{"id":"c97c1cb9-ddf9-45da-be7e-5a7f38c37189","arxiv_id":"2607.27740","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A non-reciprocal (action-reaction-violating) spring between two trapped Brownian particles produces a persistent circulating probability current from a single heat bath, mapping the model onto Brownian gyration.","lead":"This paper solves a minimal model in which two tiny particles in a warm fluid interact through a spring that pulls asymmetrically, violating Newton's third law. It shows this asymmetry alone, with no temperature difference and no driving, creates a steady circulating probability current — a permanent out-of-equilibrium state.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Finite-rest-length section rests on unquantified numerics; exact l=0 result cannot carry the general claim by itself.","rationale":"The reader's verdict correctly focuses on the undocumented finite-l simulations. I independently verified the l=0 covariance p,q,r, the current prefactor w=kBT(k2-k1)/Sigma, and the vorticity formula; these are correct and establish a genuine NESS for zero rest length. The finite-l section is the only part not backed by closed-form results or reproducible data, and the singularity at zero separation makes numerical integration nontrivial. An independent perturbative check is the most direct way to decide whether the finite-l claims are real. I do not see a mathematical error in the exact part, so the paper should remain CONDITIONAL pending documentation/verification of the numerics and a toned-down statement about 'arbitrarily far'.","tokens_in":13056,"tokens_out":17264,"duration_ms":183133,"concrete_test":"Perform a small-l perturbation expansion of the 4D Fokker-Planck equation around the l=0 Gaussian solution, computing O(l) corrections to the stationary currents and vorticity; compare the predicted topology (lobe separation, vortex-antivortex count) against the l=0.2 and l=0.5 simulations run with at least two timesteps differing by 4x and a regularization of the s=0 singularity. Agreement would validate the numerics; disagreement would show the claimed finite-l features are numerical artifacts.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The exact l=0 analysis (Eqs. 4-15) is correct; I re-derived the Lyapunov solution, the current prefactor w = kBT(k2-k1)/Sigma, and the vorticity formula. The load-bearing gap is the extension to finite rest length. For l>0 the drift is nonlinear and has a directional singularity at r1=r2 (the force term (|s|-l)s/|s| is multivalued at s=0), yet Sec. VI reports qualitative changes — translational current and additional vortex-antivortex pairs — from Langevin simulations documented only in figure captions, with no timestep, ensemble size, integration time, regularization, convergence test, or error estimates. If those numerics are unreliable, the paper's general claim ('non-reciprocity alone drives a Brownian dimer out of equilibrium') is established only in the zero-rest-length limit, and the finite-l parts of the abstract and conclusions are unsupported. This is a limitation, not a refutation: l=0 already demonstrates the mechanism.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a two-dimensional Brownian dimer made of two overdamped monomers in a common isotropic harmonic trap, coupled by a non-reciprocal harmonic spring and in contact with a single thermal bath. For zero rest length l=0, the dynamics is a linear multivariate Ornstein-Uhlenbeck process. The authors derive the exact stationary Gaussian distribution, the stationary Fokker-Planck current J = Q z P(z), with prefactor w = k_B T (k_2 - k_1)/Σ (Eq. (12)), and the vorticity field with a central vortex core surrounded by an antivortex ring (Eq. (14)). They then map this system to the Brownian gyrator, identifying non-reciprocity and temperature gradient as two independent symmetry-breaking mechanisms, and unify them in a two-temperature generalization with w = k_B(k_2 T_1 - k_1 T_2)/Σ (Eq. (21)). For finite rest length l>0, the equations are nonlinear; the paper reports numerical Langevin simulations showing a translational current component and additional vortex-antivortex pairs. The central claim is that non-reciprocity alone, without any temperature gradient, external drive, or explicit time dependence, drives the system into a nonequilibrium steady state with non-zero current.","tokens_in":13073,"tokens_out":3716,"duration_ms":42537,"significance":"If the results are correct, the l=0 solution is a clean, minimal demonstration that non-reciprocal linear coupling suffices to produce a nonzero stationary probability current in an otherwise equilibrium single-bath system. The exact Gaussian steady state, current, and vorticity are derived in closed form with no fitted parameters, and the two-temperature formula (Eq. (21)) provides