{"id":"70629829-eccf-4b51-8904-6cb352b7529c","arxiv_id":"2506.14416","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"The authors derive a linear-response formula for entropy production of slowly driven non-equilibrium Markov systems, use it to optimize protocols, and find overshoot protocols with diverging parameters and finite dissipation in the slow-driving limit.","lead":"This paper derives a formula for the entropy produced when a non-equilibrium system is driven between steady states, then uses it to find protocols that minimize dissipation. The approach generalizes a well-known near-equilibrium method, and in a three-state example it predicts optimal protocols can overshoot and keep entropy production finite in the slow-driving limit.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Optimal-protocol predictions rest on a second-order alpha-dot expansion that Appendix F shows is violated by the optimized protocols; the finite-entropy and diverging-parameter claims are therefore unsupported.","rationale":"The paper proposes a useful-looking linear-response expansion for entropy production in slowly driven non-equilibrium Markov systems and makes a genuine effort to test it against exact master-equation integration. Credit is due for the explicit Appendix F discussion and the accompanying code. However, the central physical claim, that optimized protocols have diverging parameters and finite entropy production in the slow-driving limit, requires that the second-order expansion be accurate on exactly those optimized protocols. The authors state that this condition fails: the optimized protocols are not slowly driven even at large tf, and the expansion and exact integration diverge as tf grows. Thus the headline result is an artifact of optimizing an uncontrolled approximation, not an established property of the real dynamics. The boundary-term treatment is an additional gap, but the expansion-validity problem alone is decisive. The reader's weakest-assumption analysis identified the same load-bearing issue, and no independent evidence in the manuscript repairs it.","tokens_in":18605,"tokens_out":4565,"duration_ms":50863,"concrete_test":"Reproduce the three-state model with the provided code and compute the exact master-equation entropy production for the optimized overshoot protocols at tf = 50, 100, and 200, well beyond the range in Fig. 6. Determine (i) whether max_t |alpha_dot(t)| tau_rel(alpha(t)) remains of order 1 or grows as tf increases, and (ii) whether the exact total entropy approaches the expansion prediction or continues to deviate and grow. If either (i) or (ii) holds, the claimed finite-entropy slow-driving limit is not supported. As a secondary check, repeat the variational optimization while retaining the delta[alpha_dot A(3)] boundary contribution to see whether the optimal protocol changes.","verdict_should_be":"REJECT","load_bearing_attack":"For Eqs. (23)-(24) to support the headline results, the second-order expansion of sigma in powers of alpha_dot must be accurate on the protocols that minimize it. The paper's own Appendix F and Fig. 6 contradict this in the regime that matters: for the optimized 'overshoot' protocols, the deviation between the expansion and direct Euler integration of the master equation grows with protocol duration, and the text states that increasing tf does not make alpha_dot tend to zero and that 'we are not within the alpha_dot range where a 2nd order expansion is a sufficient approximation for the dissipation.' Since the claimed finite entropy production and diverging parameters occur precisely for these long-duration protocols, the quantity being minimized is an uncontrolled approximation of the true entropy production. Minimizing an approximate objective exactly does not establish a property of the exact process, and the reported discrepancy is direct evidence that it does not. A secondary issue: the boundary term [alpha_dot A(3)] in Eq. (23) is omitted from the Euler-Lagrange equation (33) on the ground that it is 'fixed', but alpha_dot at the boundaries is not fixed in the variational problem, so this omission can also shift the minimizing protocol.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a second-order expansion of the stochastic entropy production for Markov jump processes driven by a time-dependent parameter α(t), starting from the Mandal–Jarzynski expansion of the probability distribution. It expresses the expansion coefficients in Eqs. (17)–(21), rewrites them as time-correlation functions of three observables in Eqs. (25)–(32), and then uses a Lagrangian/Euler–Lagrange formulation to find protocols that minimize the total entropy production between fixed endpoints. Applied to a three-state non-equilibrium model, the optimization produces 'direct' protocols for short durations and 'overshoot' protocols for durations longer than a threshold, with apparent finite entropy production and diverging control parameters in the long-duration limit. The paper argues that this closes a gap by extending linear-response-type optimization to systems far from equilibrium.","tokens_in":18872,"tokens_out":8764,"duration_ms":98266,"significance":"If the central claims were valid, the framework would be a valuable extension of Sivak–Crooks thermodynamic geometry to non-equilibrium steady states and would provide a computationally tractable route to optimal protocols. The manuscript contains substantial analytic work: the entropy-production coefficients, their correlation-function representation, and the exact toy-model calculations