{"id":"3455efd4-9da3-4e29-9443-c1eb7bad2e09","arxiv_id":"2509.11314","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A variational framework over optimal-transport maps determines the optimal final distribution that minimizes finite-time thermodynamic cost under task constraints.","lead":"This paper derives a general way to choose the final probability distribution, not just the path, when driving a tiny particle system in finite time, so that entropy or work cost is minimized while the task (transport, squeezing, erasure, measurement) is still achieved. It gives explicit optimal formulas for several tasks, including non-Gaussian thermal squeezing and Gaussian measurement and feedback.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Gaussian ansatz in Appendix B is not proven globally optimal, so Eqs. (42)–(45) may be upper bounds rather than the claimed minimal measurement and feedback costs.","rationale":"The paper develops a coherent variational framework: Eq. (14)-(15) is a legitimate Lagrange-multiplier formulation, and Eq. (22) is a useful general variation formula. The thermal-squeezing application is genuinely robust because the objective is convex and the covariance constraints are linear, so the affine optimal map is globally optimal even for non-Gaussian initial distributions. The particle-transport and information-erasure discussions are either straightforward or explicitly reproduce prior results. The soft spot is exactly where the reader placed it: the measurement and feedback sections claim optimality but solve Eq. (41) only under a Gaussian ansatz with no proof that the stationary point is the global minimizer over all final distributions with the required mutual information. Because the mutual-information constraint is non-convex and Eq. (41) is only necessary, the reported costs in Eqs. (42)-(45) are not yet established as true thermodynamic optima. This is an addressable gap: either prove global optimality (e.g., via a suitable transport-information inequality) or rephrase the results as 'optimal within the Gaussian family.' The recommended verdict remains CONDITIONAL, matching the reader; no new verdict adjustment is needed.","tokens_in":34678,"tokens_out":7520,"duration_ms":90586,"concrete_test":"Numerically minimize the measurement cost over non-Gaussian final conditionals. Fix p_X(x)=N(0,1), p_Y(y)=N(0,ΞYY_ini), discretize x and y on a common grid, and search over normalized conditional distributions q(y|x) satisfying q_X=p_X and I(X;Y)=ln2, minimizing ∑_x p_X(x) W_2^2(N(0,ΞYY_ini), q(·|x)) using the exact 1-D Wasserstein formula per x. Initialize from the Gaussian solution Eq. (42) and from several non-Gaussian random conditional distributions. If any feasible q gives cost below 0.75 ΞYY_ini/(Dτ), the Gaussian ansatz is not globally optimal and Eqs. (42)–(43) need qualification. An analogous discretized search for feedback with I_ini=ln3, I_f=0 would test Eqs. (44)–(45).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The advertised information-processing results rest on a global-optimality claim that the paper does not establish. In Sec. IV F, the variational equation (41) is solved only after restricting final distributions to be Gaussian and postulating the conditional transport-map forms (B3) and (B13). Equation (41) is a first-order stationarity condition for a problem whose objective (36) is convex in the conditional transport map but whose equality constraint I_fin = I_f is a level set of a mutual-information functional; such level sets are generally non-convex, so multiple stationary points can exist. The paper gives no uniqueness or global-minimality argument, and no check outside the Gaussian family is performed. Thus the values in Eqs. (42)–(45) are currently proven optimal only within the Gaussian ansatz; if a non-Gaussian conditional distribution with the same mutual information has lower partial entropy production, these formulas are upper bounds, not the claimed minima. This does not invalidate the general framework of Eqs. (14)–(15) and (22), nor the moment-constrained thermal-squeezing result, which is globally justified by convexity; it specifically undermines the 'truly optimal' wording applied to measurement and feedback.