{"id":"d9a9fcf0-47f9-4ea6-918a-0d4790421495","arxiv_id":"2505.14912","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A feedback-coupled Ehrenfest model of cyclic shear produces a genuine yielding transition, and a three-state coarse graining captures non-monotonic fatigue with a universal tan^2 scaling near the critical strain.","lead":"This paper adds a feedback term to a simple random-walk model of glasses under cyclic shear, and shows the model then develops a true yielding transition with sharply divergent timescales. It also reduces the dynamics to just three kinds of states and captures the non-monotonic fatigue behavior seen in simulations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The transition and the exponent -1 rely on W_US(P_U) being strictly linear in P_U, but the microscopic rate (3) has near-threshold states whose escape saturates at 1/tau0, making the linear coefficient N-dependent; no N-scaling check is given.","rationale":"The reader's weakest-assumption analysis identified the same general modeling assumption: the mechanical noise felt by stable blocks is taken to be proportional to the instantaneous unstable fraction and the escape rates are taken to vanish linearly in P_U. My stress-test sharpens this into a concrete technical failure mode. Eq. (3) gives the microscopic escape rate r_j(D) = 2D/[tau0(Delta_j^2 + 2D)]. This is linear in D only when D is much smaller than Delta_j^2. The threshold stable states have Delta_j approaching zero, so for any finite N there is a crossover value of P_U below which the linearization holds only for an extremely small interval, and the effective linear coefficient is controlled by the most shallow stable state. Summing over the near-threshold states yields a coefficient that grows with N, while the paper's Eq. (6) gives a coefficient proportional to q_k/a that does not grow. If the exact coefficient diverges or simply differs from Eq. (6), the bifurcation condition and the divergence exponent could shift with N, and the universal tan^2 scaling derived from the linearized model class (10)-(11) would not describe the full model in the continuum limit. This is not a contradiction proven from the text; it is an unresolved, testable sensitivity. The appropriate verdict remains CONDITIONAL: the central claims are plausible and internally consistent as written, but they depend on a linearization whose N-dependence is not established and whose derivation is relegated to an unavailable Supplemental Material. If the proposed N-scaling test shows convergence and agreement with Eq. (6), the concern is resolved and the conditional verdict can be lifted; if not, the claims about a genuine transition require revision.","tokens_in":9089,"tokens_out":14605,"duration_ms":140933,"concrete_test":"For N = 100, 400, 1600 (and 6400 if feasible), fix alpha and compute the exact small-P_U slope W'_US from Eq. (3) using the stationary distribution of the master equation, i.e. W'_US = (2/tau0) sum_{j>k} q_j / (sqrt(eps_j) - gamma)^2, and compare it with the coefficient q_k p_{k-1,k}/(tau0 a) from Eq. (6). Then locate gamma*(N) from the exact W_US(P_U) curve and check whether gamma*(N) converges to a finite limit and whether the ratio W'_US / [q_k p_{k-1,k}/(tau0 a)] tends to 1, grows with N, or depends on alpha. Also repeat the scaling collapse of P_U(t) in Fig. 3 at N = 1600 to see whether the tan^2 form holds with the same gamma_c(N).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central results—the genuine dynamical transition at gamma*, the divergence tau_US ~ (gamma - gamma*)^(-1), and the universal tan^2 scaling—all follow from the linearization W_US = alpha Y W'_US, W_SS = alpha Y W'_SS (Eqs. 6, 10-11), justified by 'the behaviour of 1/tau_j in (3) for D -> 0'. This linearization is not uniform in the number of states N. For a stable state j > k, the escape rate from Eq. (3) is r_j(D) = 2D/[tau0((sqrt(eps_j) - gamma)^2 + 2D)], which is linear in D only while D << (sqrt(eps_j) - gamma)^2. Because sqrt(eps_j) - gamma vanishes as j -> k, the most shallow stable state has Delta_min ~ 1/(4N gamma), and for any fixed small P_U the saturation r_j = 1/tau0 sets in once alpha P_U exceeds Delta^2. In the continuum limit the number of near-threshold states grows, so dW_US/dP_U at P_U = 0, computed from the exact sum over S, involves sum q_j / Delta_j^2, which grows with N, whereas the closed-form quotient in Eq. (6) has a coefficient q_k p/(tau0 a) that does not. The two-state transition condition alpha W'_US(gamma*) = W_SU(gamma*) is therefore sensitive to N unless a cancellation is shown. No N-scaling analysis or SM derivation is provided, so the reported exponent and the scaling law could be finite-N artifacts of the approximate W_US.