in the appendices are nontrivial and appear internally consistent, aided by the provided Mathematica notebook. The overshoot phenomenon is a surprising and potentially falsifiable prediction. However, because the paper's headline results in the slow-driving limit are obtained in a regime that Appendix F itself identifies as outside the validity of the second-order expansion, the significance of those claims is not currently established; the value of the work at this stage is mainly the formal expansion and its toy-model implementation for short protocols.","major_comments":[{"comment":"The central claims of finite entropy production and diverging parameters in the slow-driving limit are not supported by the presented evidence. The optimized protocols are obtained by minimizing the approximate Lagrangian in Eq. (24), which is a second-order expansion in α̇. For the optimized 'overshoot' protocols, Fig. 6 and Appendix F show that the difference between the expansion and direct Euler integration of the master equation grows with protocol duration, and the text states that increasing tf does not make α̇ tend to zero and that 'we are not within the α̇ range where a 2nd order expansion is a sufficient approximation for the dissipation.' Consequently, the leveling off of the entropy production in Fig. 5 and the statement that longer durations lead to increasingly extreme overshoots with E→∞ are properties of the approximate objective, not established properties of the exact entropy production. Minimizing an uncontrolled approximation does not determine the true optimal protocol, and the reported discrepancy is direct evidence against the headline claims. The abstract's 'finite entropy production in the slow-driving limit' therefore requires either an exact calculation or a controlled approximation valid on the optimized protocols.","section":"Sec. IV, Appendix F, Figs. 5–6"},{"comment":"The boundary term [α̇A^(3)] in Eq. (23) cannot be omitted from the variational problem on the stated grounds. Section III says this term is 'fixed' and therefore identical for all protocols with the same boundary conditions, but the variational problem fixes only α(ti), α(tf), and the duration; α̇ at the endpoints is not fixed. Unless additional boundary conditions specifying α̇(ti) and α̇(tf) are imposed, the Euler–Lagrange equation (33) is not the stationarity condition for the full functional (23). Including the boundary term would, at minimum, add natural boundary conditions, and it can shift the minimizing protocol. Since the overshoot protocols have nonzero endpoint velocities, this omission is relevant to the reported optimal protocols.","section":"Sec. III, Eqs. (23) and (33)"}],"minor_comments":[{"comment":"The phrase 'thirth order expansion' should be 'third order expansion'.","section":"Sec. IIB"},{"comment":"The sentence 'the main idea is to assumptions are that there exists an equilibrium probability distribution' contains a grammatical error and should be rewritten.","section":"Sec. IIA"},{"comment":"The expression for L appears to be missing a closing parenthesis before the final α̇² term; it should read (A^(1)(α)+A^(2)(α)−A^(3)(α))α̇².","section":"Eq. (24)"},{"comment":"The phrase 'up to rather short protocol duration (tp ≳ 1)' is ambiguous or contradictory: 'up to' suggests an upper bound, while '≳' suggests a lower bound. Please clarify which duration regime is meant.","section":"Fig. 5 caption and Sec. IV"},{"comment":"The statement 'Since W is symmetric and therefore diagonalizable' is incorrect: the transition matrix W is generally not symmetric. The spectral decomposition used in Eq. (11) requires diagonalizability, which is a generic property but not guaranteed by symmetry of W.","section":"Appendix D"},{"comment":"The reference 'see 6' should read 'see Fig. 6'.","section":"Appendix F"}],"recommendation":"reject","confidential_remarks":"The central finding of the paper is undermined by its own Appendix F: the expansion is uncontrolled precisely on the optimized protocols that produce the advertised 'remarkable properties', and the boundary-term issue compounds the problem. This is not a presentation issue; the headline claims are not established. A revision could potentially salvage the formal expansion, but the current claims would need to be substantially narrowed, and the variational treatment would need to be corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one for the expansion, not for the abstract. The second-order entropy production expansion for slowly driven non-equilibrium steady states is a real step beyond Mandal-Jarzynski, and the correlation-function forms (Eqs. 28-32) make it usable. The toy model is worked out thoroughly, the exact master equation integration is there, and the Mathematica notebook is public. That part of the paper is solid.\n\nThe problem is that the headline results are taken from a regime where the paper's own expansion is known to fail. For the optimized overshoot protocols, Appendix F says increasing the duration doesn't make alpha_dot tend to zero and that the second-order expansion is not a sufficient approximation. Figure 6 shows the deviation between expansion and exact integration growing with protocol duration. So the claims that optimal protocols have diverging parameters and finite entropy production in the slow-driving limit are not backed by the evidence; you are minimizing an approximate functional in the regime where the approximation is uncontrolled. The paper is honest about the discrepancy, but the abstract and discussion don't qualify it. There is also a secondary variational issue: the boundary term alpha_dot A(3) is omitted from the Euler-Lagrange equation because it is called 'fixed', but alpha_dot at the endpoints is free in this problem, so the boundary term may not be constant over the allowed protocols. That needs a closer look.