{"id":"ff030326-d3e3-43ad-9a80-d53c21a9a06f","arxiv_id":"2501.17849","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":9,"one_line_summary":"A dissipation-functional framework is derived from entropy production and used to directly compute stationary shapes and flows of active fluid surfaces, revealing first-order shape transitions and hydrodynamic-screening effects.","lead":"This paper builds a variational principle for active fluid surfaces, letting researchers compute stationary cell-like shapes directly instead of simulating full dynamics. It maps out shape transitions in membranes and division-like surfaces, connecting activity, friction, and geometry.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The rp flux/force swap requires Λ>0; Onsager reciprocity alone does not guarantee invertibility, though second-law preservation is exact for this model because Θ̂int=Θint.","rationale":"The reader's weakest assumption identifies the invertibility of the Onsager matrix as the load-bearing point, and that is indeed the right region of the argument. However, the reader's specific objection is partly misdirected: the relevant Onsager matrix in this model is not a symmetric positive-semidefinite matrix, because the reactive coupling ξ enters with opposite signs in Eqs. (27) and (33), making the matrix [[η_b,ξ],[-ξ,Λ]] with determinant η_bΛ+ξ². The actual vulnerability is the division by Λ in Eqs. (35)–(36). The paper's universal statement that Onsager relations guarantee invertibility is false in general, and even invertibility is insufficient if Λ=0: the matrix can be invertible while the constructed functional diverges. The second-law preservation, which the reader doubted, is actually exact because the transformed entropy production equals the original one after substituting the constitutive laws. The central equivalence for the constitutive laws used in the examples is therefore sound provided Λ>0, and the numerical results, which prescribe ξΔµ(c) without invoking rp or Λ, are not endangered. The CONDITIONAL verdict remains appropriate, but the condition should be stated as Λ>0 and positive definite dissipative coefficients, not as a general Onsager-invertibility guarantee.","tokens_in":92,"tokens_out":18929,"duration_ms":329557,"concrete_test":"Set Λ=0, ξ≠0 in Eqs. (27) and (33) and attempt to construct R from Eqs. (35)–(39). The coefficients ξ/Λ and 1/Λ diverge, so no finite dissipation functional exists even though the 2×2 Onsager matrix is invertible (det=ξ²). Adding the missing hypothesis Λ>0 to the statement in Sec. IV D would resolve this. Independently, symbolically substitute Eqs. (27),(33) into (32) and Eqs. (35),(36) into (37); the resulting expressions are identical, settling the second-law question in the paper's favor.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The framework's central equivalence (App. E) rests on Eqs. (35)–(36), which express Δµ through rp by dividing by Λ. The paper claims that 'Onsager relations guarantee an invertible coefficient matrix' and therefore that the transformation always exists. That guarantee is false: the second law only forces the symmetric part of the Onsager matrix to be positive semidefinite, and a linear constitutive law with L=0 is singular. More importantly, invertibility is not the right condition: with Λ=0 and ξ≠0, the matrix [[η_b, ξ],[-ξ, 0]] has determinant ξ²>0, yet the specific inversion used in Eqs. (35)–(36) is undefined because Λ appears in the denominator. Thus the variational construction is not guaranteed for every linear Onsager constitutive law; it requires Λ>0, with η_b and η_s nonnegative for convexity. The reader's second-law worry, by contrast, does not land: substituting Eqs. (27),(33) into (32) and Eqs. (35),(36) into (37) gives the same expression, so Θ̂int=Θint exactly. Since the numerical applications prescribe ξΔµ(c) and never use rp or Λ, the specific phase diagrams are unaffected; the concern is about the advertised generality of the method.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a variational (Rayleigh-functional) formulation for active isotropic fluid surfaces. Starting from the internal entropy production of previous surface-hydrodynamics frameworks, the authors rewrite the linear Onsager constitutive laws by exchanging the thermodynamic force Δμ for the reaction rate r_p, obtaining a dissipation functional R whose stationary points are claimed to reproduce force and torque balance, including reactive non-dissipative couplings. Geometric relations and a scaling Lagrangian-Eulerian (SLE) parameterization are imposed through Lagrange multipliers, reducing stationary shape and flow computation to boundary value problems. The method is applied to open