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a mean-field model of amorphous solids under cyclic shear, generalizing the Ehrenfest model of Mungan and Sastry by adding self-consistent mechanical noise D(t)=alpha P_U(t). In a two-state coarse graining, the authors find a dynamical transition at a critical strain amplitude gamma*: below gamma* only the frozen state P_U=0 exists, above it a fluidized state appears, and the escape time diverges as (gamma - gamma*)^(-1) (Eq. 9). They introduce a three-state model (unstable, threshold, absorbing) and show that it yields non-monotonic fatigue-failure trajectories and a basin of attraction that shrinks with increasing gamma, so that well-annealed samples have a larger critical strain gamma_c. Near gamma_c they propose a universal scaling P_U = delta_gamma [1 + tan^2(a sqrt(delta_gamma) delta_t)] (Eq. 19). Numerical trajectories of the full and coarse-grained models are presented in Figs. 1-3 to support these claims.","tokens_in":9434,"tokens_out":7487,"duration_ms":62518,"significance":"If substantiated, the paper would provide a minimal mean-field explanation of several simulation observations: the divergence of cycles near yielding, the annealing-dependent yield point, the non-monotonic property changes, and the spectrum of frozen states. Its strengths are the explicit master equations, the transparent fixed-point analysis in the two-state model, and the numerical demonstrations in Figs. 1-3. However, the manuscript's central derivations are almost entirely deferred to an unavailable Supplemental Material, and the linearization underlying the exponent might be sensitive to the number of states N; these issues must be addressed before the claims can be accepted as established.","major_comments":[{"comment":"The formulas for W_US in Eq. (6), the three-state coarse-grained rates in Eqs. (16)-(18), and the derivation of the universal scaling (19) are all referenced to Ref. [31], which is listed as 'Supplemental Material (To be included) (2025)' and is not part of the submitted manuscript. These are load-bearing derivations: without them, Eq. (6) is an assertion, and the claimed 'full generality' of Eq. (19) is unverifiable. Please include the SM in the submission or move the key steps to the main text.","section":"Eq. (6), Eqs. (16-18), Eq. (19); Ref. [31]"},{"comment":"The linearization W_US = alpha Y W'_US and W_SS = alpha Y W'_SS is justified by the D->0 behavior of 1/tau_j in Eq. (3). For a stable state j > k, r_j(D) = 2D/[tau0((sqrt(eps_j)-gamma)^2 + 2D)], which is linear only for D << (sqrt(eps_j)-gamma)^2. Since the gap (sqrt(eps_j)-gamma) vanishes as j -> k, the number of states for which the linearization fails at a given small P_U grows with N. The exact derivative dW_US/dP_U at P_U=0 involves a sum over q_j/(sqrt(eps_j)-gamma)^2, which is expected to grow with N, whereas the closed-form Eq. (6) gives a coefficient q_k p_{k-1,k}/(tau0 a) that does not exhibit such growth. Unless a cancellation is demonstrated, the transition condition alpha W'_US(gamma*) = W_SU(gamma*) and the exponent -1 in Eq. (9) may be finite-N artifacts. Please provide the SM derivation of Eq. (6) and an N-scaling check, for example by varying N in simulations of the full master equation.","section":"Divergence of timescales; Eqs. (6), (10)-(11)"},{"comment":"The paper claims a power-law divergence of cycles on both sides of the yielding point, but the only derived behavior is the tau_US ~ (gamma - gamma*)^(-1) divergence for gamma > gamma* (Eq. (9)). Footnote [18] acknowledges that elasto-plastic simulations instead indicate a logarithmic divergence