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a unified variational framework for finite-time thermodynamic optimization in overdamped Langevin systems. Instead of fixing the final distribution, as in the standard Benamou–Brenier/Wasserstein approach, the authors impose task-specific equality constraints A[p_fin]=A_f and minimize the thermodynamic cost over both the transport map and the final distribution. The central result is the Lagrange multiplier system in Eqs. (14)–(15), together with the general variation formula Eq. (22) for functionals of the final distribution. The framework is applied to particle transport, thermal squeezing, information erasure, work optimization in quadratic potentials, free-energy control, and measurement/feedback. For measurement and feedback, explicit formulas are obtained for Gaussian final distributions, Eqs. (42)–(45), using the ansatz in Appendix B.","tokens_in":34923,"tokens_out":8713,"duration_ms":110890,"significance":"If the main claims hold, the paper provides a useful unified treatment of finite-time task optimization that goes beyond fixing the final distribution, and it generalizes earlier results on erasure [39,40] and quadratic-potential work [41]. The derivation of the variational equations from first principles is clean, and the thermal-squeezing result is genuinely non-Gaussian and globally justified by the Gelbrich/Wasserstein lower bound. The general variation formula Eq. (22) is a valuable technical contribution. The information-processing applications are potentially significant, but their current proof of optimality is restricted to a Gaussian ansatz; without a global optimality argument, Eqs. (42)–(45) are only proven optimal within that family. The paper is therefore not yet at the level of its advertised 'truly optimal' claims for measurement and feedback.","major_comments":[{"comment":"The measurement and feedback results are derived under an explicit Gaussian restriction: 'We restrict the final distributions to be Gaussian' (Appendix B1) and the ansatz forms (B3), (B13) for the conditional transport maps. Equation (41) is only a first-order stationarity condition for a problem whose constraint I_fin=I_f is a level set of mutual information; such level sets are generally non-convex, and stationarity does not imply global optimality. No uniqueness, second-order, or non-Gaussian check is provided. Figures 3 and 4 compare only Gaussian candidates, so they do not test the ansatz. Consequently, Eqs. (42)–(45) are currently proven optimal only within the Gaussian family; if a non-Gaussian final distribution with the same mutual information has lower partial entropy production, these formulas are upper bounds rather than the claimed minima. This affects the load-bearing 'trul","section":"Appendix B and Sec. IV F, Eqs. (42)–(45)"},{"comment":"Even within the Gaussian family, the paper solves only the first-order condition (41) after postulating the form of the conditional map. For a non-convex constrained problem, multiple stationary points can exist, and the solved point could be a saddle point or local maximum. The authors should at least verify that the solution satisfies a second-order sufficient condition, or provide independent evidence that the Gaussian solution is the global minimizer. This is a separate, more restricted concern, but it reinforces the need to calibrate the optimality claims for measurement and feedback.","section":"Sec. IV F, Eq. (41) and stationarity"}],"minor_comments":[{"comment":"Equation (18) has 'F_fin[T] = β ∫ dr ...'; from the preceding definition F = β^{-1} D + F_eq, the prefactor should be β^{-1}. The following equation (19) uses β^{-1}, so this appears to be a typo.","section":"Eq. (18)"},{"comment":"In Eq. (B13), the last term is written as (Ξ^{XY}_{fin}/Ξ^{YY}) x, but the conditional mean of X|Y should involve y, not x. Please check whether this is a typo; the same applies to the surrounding discussion.","section":"Eq. (B13)"},{"comment":"Equation (A23) applies the local formula (22) to the mutual information, which is a nonlocal functional of p_fin. The derivation is valid because Eq. (A21) reduces δI_fin to ∫ (δp_fin) ι_fin, but the text should state this explicitly. The current wording 'Using Eq. (22), we have' is too terse for a reader checking the validity of Eq. (41).","section":"Appendix A3"},{"comment":"The index notation in Eq. (41) is ambiguous: T^Y_{meas,i} and r_i should be specified to run only over the Y components, since the X component of the cost variation is identically zero. A short sentence clarifying the index range would improve readability.","section":"Sec. IV F, Eq. (41)"}],"recommendation":"major_revision","confidential_remarks":"The central variational framework is sound and valuable; the main revision should focus on the global optimality claim for measurement and feedback. If the authors can prove the Gaussian solution is globally optimal, or if they soften the claim to Gaussian-optimal, the paper would be suitable for publication. The Gaussian ansatz should also be disclosed earlier in the presentation so that readers are not misled by the unqualified 'optimal' statements in the abstract and introduction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid paper that usefully unifies a set of final-distribution optimization problems in finite-time stochastic thermodynamics. The general variational equations (14)–(15) and the functional variation formula (22) are cleanly derived, and the non-Gaussian thermal squeezing example is a genuine new result. The main weakness is that the advertised information-processing applications (measurement and feedback) are solved only within a Gaussian ansatz, so the 'truly optimal' language overshoots for those cases.