Helfrich-type membranes with spontaneous-curvature coatings and pulling forces, and to closed active surfaces with concentration-dependent active tensions, yielding phase diagrams for protrusion, polarization/propagation, and guided division. Appendix E contains an explicit derivation of the equivalence between the variational equations and force balance for axisymmetric surfaces.","tokens_in":31473,"tokens_out":10376,"duration_ms":106877,"significance":"If correct, the construction provides a systematic way to generate variational principles for active surface theories, extending the equilibrium membrane variational approach to non-equilibrium settings while retaining reactive couplings. The paper is strong on concreteness: Appendix E gives a detailed axisymmetric derivation, the numerical code is released on GitHub, and the applications are tested against known results such as the Derenyi et al. tube-pulling force and previous mechanochemical polarization instabilities. The dense parameter sweeps and explicit stability checks add substantial value. The main caveat is that the flux-force transformation underlying the functional is not as general as claimed, and the general (non-axisymmetric) equivalence is not proven; these issues affect the advertised scope of the method but not the specific numerical phase diagrams.","major_comments":[{"comment":"The transformation from the (v, Δμ) to the (v, r_p) description divides by the coefficient Λ. The paper states that Onsager relations guarantee an invertible coefficient matrix, but positive-semidefinite symmetric matrices are not generally invertible, and invertibility of the full matrix is also not sufficient for this specific swap: for Λ=0 and ξ≠0 the constitutive matrix [[η_b, ξ],[-ξ, Λ]] has determinant η_bΛ+ξ² > 0, yet Eqs. (35)–(36) are undefined because Λ appears in the denominator. The existence of the Rayleigh functional therefore requires Λ>0 (and η_b, η_s ≥ 0 for convexity), not merely Onsager reciprocity. Since the numerical applications prescribe ξΔμ(c) and never use r_p or Λ, the phase diagrams are unaffected, but the advertised generality for \"the full set of a priori defined constitutive laws\" is overclaimed. Please either impose and state the condition Λ>0 or show how singular/diffusionless cases are to be handled.","section":"Sec. IV D, Eqs. (35)–(36)"},{"comment":"The central equivalence claim is proven only for axisymmetric surfaces. Appendix E derives the Euler-Lagrange equations for the meridional, azimuthal, and normal variations in the axisymmetric setting. The unconstrained dissipation functional is stated to be valid for arbitrary surfaces, but no general tensor derivation of δR/δv = 0 from Eqs. (20)–(21) is given. If the general statement is to be retained, a coordinate-free or general-coordinate proof should be added; otherwise the claim should be qualified to axisymmetric surfaces. This does not affect the examples, all of which are axisymmetric, but it matters for the advertised scope of the method.","section":"Sec. IV D and App. E"},{"comment":"The paper does not specify how the stationary concentration field c(u) is determined in the direct stationary computations that produce Figs. 5 and 7. The variation (45) is over velocities and geometric time-derivatives, not over c, and the reaction-diffusion dynamics (52) is not listed among the ODEs in Table II. If the steady-state concentration is obtained by appending Eq. (52) with ∂_t c = 0 to the boundary value problem, that should be stated explicitly; otherwise the phase diagrams cannot be reproduced from the text alone.","section":"Sec. IV F 2 and Sec. V B"}],"minor_comments":[{"comment":"The sentence \"One can directly verify that if coefficients η, η_b, ξ are such that the second law is respected in the original entropy production ... then it will also be respected by Θ̂int\" is true but not demonstrated; adding the explicit identity Θ̂int(v, r_p) = Θint(v, Δμ), which follows by substituting Eqs. (27),(33) into (32) and Eqs. (35),(36) into (37), would remove the appearance of an unproved assertion.","section":"Sec. IV D"},{"comment":"The claimed quantitative agreement with ref. [35] appears arithmetically inconsistent: with γ0 = 12.5 κ/R_b², the formula f0 = 2π√(γ0κ) gives f0 ≈ 22.2 κ/R_b, so κ/R_b ≈ f0/22 rather than approximately f0/5. Please clarify the comparison or correct the stated factor.","section":"Sec. V A 2"},{"comment":"In the ∂th row it is unclear whether the boundary conditions ζ(0)=ζ(1)=0 apply to both open and closed surfaces; the text should state explicitly what replaces these conditions for closed surfaces, where no physical boundary exists.","section":"Table III"},{"comment":"The notation switches between v_u and normalized