when the yielding point is approached from below. The model's prediction for the approach from below is not presented, and the discrepancy is not discussed. Clarify which side the model addresses and how the logarithmic behavior might arise within the same framework.","section":"Introduction, first paragraph and footnote [18]"},{"comment":"The non-monotonic fatigue-failure dynamics of the three-state model is a built-in consequence of the assumption that stable-state escape rates are proportional to P_U: every term on the right-hand side of Eqs. (16)-(18) carries a factor P_U, so an initial decay of P_U is automatic unless the threshold population P_T grows. The statement that the model 'predicts' fatigue failure dynamics (page 4) should be tempered; the novel content is the existence of the critical frozen state, the basin-of-attraction dependence on annealing, and the scaling form (19). I suggest rephrasing to emphasize the consistency-check role of the three-state construction.","section":"Generalised Master Equation with mechanical noise; Eqs. (16)-(18)"}],"minor_comments":[{"comment":"The steady-state relation P*_U = W_US/(W_US + W_SU) should be written with the argument of W_US made explicit, e.g., W_US(P*_U), to avoid an apparent algebraic definition of the fixed point.","section":"Eq. (7)"},{"comment":"The caption refers to tau_lin without defining it in the text; please define tau_lin, presumably the timescale obtained from linearizing Eq. (5).","section":"Fig. 1(d)"},{"comment":"The notation W'_US and W'_SS is introduced without explicit definitions; please provide the definitions that are claimed to be in the SM.","section":"Eq. (12) and text before it"},{"comment":"The counting argument gives a solution space of dimension N_s - 2, which for N_s = 2 yields isolated solutions but does not by itself prove uniqueness of the critical frozen state; please clarify whether uniqueness has been checked.","section":"Just after Eq. (14)"},{"comment":"The denominator (sqrt(eps_j)-gamma)^2 is formatted with a superscript that may be misread; please use consistent notation, e.g., (\\sqrt{\\epsilon_j}-\\gamma)^2.","section":"Eq. (3)"},{"comment":"The paper chooses a half-Gaussian reference distribution (footnote [30]) but does not discuss the sensitivity of the transition and scaling results to this choice; a sentence on robustness across distributions would strengthen the claim.","section":"General formatting"}],"recommendation":"major_revision","confidential_remarks":"The absence of the Supplemental Material is a serious obstacle to evaluating this submission; the editor should request the SM as part of the revision. The N-scaling concern about the linearization is central and should be resolved before acceptance. The paper presents an interesting and potentially significant minimal model, but in its present form it is not yet verifiable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a useful, clearly written paper that adds self-consistent mechanical noise to the Ehrenfest model for cyclic shear, and shows—within that model—a genuine dynamical transition with a linear divergence of the escape time, plus a three-state coarse-graining that reproduces non-monotonic fatigue and a tan^2 scaling near the critical strain. The two-state part is clean and checkable from the main text. The three-state construction and the scaling law are new relative to [24–26], and the paper is honest that [25] already had a mean-field transition.\n\nThe soft spots are where the model assumptions do the work. The transition and the exponent −1 rely on the escape rates from stable states being strictly linear in the unstable fraction P_U. The paper justifies this from the D→0 limit of Eq. (3), but the stress-test note is right that this linearization is not uniform: for the shallowest stable states, √ε_j − γ ~ 1/(4Nγ), and for any fixed small P_U, their rates saturate at 1/τ0 when αP_U exceeds Δ_j^2. The closed-form W_US in Eq. (6) has a specific coefficient, but the exact sum over stable states gives a derivative at P_U=0 that grows with N unless a cancellation occurs. The paper gives no N-scaling analysis and the SM is unavailable, so we can't check whether the exponent and the tan^2 scaling are robust or finite-N artifacts. This is a meaningful gap, but not a demonstrated contradiction.