\n\nStrengths first. The reduction of the task to a constrained transport problem is natural, and the calculus is careful: they derive the variation of a general functional of the final distribution, which is the backbone of the framework. They also check against earlier independent results (erasure [39,40], quadratic-potential work [41]), which is reassuring. The thermal squeezing results for arbitrary (possibly non-Gaussian) initial distributions, with covariance constraints, are new and correct: the optimal map is the affine map given in Eq. (32), and the example with a Laguerre–Gaussian profile is a nice demonstration that the framework goes beyond Gaussian assumptions.\n\nNow the soft spots. In Sec. IV F and Appendix B, the measurement and feedback problems are solved by restricting the final distributions to be Gaussian and postulating the conditional transport maps (B3) and (B13). The variational equation (41) is only a first-order stationarity condition, and the mutual-information constraint I_fin = I_f is non-convex in the final distribution, so multiple stationary points can exist. There is no proof that the Gaussian solution is globally optimal among all final distributions with that mutual information. Thus Eqs. (42)–(45) are proven optima within the Gaussian family, not unconditional minima. The main text calls these 'the optimal final distribution' and 'the optimal value' without that qualification. That should be fixed either by a global optimality proof or by adding the qualifier 'within the Gaussian ansatz.'\n\nOther minor points: the free-energy control subsection is only sketched, with no worked example, and the erasure and quadratic-potential sections state reproduction of earlier results without showing the calculation. These are addressable.\n\nOverall: the general framework and the thermal squeezing result are worth publishing. The measurement/feedback claims need either a proof of global optimality or an explicit qualifier. This paper deserves peer review, not desk rejection.","headline":"Clean variational framework for optimizing final distributions, but the measurement/feedback results are Gaussian-ansatz optima, not proven global optima.","tokens_in":35399,"tokens_out":3274,"would_cite":true,"duration_ms":38802,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q22","82C31","60J70"],"pacs":["05.70.Ln","05.40.-a"],"model":"deepseek-v4-flash","headline":"A variational framework shows that relaxing the final distribution lowers the minimal thermodynamic cost of finite-time tasks in overdamped Langevin systems, with the optimal final distribution determined by Lagrange multiplier equations ov","keywords":["optimal transport","stochastic thermodynamics","finite-time processes","entropy production","Lagrange multipliers","final distribution optimization","measurement and feedback","thermal squeezing"],"falsifier":"Construct a non-Gaussian joint distribution for (X,Y) with mutual information I_f = ln 2 and compare its required partial entropy production (obtained by numerically solving Eq. (41) without the Gaussian ansatz) against the Gaussian result (43). If a non-Gaussian solution gives a lower cost than Ξ_ini^YY ρ_f²/(Dτ), the optimality claim fails.","tokens_in":34536,"feed_emoji":"⚙️","tokens_out":1593,"duration_ms":22905,"temperature":0.7,"pith_summary":"The paper argues that in finite-time stochastic thermodynamics, fixing the final distribution—as standard optimal transport theory does—is unnecessary and wasteful. Any task that can be expressed as an equality constraint on a functional of the final distribution leaves many admissible final distributions, and the paper proposes to optimize over them using Lagrange multipliers, yielding variational equations whose solution gives the true optimal transport map, the optimal final distribution, and the optimal protocol. The authors show that this unified framework reproduces known results for particle transport, information erasure, and quadratic-potential work optimization, while producing new analytic results for thermal squeezing with non-Gaussian distributions and for measurement and feedback processes. If correct, this provides design principles for fast, energy-efficient information-processing devices and thermodynamic machines.","feed_headline":"Optimizing the final state cuts thermodynamic cost","feed_subtitle":"A variational framework over optimal transport maps gives the true minimum cost for finite-time tasks, from erasure to feedback.","key_machinery":"The key object is the transport map T that pushes the initial distribution p_ini forward to the final distribution p_fin. The paper replaces the standard fixed-final-distribution optimal transport problem with a variational problem over T, using Lagrange multipliers λ to enforce the task constraint A[T]=A_f. The central identity is the variation formula δA[T]/δT_i = ∂_{r_j}{ĝ(T(r))} J̃_{ji}(r) p_ini(r) (Eq. (22)), where ĝ is the derivative of the task functional's density with respect to the final density. This reduces the constrained optimization to solving a system of algebraic equations for T and λ, from which the optimal protocol follows via the Benamou–Brenier velocity field.","core_discovery":"The central claim is that the minimum thermodynamic cost for a task specified by a constraint A[p_fin]=A_f is obtained by solving the variational equations δC̃_τ/δT + λ·δA/δT = 0 and A[T]=A_f, where T is the optimal transport map from the initial to the final distribution, C̃_τ[T] is the cost functional (entropy production, work, or partial entropy production), and A[T] is the task functional evaluated after transport. The paper derives the general variation of any final-distribution functional (Eq. (22)), applies it to several tasks, and obtains closed-form optimal final distributions and minimal costs, including the analytic Gaussian results for measurement (Eqs. (42)–(43)) and feedback (E","pith_inferences":["A direct testable extension is to numerically solve the full variational equation (41) for measurement in non-Gaussian settings and check whether a non-Gaussian final distribution with the same mutual information I_f achieves a lower partial entropy production than the Gaussian-ansatz result (43).","The framework suggests that any 'thermodynamic speed limit' that fixes both endpoints may be systematically tightened by optimizing the terminal distribution; a similar hierarchy might exist for Markov jump processes, where the optimal transport formulation is less straightforward.","The variational equation for mutual information in measurement is nonlinear and may admit multiple solutions; the paper's Gaussian ansatz picks one branch, but other branches could correspond to genuinely different—possibly better—protocols, a point the paper leaves open.","The result for free-energy control implies that the minimal entropy production to reach a target nonequilibrium free energy may be significantly lower than the cost of reaching any specific final state, a fact relevant for designing work-storage devices."],"forward_implications":["If the framework is correct, any thermodynamic task that can be expressed as an equality constraint on a final-distribution functional has an explicit variational characterization of its minimal finite-time cost.","The optimal final distribution for thermal squeezing is generally non-Gaussian, and its minimal entropy production is given by Eq. (33), which extends prior Gaussian-only results.","For measurement and feedback, the optimal final distribution and minimal partial entropy production are given by Eqs. (42)–(45), enabling quantitative design of finite-time information processors.","The framework unifies and generalizes earlier results on optimal finite-time erasure and work in quadratic potentials, showing that these are special cases of the same variational scheme.","The method applies equally to entropy production, work, and partial entropy production, covering both conservative and nonconservative driving, and extends naturally to far-from-equilibrium regimes."],"fun_headline_variants":["Final distribution choice cuts thermodynamic cost","Variational framework optimizes finite-time tasks","Optimal final states minimize thermodynamic cost","Tuning final distribution reduces dissipation","Finite-time thermodynamics: choose final state wisely"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"For the measurement and feedback results, the paper assumes the optimal final distributions are Gaussian (Appendix B); if a non-Gaussian final distribution with the same mutual information yields lower partial entropy production, the stated costs are upper bounds rather than true optima.","fun_headline_variants_meta":{"raw":{"variants":["Final distribution choice cuts thermodynamic cost","Variational framework optimizes finite-time tasks","Optimal final states minimize thermodynamic cost","Tuning final distribution reduces dissipation","Finite-time thermodynamics: choose final state wisely"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000388,"raw_usage":{"total_tokens":1844,"prompt_tokens":667,"completion_tokens":1177,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":411,"completion_tokens_details":{"reasoning_tokens":1115}},"tokens_in":411,"tokens_out":1177,"duration_ms":10593,"temperature":1.0,"reasoning_tokens":1115,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T16:47:16.907168+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a non-Gaussian joint distribution for (X,Y) with mutual information I_f = ln 2 and compare its required partial entropy production (obtained by numerically solving Eq. (41) without the Gaussian ansatz) against the Gaussian result (43). If a non-Gaussian solution gives a lower cost than Ξ_ini^YY ρ_f²/(Dτ), the optimality claim fails.","supporting_citations":[],"review_version":1}