components such as ̄v_u without always recalling that barred components refer to the normalized basis ̄e_u = e_u/h; a brief notational remark at the start of Appendix E would improve readability.","section":"App. E"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a substantial and useful numerical study, and the axisymmetric derivation in Appendix E is a genuine contribution. The main technical concern is the overclaimed generality of the flux-force swap in Section IV D; this is fixable by adding the explicit condition Λ>0 and qualifying the general statements. I would not require new numerical experiments, but the authors should also clarify how stationary concentration fields are computed. With these revisions, the paper could be suitable for publication in this journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: real advance, but the advertised generality rests on an overstrong invertibility claim; the specific applications survive.\n\nWhat's actually new: the SLE parameterization, the construction of a Rayleigh functional from entropy production that includes reactive couplings via a flux-force swap, and the Lagrange-multiplier formulation that turns stationary active surface shape equations into a boundary value problem. App. E is detailed and shows equivalence to force balance for axisymmetric surfaces. The applications produce new physics: Gibbs loops for spontaneous-curvature coated membranes, disconnected branches under pulling, and a hydrodynamic-screening-induced reentrant instability in the division-like model. Code is on GitHub. That is real, reproducible work.\n\nSoft spots: the paper claims Onsager relations guarantee an invertible coefficient matrix. That is false; positive semidefinite is enough for the second law, not invertibility. More precisely, the specific transformation from Δμ to rp divides by Λ, so it requires Λ>0 (with ηb,ηs nonnegative for convexity). For a model with Λ=0 but a nonzero reactive coupling, the functional doesn't exist, even though the Onsager matrix determinant could be nonzero. So the 'generic symmetries' claim is too strong. That said, the stress-test note is right that the second-law preservation worry does not land: substituting the transformed laws into the entropy production gives exactly Θ̂int=Θint. And the worked examples never use rp or Λ — they prescribe ξΔμ(c) directly — so the numerical results stand. The fix is a caveat, not a rework.\n\nAlso under-explained: the relationship between the two-species (fuel/product) model used to derive the framework and the single concentration field c used in the closed-surface applications. The text asserts the active stress ξΔμ(c) and the turnover dynamics Eq. (52) without a careful mapping. This is not fatal but a referee should ask for it.\n\nProportion: the central equivalence is solid for axisymmetric surfaces, the method works, and the overstatement is in the generality. I would send to peer review with major-but-addressable comments.","headline":"A genuinely useful variational method for active surface shape spaces, with an overbroad invertibility claim that needs a caveat but doesn't break the main results.","tokens_in":32019,"tokens_out":2152,"would_cite":true,"duration_ms":21878,"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":"This paper establishes a variational formulation of active isotropic fluid surfaces: the stationary points of a dissipation functional built from entropy production, supplemented by Lagrange multipliers that enforce geometric constraints…","keywords":["active fluid surfaces","dissipation functional","Onsager relations","Rayleigh functional","shape space analysis","membrane mechanics","hydrodynamic screening","variational method"],"falsifier":"Take a linear constitutive law with a positive semidefinite but singular Onsager matrix—for example, Eqs. (27) and (33) with $\\Lambda=0$ and $\\xi\\neq 0$—and compare the direct force-balance solution for an axisymmetric surface with the stationary points of any candidate Rayleigh functional; if the variational equations fail to reproduce the same geometry and flow, or if the flux-force transformation violates the second law, the claimed equivalence is false.","tokens_in":30946,"feed_emoji":"🧫","tokens_out":9210,"duration_ms":77969,"temperature":0.7,"pith_summary":"This paper claims that the shape dynamics of active fluid surfaces—minimal models for membranes, cell cortices, and tissues—can be recast as a variational problem. The authors construct a Rayleigh dissipation functional from the entropy production of the surface and show that, after changing the thermodynamic flux-force pairing, even reactive (non-dissipative) active couplings contribute to the functional. The stationary