\n\nAlso, the non-monotonic fatigue is built into the three-state model via the feedback D = αP_U, so seeing it emerge is a consistency check, not a prediction. That's okay, but it should be read that way. The paper also doesn't make quantitative contact with atomistic simulations; it's a mechanism paper, and it's useful on those terms.\n\nWho it's for: anyone working on elastoplastic mean-field theories or cyclic shear simulations who wants a minimal, analytically tractable toy that reproduces several observed features. The tan^2 scaling is a concrete prediction that could be tested in simulations.\n\nRecommendation: send it to a serious referee. The two-state derivation deserves to be checked, and the N-sensitivity issue needs to be resolved before the quantitative claims (exponent −1, tan^2) are accepted. The paper is not discredited by the concern, but the referees should ask for the SM derivation and an N-scaling check. My own verdict is conditional.","headline":"A clean mean-field feedback model for cyclic shear yields a transition and a tan^2 scaling, but the key linearization is not checked for N-sensitivity, so the quantitative claims are conditional.","tokens_in":10015,"tokens_out":4404,"would_cite":true,"duration_ms":37925,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Adding mechanical noise feedback to the Ehrenfest model makes cyclic shear yielding a genuine dynamical transition, with a diverging timescale and a universal scaling form.","keywords":["cyclic shear","amorphous solids","yielding","Ehrenfest model","mechanical noise","dynamical transition","fatigue failure","coarse graining"],"falsifier":"Measure in atomistic simulations of cyclic shear the instantaneous strain diffusion coefficient $D(t)$ and the unstable fraction $P_U(t)$ near the yield amplitude; if $D/P_U$ is not approximately constant, or if the number of cycles to steady state does not diverge as $(\\gamma-\\gamma^*)^{-1}$, the transition mechanism does not hold. The universal scaling $P_U/\\delta\\gamma = 1+\\tan^2(a\\sqrt{\\delta\\gamma}\\,\\delta t)$ is directly testable: a failure to collapse data onto this curve near $\\gamma_c$ would falsify the claimed universality.","tokens_in":8895,"feed_emoji":"🔁","tokens_out":7009,"duration_ms":57695,"temperature":0.7,"pith_summary":"This paper claims that the puzzling many-cycle behaviour of amorphous solids under cyclic shear (the divergence of the number of cycles to steady state, non-monotonic fatigue-failure dynamics, and coexisting frozen and fluidized states) can be explained by a coarse-grained description in which mechanical noise feeds back through the fraction of unstable blocks, $D(t)=\\alpha P_U(t)$. The feedback turns the Ehrenfest-type random-walk model into one with a genuine dynamical transition at a critical strain amplitude $\\gamma^*$, below which only a frozen state with $P_U=0$ exists and above which a fluidized state appears, with the escape time from stable states diverging as $(\\gamma-\\gamma^*)^{-1}$. A minimal three-state coarse graining (unstable, threshold, absorbing) is shown to reproduce the non-monotonic fatigue-failure dynamics that a two-state description cannot capture. Near the annealing-dependent critical strain $\\gamma_c$, the unstable fraction follows the universal scaling $P_U=\\delta\\gamma[1+\\tan^2(a\\sqrt{\\delta\\gamma}\\,\\delta t)]$. If correct, these results identify mechanical noise feedback, rather than thermal activation, as the mechanism that sets the yield point and controls the fatigue limit.","feed_headline":"Mechanical noise turns cyclic shear yielding into a phase transition","feed_subtitle":"Feedback from unstable regions gives a sharp yield point, diverging failure times, and a universal scaling curve.","key_machinery":"The load-bearing object is the feedback relation $D(t)=\\alpha P_U(t)$, which makes the transition rates out of stable states depend on the instantaneous fraction of unstable blocks. With this feedback the coarse-grained master equation becomes nonlinear, and a fixed-point analysis yields the