an elegant interpolation between the non-reciprocal and thermal-gradient mechanisms. The analytic l=0 part is strong and, as the reader's report confirms, algebraically sound. However, the paper's broader claims about finite rest length — the appearance of a translational current and additional vortex-antivortex pairs — rest entirely on numerical simulations whose protocol and convergence are not documented. The l=0 exact result already establishes the core mechanism, but the finite-l section needs substantial strengthening before the general conclusion is fully supported.","major_comments":[{"comment":"The finite-rest-length claims are unsupported by the numerical evidence as presented. The text reports qualitatively new behavior for l>0 — a translational current component and an increased number of vortex-antivortex pairs — but the Langevin simulations are described only by figure captions. No timestep, integration scheme, ensemble size, integration time, steady-state convergence criterion, or error estimates are given. Since the nonlinear equations for l>0 have no closed-form solution, these claims rest entirely on those simulations. Please provide the full numerical protocol and show convergence tests (e.g., current/vorticity profiles at increasing resolution and simulation time), or explicitly restrict the claims to l=0 and mark the finite-l results as preliminary.","section":"§VI, Figs. 7 and 8"},{"comment":"For l>0, the force term (|s|-l) s/|s| is multivalued at s = r1 - r2 = 0, making the drift not Lipschitz at that point. The numerical integration therefore requires a regularization or a well-defined limiting convention. The manuscript does not state how this singularity was handled. If no regularization was used, the finite-l simulations may be ill-defined for trajectories that pass through s=0; if a regularization was used, it should be specified and its influence on the reported current/vorticity should be assessed. This is a technical load-bearing issue for §VI.","section":"§II and §VI, Eqs. (1)-(2)"},{"comment":"The statement that non-reciprocity alone can drive the system 'arbitrarily far from equilibrium' is supported by the l=0 exact result in the sense of a nonzero stationary current for any k1≠k2, but the paper does not quantify 'far from equilibrium' via, for example, entropy production rate. More importantly, the concluding sentence extends the claim to the finite-l case: 'For l≠0, all these quantities are calculated numerically.' Since the numerical details are absent, the general conclusion is stronger than the evidence currently supports. Please either supply the missing numerical substantiation or soften the finite-l part of the conclusion.","section":"§VIII, concluding remarks"}],"minor_comments":[{"comment":"The symbol Q is used both for the current matrix Q in Eq. (9) and for the quadratic form Q(x1,x2) in Eq. (14). This is confusing. Please rename the quadratic form, e.g., R(x1,x2) or G(x1,x2).","section":"§V, Eq. (14)"},{"comment":"The figure captions for Figs. 7 and 8 state 'The parameter values used in simulation are kept the same as before,' but the earlier figures refer to l=0. For finite-l runs, the values of l are listed in the captions, but no other numerical parameters (e.g., k0, T, γ) are restated. Please make each caption self-contained and report the simulation parameters and any requested accuracy.","section":"§VI and figure captions"},{"comment":"The sentence 'We verified Eqs. (S8)-(S10) ... from numerical simulations (Fig. 1,3 of the main text)' reads as a self-check. While it is fine to mention consistency with the main-text figures, the analytical derivation is itself sufficient; consider clarifying that the figures are illustrative comparisons rather than independent verification.","section":"Supplementary Material, S2"},{"comment":"There are minor typographical issues, such as inconsistent spacing around the '≠' symbol and occasional missing spaces after commas. A careful proofread is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The exact l=0 analysis is elegant and likely correct, and it alone demonstrates the key mechanism. The finite-rest-length section is the main weakness: without simulation details, the novelty of the l>0 qualitative changes cannot be assessed. I recommend major revision with a request for full numerical protocols and convergence evidence. If the authors cannot provide those details, the finite-l claims should be either omitted or explicitly labeled as preliminary."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Julia — the core of this paper holds up. For zero rest length, the non-reciprocal dimer is a linear OU system, and the authors correctly solve the Lyapunov equation and derive a nonequilibrium steady-state current proportional to k2−k1. I re-derived Eqs. (6)–(14) and the two-temperature generalization; all check out. The superposition formula w = kB(k2T1 − k1T2)/Σ is a neat unification, and the cancellation condition is a nice touch. The paper is self-contained, with no fitted parameters and no dependence on the authors' prior results. That is real work and it deserves credit.