points of the constrained functional are shown to yield exactly the geometries and flows that satisfy force and torque balance for the prescribed linear constitutive laws. This equivalence turns the search for stationary shapes into a boundary value problem, so emergent shape spaces can be computed directly rather than only explored by time-stepping dynamics. The paper demonstrates the framework on open membranes and closed active surfaces, revealing first-order shape transitions, degenerate solution branches, and a role for hydrodynamic screening in division-like geometry.","feed_headline":"Stationary shapes of active fluid surfaces now directly computable","feed_subtitle":"A dissipation functional turns force balance into a variational problem for active surfaces.","key_machinery":"The load-bearing object is the constrained Rayleigh functional $\\bar{R}=R+L$, where $R$ is assembled from the free-energy change rate and half the total entropy production, and $L$ uses Lagrange multipliers $\\alpha,\\beta,\\zeta$ to impose the geometric identities $r'=h\\cos\\psi$, $z'=-h\\sin\\psi$, and the scaling Lagrangian-Eulerian (SLE) parameterization. In the SLE parameterization, fixed mesh coordinates map to physical arc length through a single global scale factor $h(t)$, so numerical resolution is controlled independently of local flows. Variation of $\\bar{R}$ with respect to velocities and shape-derivative fields produces a coupled system of first-order ordinary differential equations whose boundary terms become the boundary conditions of the surface problem. The identity that makes the construction work is the flux-force transformation (35)–(36): choosing $r_p$ instead of $\\Delta\\mu$ as the thermodynamic force makes the active cross-coupling $\\xi/\\Lambda$ appear with the same sign in both constitutive laws, so it contributes to dissipation and the entropy production becomes a valid variational principle.","core_discovery":"The central claim is an equivalence: the functional $R(v,r_p)=dF/dt + D(v,r_p)$ in Eq. (39), together with the Lagrange-multiplier terms $L$ in Eq. (43), has stationary points that are precisely the surfaces and flows satisfying force balance $\\mathrm{div}(T)=-f^{\\mathrm{ext}}$ and torque balance $\\mathrm{div}(M)=-\\epsilon:T$ with the tension $T=T_e+T_d$ and moment $M=M_e$ defined by the constitutive laws (27)–(29). The equivalence is verified explicitly for axisymmetric surfaces in Appendix E of the paper. The crucial step is replacing the chemical potential difference $\\Delta\\mu$ by the reaction rate $r_p$ as the thermodynamic force, which turns the reactive Onsager coefficient $\\xi$ into a dissipative coupling so that the entropy production defines a genuine Rayleigh functional. As a result, stationary geometries and flows of active surfaces can be computed directly as solutions of a boundary value problem, without first simulating the full time-dependent dynamics.","pith_inferences":["Editorial inference: the variational equivalence may extend beyond axisymmetry, but the analog of the SLE parameterization would need to control shearing and twisting of the surface map, not just meridional stretching; the paper identifies this as an open question.","Editorial inference: if the dissipation functional is convex near a stationary solution, its second variation could supply a direct linear-stability criterion, complementing the perturb-and-simulate stability tests used in the paper.","Editorial inference: the first-order transitions and hysteresis found in the protrusion and division models suggest that biological systems could exploit bistable shape spaces for irreversible developmental decisions, a functional speculation the paper does not pursue."],"forward_implications":["Stationary geometries and flows of an active fluid surface can be obtained directly from a boundary value problem, without time-stepping through transients.","Nonlinear shape spaces can be mapped on dense parameter grids, exposing Gibbs loops, disconnected solution branches, and first-order transitions between protrusion geometries.","For closed mechanochemically active surfaces, the framework predicts multi-stable stationary shape regions and a reentrant symmetry-breaking and restoring instability controlled by hydrodynamic screening.","Hydrodynamic screening, set by the ratio $R_0/L_h$, controls whether contractile rings produce sharply divided dumbbells or broader, more elongated ingressions during division-like transformations.","The same variational construction applies to other linear constitutive laws for active surfaces with different broken symmetries, providing a route to classify their stationary shape spaces."],"supporting_citations":[{"why":"Provides the