\n\nThe framework may be salvageable, and the overshoot idea is interesting, but as written the central physical claims are unsupported. I'd send it to review because the formal expansion and the reproducible code deserve referee time; I'd ask the authors to either prove error control for the optimized protocols or drop the diverging-parameter/finite-entropy statements from the abstract. The citation pattern looks fine, and the Appendix F transparency is a point in its favor.","headline":"The expansion is a real contribution, but the headline claims come from a regime the paper itself says the expansion cannot handle.","tokens_in":19340,"tokens_out":3302,"would_cite":true,"duration_ms":36046,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82C31","60J27","49K15"],"pacs":["05.70.Ln"],"model":"deepseek-v4-flash","headline":"Far-from-equilibrium drives can overshoot and still save entropy.","keywords":["stochastic thermodynamics","entropy production","non-equilibrium steady states","linear response","optimal control","master equation","protocol optimization","thermodynamic geometry"],"falsifier":"For the three-state model, numerically integrate the full master equation for an optimized overshoot protocol of duration, say, $t_p = 10$ and compare the exact entropy production with the expansion; the central claim would be settled if the exact dissipation shows a finite plateau as $t_p$ grows and if including third-order corrections in $\\dot\\alpha$ leaves that plateau unchanged.","tokens_in":18382,"feed_emoji":"⚙️","tokens_out":7404,"duration_ms":74813,"temperature":0.7,"pith_summary":"This paper tries to close a gap in stochastic thermodynamics: how to compute and minimize the entropy cost of slowly driving a system from one non-equilibrium steady state to another. It derives a general expansion of the total entropy production in powers of the driving rate $\\dot\\alpha$, with coefficients built from the pseudo-inverse of the transition matrix, and turns the minimization into a Euler-Lagrange problem. For a three-state model driven by an energy level, the optimal protocol is not monotone: for long enough protocol durations it first overshoots far past the target, then returns, producing less entropy than a direct or linear protocol. The authors further claim that in the slow-driving limit the optimal parameters diverge while the entropy production stays finite, in contrast to near-equilibrium systems where the optimal path is independent of duration and dissipation falls off as $t_f^{-1}$. A sympathetic reader would care because this would give a practical optimization scheme for molecular motors, memories, and other far-from-equilibrium devices.","feed_headline":"Far-from-equilibrium drives can overshoot and still save entropy","feed_subtitle":"A linear-response scheme predicts finite dissipation and diverging control parameters in the slow-driving limit.","key_machinery":"The load-bearing device is the second-order adiabatic expansion of the probability distribution, $p(t) = p_{\\rm SS} + \\dot\\alpha W^\\dagger \\partial_\\alpha p_{\\rm SS} + \\dot\\alpha^2 W^\\dagger(\\partial_\\alpha W^\\dagger \\partial_\\alpha p_{\\rm SS} + W^\\dagger \\partial_\\alpha^2 p_{\\rm SS}) + \\ddot\\alpha (W^\\dagger)^2 \\partial_\\alpha p_{\\rm SS}$, in which $W^\\dagger$ is the pseudo-inverse of the master-operator transition matrix. Inserting this expansion into the entropy-production rate makes the dissipation depend only on the instantaneous parameter, its velocity, and the steady state, not on the full trajectory; the coefficients can be rewritten as time-correlation functions of three observables, making them measurable in simulations or experiments. The same expansion plus the Euler-Lagrange first integral turns protocol design into a quadrature, and the non-zero steady-state term $\\sigma_{\\rm SS}$ acts like a potential landscape that makes overshooting optimal.","core_discovery":"The central claim is that the entropy production of a slowly driven Markov system far from equilibrium can be written as $\\Delta S_{\\rm tot} = \\int_{t_i}^{t_f} L(\\alpha,\\dot\\alpha)\\,dt + [\\dot\\alpha A^{(3)}]_{t_i}^{t_f}$ with $L = \\sigma_{\\rm SS} + F\\dot\\alpha + (A^{(1)}+A^{(2)}-A^{(3)})\\dot\\alpha^2$, where $\\sigma_{\\rm SS}$ is the steady-state dissipation and $F,A^{(1)},A^{(2)},A^{(3)}$ are correlation functions of three observables: the entropy production per state, the logarithmic derivative of the steady-state distribution, and the logarithmic derivative of the transition rates. The Euler-Lagrange equation for this Lagrangian reduces to a first integral giving $\\dot\\alpha = \\pm \\sqrt{(E+\\sigma_{\\rm SS})/(A^{(1)}+A^{(2)}-A^{(3)})}$, which can be integrated numerically to find protocols of any duration. Applied to a three-state model with a non-conservative force, the optimized protocols split into direct protocols and overshoot protocols for durations beyond $t_p^* \\approx 2.70$; the overshoot spends most of the time in a region of low steady-state dissipation and beats the linear protocol, with entropy production leveling off at long durations while the parameter excursion grows without bound.","pith_inferences":["If the finite-entropy plateau survives higher-order corrections, the