dynamical transition and the exponent $-1$ divergence of the escape time. A slow-manifold reduction near the critical frozen state produces the $\\tan^2$ scaling (19). The three-state coarse graining (unstable, threshold, absorbing) is the minimal description that retains the non-monotonic fatigue-failure dynamics, which the two-state description cannot show.","core_discovery":"The central claim is that a self-consistent mechanical-noise feedback term converts the previously studied Ehrenfest model of cyclic shear yielding from a model with a crossover into one with a true dynamical phase transition. The transition is characterized by a critical strain amplitude $\\gamma^*$: for $\\gamma<\\gamma^*$ the only steady state has $P_U=0$ (frozen), while for $\\gamma>\\gamma^*$ a fluidized fixed point with $P_U\\neq 0$ emerges, and the mean escape time from stable states diverges as $(\\gamma-\\gamma^*)^{-1}$ approaching the transition from above. The paper further claims that a three-state coarse graining (unstable, threshold, absorbing) is the minimal description that shows non-monotonic 'fatigue failure' dynamics (plastic activity first drops, then rises to failure), and that near the critical strain the unstable fraction follows the universal form $P_U=\\delta\\gamma[1+\\tan^2(a\\sqrt{\\delta\\gamma}\\,\\delta t)]$. These results are presented as explaining the divergence of cycles, the annealing-dependent yield strain, and the coexistence of frozen and fluidized states seen in simulations.","pith_inferences":["If the linear feedback assumption $D=\\alpha P_U$ fails in real systems (for instance because Eshelby stress fluctuations from distant events are delayed or nonlocal), the exponent $-1$ and the $\\tan^2$ form would likely change, but a qualitatively similar frozen-fluidized transition might persist; measuring $D(t)$ against $P_U(t)$ in simulations would distinguish these cases.","The $\\tan^2$ scaling is derived for a broad class of master equations, so it may apply to other driven disordered systems where activity feeds back into local stability, not just cyclic shear of glasses.","The three-state picture suggests that fatigue failure is preceded by an accumulation of probability in threshold (marginally stable) states; this could be probed in simulations by tracking the population of such states as a function of cycle number.","The same feedback mechanism could be incorporated into continuum elasto-plastic models, where making the local yield stress depend on instantaneous plastic activity might reproduce the fatigue-limit divergence without invoking thermal activation."],"forward_implications":["The number of cycles to reach a steady state diverges as $(\\gamma-\\gamma^*)^{-1}$ on either side of the yield amplitude, matching the fatigue-limit behaviour reported in simulations.","Below $\\gamma^*$ the system always freezes, but different initial annealing states lead to different frozen configurations; above $\\gamma^*$ a fluidized state coexists with frozen attractors whose basin shrinks as $\\gamma$ grows.","The critical strain $\\gamma_c$ for a well-annealed sample is larger the better the sample is annealed, consistent with the observation that well-annealed glasses yield at higher amplitudes.","Near $\\gamma_c$, the relaxation time to reach the fatigue minimum and the time to re-fluidize both scale as $\\delta\\gamma^{-1/2}$, following from the $\\tan^2$ form.","A two-state description cannot produce non-monotonic fatigue-failure dynamics; three state classes (unstable, threshold, absorbing) are necessary, with threshold-state accumulation preceding failure."],"supporting_citations":[{"why":"Introduces the Ehrenfest urn mapping of cyclic shear yielding and the notion of stable and unstable states that this work extends with mechanical noise.","marker":"[26]"},{"why":"Mean-field treatment of mechanical noise in a mesoscopic-block model that exhibits a dynamical transition, the approach the paper builds on and generalizes.","marker":"[25]"},{"why":"Single-block energy-landscape model whose qualitative agreement with simulation motivated the coarse-grained descriptions extended here.","marker":"[24]"},{"why":"Simulations showing