a three-state Markov network with known transition rates and a coarse-grained pair of observables; compute the two-time correlations exactly or with high statistics, evaluate $\\hat\\sigma_{A,B}(\\tau)$, and compare it with the true entropy production rate. Any case with $\\hat\\sigma_{A,B}(\\tau)>\\sigma$ (or $\\hat\\sigma_{A,B}(t)>\\sigma(t)$ in the time-dependent limit) would refute the bound.","supporting_citations":[{"cited_title":"1(a) to be fully observable and driven periodically in time","cited_arxiv_id":null,"evidence_quote":"Provides the Markov-network setup and local detailed balance condition underlying the proof."},{"cited_title":"1(a) with cyclesC 1 = 1→2→3→1 andC 2 = 1→4→3→1","cited_arxiv_id":null,"evidence_quote":"Defines the entropy production rate in terms of currents and affinities in stochastic thermodynamics."}],"review_version":1}