\n\nThe soft spot is the finite-rest-length extension at the end. For l>0 the dynamics are nonlinear, and the paper reports qualitative changes — translational current, extra vortex-antivortex pairs — based on Langevin simulations described only in figure captions. No timestep, ensemble size, integration time, or error bars. The force term (|s|−l)s/|s| is multivalued at s=0, so the numerics need regularization. That is a gap, not a refutation: the l=0 mechanism is already established. But the abstract and conclusions claim the non-reciprocal dimer drives currents for general l, and that part is unsupported as written.\n\nTwo smaller issues. The phrase 'arbitrarily far from equilibrium' overstates: the current prefactor w is bounded by kBT and the current lives in the (x1,x2) or (y1,y2) shape plane, not in the real-space motion of either monomer. And the novelty discussion is thin: ref [30] (Loos-Klapp) is cited only for non-potentiality, but its title covers irreversibility induced by non-reciprocal interactions. The authors should say explicitly what their exact calculation adds beyond that framework — the closed forms are new, but the phenomenon class is not.\n\nThe reader's conditional verdict is about right. With the numerics documented and the claims toned down, this is a solid paper. Without that, it is a correct calculation with an unsubstantiated extension. I would send it to peer review — a referee can sort out the finite-l claims — but I wouldn't accept it in the current form. I would cite the l=0 results in the next year. Maybe useful in a reading group on non-equilibrium steady states; it's a clean example but not a central one.","headline":"The exact l=0 result is correct and worth knowing; the finite-rest-length section needs substantial cleanup before this is publishable.","tokens_in":13798,"tokens_out":2308,"would_cite":true,"duration_ms":22402,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["05.40.-a","05.70.Ln"],"model":"deepseek-v4-flash","headline":"This paper proves that a two-monomer Brownian dimer with a non-reciprocal harmonic spring—different stiffness on each monomer—reaches a genuinely non-equilibrium steady state with a non-zero probability current for any k1≠k2, even in a sing","keywords":["Non-reciprocal interactions","Brownian dimer","Non-equilibrium steady state","Probability current","Vorticity","Brownian gyration","Overdamped Langevin dynamics","Lyapunov equation"],"falsifier":"Measure the steady-state current in a realization of the model at k1=k2 and confirm it is exactly zero; then vary k2−k1 and check that the current magnitude scales linearly with |k2−k1| and its direction reverses with the sign, as Eq. (12) predicts. For the finite-rest-length claims, repeat the Langevin integration with decreasing timestep and increasing ensemble size: the additional vortex-antivortex pairs in Fig. 8 must persist under convergence, or they are numerical artifacts.","tokens_in":12719,"feed_emoji":"🌀","tokens_out":4517,"duration_ms":47209,"temperature":0.7,"pith_summary":"The paper claims that a minimal overdamped Brownian dimer—two monomers in a common isotropic harmonic trap, linked by a harmonic spring whose stiffness differs for each monomer, and sharing one thermal bath—settles into a non-equilibrium steady state with a persistent probability current whenever the two stiffnesses differ. For the exactly solvable zero-rest-length limit, the stationary distribution is Gaussian and the current is proportional to k2−k1, so any imbalance, however small, drives autonomous gyration with no temperature gradient, external force, or time-dependent drive. This matters because it identifies non-reciprocity of inter-particle forces as a standalone mechanism for generating non-equilibrium currents, unifying it with the physics of the Brownian gyrator. The finite-rest-length case, treated numerically, shows the current gaining a translational component and the vorticity developing additional vortex-antivortex pairs.","feed_headline":"Non-reciprocal spring alone drives Brownian dimer out of equilibrium","feed_subtitle":"No temperature gradient or external drive: the stiffness imbalance k1≠k2 sets a steady gyration current.","key_machinery":"The central object is the stationary Lyapunov equation for a multivariate Ornstein-Uhlenbeck process, AC + CAᵀ = 2D, whose solution fixes the Gaussian covariance C and hence the probability current through Q = D C⁻¹ − A. The current is controlled by the antisymmetric part of QC, whose single independent component is w = kB(k2T1 − k1T2)/Σ; this scalar w is the pivot on which all non-equilibrium behavior hangs.","core_discovery":"For a dimer governed by the overdamped Langevin equations with drift matrix A = [[k0+k1, -k1],[-k2, k0+k2]] and a single-temperature bath, the steady-state covariance solves the Lyapunov equation AC + CAᵀ = 2kBT I, giving an exact Gaussian distribution. The stationary probability current is J = Q z P with Q = kBT C⁻¹ − A, and its overall prefactor is w = kBT(k2−k1)/Σ, where Σ = 2k0+k1+k2. Consequently, a non-zero current exists for every k1≠k2, vanishing only for the reciprocal case; the vorticity has a positive central core surrounded by a negative ring (a vortex-antivortex pair). The same algebra with two bath temperatures T1,T2 replaces w by kB(k2T1 − k1T2)/Σ, unifying the non-reciprocal","pith_inferences":["If the finite-rest-length predictions survive careful numerical scrutiny, a direct experimental test could be built: measuring the steady-state current while sweeping k2−k1 and checking the predicted linear dependence and sign reversal.","The exact solvability at l=0 hints that any two-degree-of-freedom linear system with asymmetric drift and isotropic diffusion will display a current proportional to the antisymmetric part of the drift; the mechanism may extend to larger non-reciprocal networks.","The cancellation condition k1/k2 = T1/T2 suggests a zero-current 'effective equilibrium' surface in parameter space; an editor's extrapolation is that an entropy-production or fluctuation-dissipation characterization could be formulated along that surface.","The unquantified numerics for l>0 (no timestep, ensemble size, or convergence analysis) leave the translational current and extra vortex pairs as provisional; they require independent confirmation before being treated as established."],"forward_implications":["Non-reciprocal coupling alone is sufficient to generate a non-equilibrium steady state in a passive, time-independent, single-bath system.","The direction of the gyration current is set by the sign of k2−k1, and its magnitude scales linearly with the stiffness difference, so even an arbitrarily small asymmetry yields a measurable current.","The exact l=0 solution provides a benchmark for designing experiments or simulations that probe non-reciprocal forces in colloidal or active-matter settings.","Coupling the monomers to two different temperatures yields a unified current prefactor, showing that mechanical asymmetry and thermal asymmetry are interchangeable and can be tuned to cancel exactly.","For finite rest length, the numerical results predict a translational current component and additional vortex-antivortex pairs, qualitatively extending the mechanism beyond linear couplings."],"fun_headline_variants":["Non-reciprocal spring spins Brownian dimer into gyration","No external force: stiffness imbalance drives dimer circulation","Non-reciprocity in a spring gives a dimer a perpetual spin","Single thermal bath, no drive: how a spring sets a current","One unbalanced spring pushes a Brownian dimer into motion"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The finite-rest-length (l>0) results—the translational current component and the additional vortex-antivortex pairs—rest entirely on numerical simulations described only in figure captions, with no timestep, ensemble size, integration time, or convergence analysis given; if those simulations are unreliable, the mechanism is rigorously established only for the zero-rest-length linear spring.","fun_headline_variants_meta":{"raw":{"variants":["Non-reciprocal spring spins Brownian dimer into gyration","No external force: stiffness imbalance drives dimer circulation","Non-reciprocity in a spring gives a dimer a perpetual spin","Single thermal bath, no drive: how a spring sets a current","One unbalanced spring pushes a Brownian dimer into motion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000416,"raw_usage":{"total_tokens":1958,"prompt_tokens":692,"completion_tokens":1266,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":436,"completion_tokens_details":{"reasoning_tokens":1182}},"tokens_in":436,"tokens_out":1266,"duration_ms":11031,"temperature":1.0,"reasoning_tokens":1182,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T02:12:31.846480+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the steady-state current in a realization of the model at k1=k2 and confirm it is exactly zero; then vary k2−k1 and check that the current magnitude scales linearly with |k2−k1| and its direction reverses with the sign, as Eq. (12) predicts. For the finite-rest-length claims, repeat the Langevin integration with decreasing timestep and increasing ensemble size: the additional vortex-antivortex pairs in Fig. 8 must persist under convergence, or they are numerical artifacts.","supporting_citations":[],"review_version":1}