annealing-dependent yield strain and a threshold energy, which the model's annealing parameter $\\beta$ is meant to capture.","marker":"[3]"},{"why":"Simulations revealing divergence of cycles to steady state near yielding and property evolution under cyclic shear, the empirical target of the model.","marker":"[1]"},{"why":"The fatigue-limit concept (Basquin) to which the inverse-power divergence of cycles is compared.","marker":"[21]"},{"why":"Review of elasto-plastic models that motivates the mean-field mechanical-noise feedback $D(t)=\\alpha P_U(t)$.","marker":"[27]"},{"why":"Derives the Eshelby stress propagator linking plastic activity to local stress fluctuations, justifying the proportionality of noise to unstable fraction.","marker":"[33]"}],"fun_headline_variants":["Cyclic shear yielding becomes a true phase transition","Feedback noise induces critical yield transition","Critical strain emerges in cyclic shear yielding","Shear yielding's hidden critical point revealed","Non-monotonic failure from feedback noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument rests on the assumption that the mechanical noise a stable region feels is strictly proportional to how many regions are currently unstable, so that escape rates vanish linearly as the unstable population shrinks.","fun_headline_variants_meta":{"raw":{"variants":["Cyclic shear yielding becomes a true phase transition","Feedback noise induces critical yield transition","Critical strain emerges in cyclic shear yielding","Shear yielding's hidden critical point revealed","Non-monotonic failure from feedback noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000357,"raw_usage":{"total_tokens":1909,"prompt_tokens":891,"completion_tokens":1018,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":507,"completion_tokens_details":{"reasoning_tokens":968}},"tokens_in":507,"tokens_out":1018,"duration_ms":8554,"temperature":1.0,"reasoning_tokens":968,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:27:42.258059+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure in atomistic simulations of cyclic shear the instantaneous strain diffusion coefficient $D(t)$ and the unstable fraction $P_U(t)$ near the yield amplitude; if $D/P_U$ is not approximately constant, or if the number of cycles to steady state does not diverge as $(\\gamma-\\gamma^*)^{-1}$, the transition mechanism does not hold. The universal scaling $P_U/\\delta\\gamma = 1+\\tan^2(a\\sqrt{\\delta\\gamma}\\,\\delta t)$ is directly testable: a failure to collapse data onto this curve near $\\gamma_c$ would falsify the claimed universality.","supporting_citations":[{"cited_title":"Mungan and S","cited_arxiv_id":null,"evidence_quote":"Introduces the Ehrenfest urn mapping of cyclic shear yielding and the notion of stable and unstable states that this work extends with mechanical noise."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Mean-field treatment of mechanical noise in a mesoscopic-block model that exhibits a dynamical transition, the approach the paper builds on and generalizes."},{"cited_title":"Sastry, Physical Review Letters126, 255501 (2021)","cited_arxiv_id":null,"evidence_quote":"Single-block energy-landscape model whose qualitative agreement with simulation motivated the coarse-grained descriptions extended here."},{"cited_title":"Bhaumik, G","cited_arxiv_id":null,"evidence_quote":"Simulations showing annealing-dependent yield strain and a threshold energy, which the model's annealing parameter $\\beta$ is meant to capture."},{"cited_title":"Leishangthem, A","cited_arxiv_id":null,"evidence_quote":"Simulations revealing divergence of cycles to steady state near yielding and property evolution under cyclic shear, the empirical target of the model."},{"cited_title":"Basquin, Proc Am Soc Test Mater10, 625 (1910)","cited_arxiv_id":null,"evidence_quote":"The fatigue-limit concept (Basquin) to which the inverse-power divergence of cycles is compared."},{"cited_title":"Agoritsas, E","cited_arxiv_id":null,"evidence_quote":"Derives the Eshelby stress propagator linking plastic activity to local stress fluctuations, justifying the proportionality of noise to unstable fraction."}],"review_version":1}