practical lesson is that deliberately moving a control parameter outside its operating range can be thermodynamically cheaper than a monotone interpolation; this should be looked for in molecular-motor and memory-reset experiments.","The correlation-function form suggests the optimal protocol could be estimated from a single long trajectory at fixed parameter values, without knowing the transition rates, which is a testable route the paper does not spell out.","For multiple control parameters, the matrix structure of the $A$-coefficients means overshoots in one parameter should generically couple to excursions in others, so optimal non-equilibrium protocols in higher dimensions are likely to be curved and non-monotone rather than straight lines."],"forward_implications":["Entropy production for slow far-from-equilibrium driving is accessible from correlation functions of three local observables, so optimal protocols can be computed numerically for arbitrary Markov models, not only near-equilibrium ones.","In the model, optimal protocols are non-monotone for durations beyond about 2.70, and their entropy production grows much more slowly with duration than a linear protocol, plateauing at long times.","In the slow-driving limit the optimal parameter excursion diverges while the entropy production remains finite, so slowness no longer implies small amplitude for non-equilibrium control.","The framework reduces to the known near-equilibrium result when detailed balance holds, because the terms $F$, $A^{(2)}$, and $A^{(3)}$ vanish while $A^{(1)}$ becomes the familiar thermodynamic metric.","The appendix extends the construction to several control parameters, allowing simultaneous optimization of multi-parameter protocols."],"supporting_citations":[{"why":"Supplies the slow-transition expansion of the probability distribution in powers of the driving rate that the general formula is built on.","marker":"[20]"},{"why":"The near-equilibrium thermodynamic-metric framework and optimal-path scheme that this paper generalizes to far-from-equilibrium systems.","marker":"[15]"},{"why":"Establishes the thermodynamic-length picture of controlling dissipation that the optimization approach extends.","marker":"[14]"},{"why":"Provides the entropy-production-rate expression used to define the cost being minimized.","marker":"[22]"},{"why":"States the near-equilibrium properties, such as dissipation scaling as the inverse duration and duration-independent optimal paths, that the non-equilibrium results are contrasted against.","marker":"[23]"},{"why":"Supplies the Green's-function time-integral form used to rewrite the pseudo-inverse and its square as correlation integrals.","marker":"[34]"}],"fun_headline_variants":["Overshoot protocol wins slow non-equilibrium drives","Finite entropy cost via parameter overshoot","Slow drives: overshoot saves entropy","Diverging controls, bounded dissipation","Overshoots reduce entropy in slow transports"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results assume that the second-order approximation in the driving speed is accurate for the optimized protocols, even though the paper itself notes those protocols drive fast enough to leave the regime where the approximation is reliable.","fun_headline_variants_meta":{"raw":{"variants":["Overshoot protocol wins slow non-equilibrium drives","Finite entropy cost via parameter overshoot","Slow drives: overshoot saves entropy","Diverging controls, bounded dissipation","Overshoots reduce entropy in slow transports"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000152,"raw_usage":{"total_tokens":1171,"prompt_tokens":883,"completion_tokens":288,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":221}},"tokens_in":499,"tokens_out":288,"duration_ms":3295,"temperature":1.0,"reasoning_tokens":221,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:17:50.262682+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the three-state model, numerically integrate the full master equation for an optimized overshoot protocol of duration, say, $t_p = 10$ and compare the exact entropy production with the expansion; the central claim would be settled if the exact dissipation shows a finite plateau as $t_p$ grows and if including third-order corrections in $\\dot\\alpha$ leaves that plateau unchanged.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the slow-transition expansion of the probability distribution in powers of the driving rate that the general formula is built on."},{"cited_title":"Aurell, C","cited_arxiv_id":null,"evidence_quote":"The near-equilibrium thermodynamic-metric framework and optimal-path scheme that this paper generalizes to far-from-equilibrium systems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the thermodynamic-length picture of controlling dissipation that the optimization approach extends."},{"cited_title":"Reliability and entropy production in non-equilibrium electronic memories","cited_arxiv_id":"2103.01184","evidence_quote":"Provides the entropy-production-rate expression used to define the cost being minimized."},{"cited_title":"Mandal and C","cited_arxiv_id":null,"evidence_quote":"States the near-equilibrium properties, such as dissipation scaling as the inverse duration and duration-independent optimal paths, that the non-equilibrium results are contrasted against."},{"cited_title":"Herpich, J","cited_arxiv_id":null,"evidence_quote":"Supplies the Green's-function time-integral form used to rewrite the pseudo-inverse and its square as correlation integrals."}],"review_version":1}