irreversible-thermodynamics formulation of mass, force, and torque balance on curved deforming surfaces from which the entropy production and constitutive laws are taken.","marker":"[18]"},{"why":"Supplies the general non-equilibrium constitutive laws for active surfaces that the paper's variational construction is designed to reproduce.","marker":"[43]"},{"why":"Pioneered Onsager variational principles for membranes, the route this work systematizes into a dissipation functional.","marker":"[40]"},{"why":"Introduced the Lagrange-multiplier variational treatment of geometric constraints for passive membranes that this paper extends to active surfaces.","marker":"[30]"},{"why":"Onsager reciprocity relations are the symmetry the paper exploits to rewrite reactive couplings as dissipative ones.","marker":"[45–47]"},{"why":"Helfrich bending energy defines the passive curvature contribution to tension and moments in the constitutive laws.","marker":"[25]"},{"why":"Point-force membrane pulling model whose characteristic first-order transition and force scale the paper reproduces quantitatively for open membranes.","marker":"[35]"},{"why":"Mechanochemical shape-instability model of closed active surfaces that the polarization, propagation, and division examples build upon.","marker":"[19]"}],"fun_headline_variants":["Variational shortcut to stationary active fluid surface shapes","Active fluid surface shapes computed without time evolution","Direct variational computation of stationary active shapes","Stationary shapes of active fluid surfaces: now direct","No dynamics needed: stationary active fluid shapes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction presupposes that the Onsager coefficient matrix is invertible and that exchanging $\\Delta\\mu$ for the reaction rate $r_p$ as the thermodynamic force preserves the second law, so every linear constitutive law can be recast in purely dissipative form and a Rayleigh functional exists.","fun_headline_variants_meta":{"raw":{"variants":["Variational shortcut to stationary active fluid surface shapes","Active fluid surface shapes computed without time evolution","Direct variational computation of stationary active shapes","Stationary shapes of active fluid surfaces: now direct","No dynamics needed: stationary active fluid shapes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001194,"raw_usage":{"total_tokens":4952,"prompt_tokens":996,"completion_tokens":3956,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":612,"completion_tokens_details":{"reasoning_tokens":3888}},"tokens_in":612,"tokens_out":3956,"duration_ms":27251,"temperature":1.0,"reasoning_tokens":3888,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T04:32:47.777969+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a linear constitutive law with a positive semidefinite but singular Onsager matrix—for example, Eqs. (27) and (33) with $\\Lambda=0$ and $\\xi\\neq 0$—and compare the direct force-balance solution for an axisymmetric surface with the stationary points of any candidate Rayleigh functional; if the variational equations fail to reproduce the same geometry and flow, or if the flux-force transformation violates the second law, the claimed equivalence is false.","supporting_citations":[{"cited_title":"Turlier, B","cited_arxiv_id":null,"evidence_quote":"Provides the irreversible-thermodynamics formulation of mass, force, and torque balance on curved deforming surfaces from which the entropy production and constitutive laws are taken."},{"cited_title":"Hassinger, G","cited_arxiv_id":null,"evidence_quote":"Supplies the general non-equilibrium constitutive laws for active surfaces that the paper's variational construction is designed to reproduce."},{"cited_title":"J ¨ulicher and R","cited_arxiv_id":null,"evidence_quote":"Pioneered Onsager variational principles for membranes, the route this work systematizes into a dissipation functional."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the Lagrange-multiplier variational treatment of geometric constraints for passive membranes that this paper extends to active surfaces."},{"cited_title":"Mietke, F","cited_arxiv_id":null,"evidence_quote":"Helfrich bending energy defines the passive curvature contribution to tension and moments in the constitutive laws."},{"cited_title":"Lipowsky, Nature 349, 475 (1991)","cited_arxiv_id":null,"evidence_quote":"Point-force membrane pulling model whose characteristic first-order transition and force scale the paper reproduces quantitatively for open membranes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Mechanochemical shape-instability model of closed active surfaces that the polarization, propagation, and division examples build upon."}],"review_version":1}