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The paper claims that the thermal decay rate of a metastable phase is exactly the equilibrium transition-state rate times a dynamical factor (1 minus the re-crossing probability), unifying previously conflicting formulas in thermal field th

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2026-08-04 01:07 UTC pith:L7C3LKJV

load-bearing objection A serious, mostly convincing unification of thermal nucleation-rate formulas, with the exactness claim stated a bit stronger than the proof warrants. the 4 major comments →

arxiv 2608.00169 v1 pith:L7C3LKJV submitted 2026-07-31 hep-th cond-mat.quant-gascond-mat.softhep-lathep-ph

Dynamics of nucleation in thermal phase transitions

Oliver Gould , Joonas Hirvonen , Andrey Shkerin , Sergey Sibiryakov This is my paper
classification hep-th cond-mat.quant-gascond-mat.softhep-lathep-ph
keywords thermal nucleationdynamical prefactortransition state theoryre-crossing probabilityfalse vacuum decayoscillonsstochastic mechanicssurface-flux ensemble
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish that the thermal decay rate of a metastable phase is exactly Γ_TST·(1−R), where Γ_TST is the equilibrium transition-state flux and R is the probability that a trajectory crossing the barrier re-crosses it and does not nucleate. If true, this resolves the long-standing mismatch between several competing formulas for the dynamical prefactor in thermal field theory, and it explains why oscillons suppress rather than enhance nucleation relative to the equilibrium prediction. The authors give two derivations, reproduce known limits, and test the formula numerically in (1+1)-dimensional scalar field theories, finding that non-perturbative 'sloshing' re-crossings can dominate the prefactor and that the resulting rate agrees with direct-decay simulations where those are feasible. The practical payoff is a numerical recipe whose cost is set by the thermalization time, not the exponentially long decay time.

Core claim

The paper’s central claim is Eq. (13): in the stationary-flux regime, the physical thermal decay rate equals the equilibrium transition-state rate multiplied by a dynamical factor, Γ = Γ_TST (1 − R), with R = R^(+) + R^(−). Here R^(+)/R^(−) are the probabilities that a trajectory crossing the dividing surface outward/inward turns around and ends up on its starting side. The relation is argued to be exact up to corrections of order e^{−E_ts/T} whenever the thermostat is large and fast, and to hold for any dividing surface connecting the deep metastable and stable regions. Because R is a probability, the real rate never exceeds the TST rate. The factorisation is derived twice: by following the

What carries the argument

The load-bearing object is the re-crossing probability R, evaluated from surface-flux ensembles: phase-space initial conditions placed on the dividing surface with weight |n·ż| e^{−H/T}, evolved for times long compared with the dynamical time but short compared with the decay time; the plateau value of the returning fraction defines R^(±). Sloshing re-crossings — trajectories that oscillate one or more times in the false vacuum before finally escaping — constitute the non-perturbative part of R and are the mechanism behind both the weak-damping suppression and the oscillon effect. Dividing-surface independence follows because a change of surface changes Γ_TST and R in compensating direction

Load-bearing premise

That a stationary-flux plateau exists between the dynamical time and the decay time whenever the bath is large and thermalization fast; if the plateau never forms, the 'thermal decay rate' in Eq. (13) is not a well-defined constant.

What would settle it

Measure R from surface-flux ensembles prepared with the fully non-perturbative criticality condition (Appendix C.2) and compare with the quadratic-ensemble results at, say, T/E_ts = 0.02 in the quartic model; a statistically significant difference would show the Gaussian initial-state approximation misses re-crossings that Eq. (13) must include. Equivalently, test the claimed dividing-surface independence by evaluating Γ_TST·(1−R) on a surface moved well away from the TS, where any residual dependence signals a breakdown of the plateau argument.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The dynamical prefactor is always ≤ 1; oscillons and other long-lived sloshing states lower the rate relative to the equilibrium TST prediction, not raise it.
  • The steady-state rate is independent of the dividing surface over a wide range, so the troublesome question of choosing the right transition state is circumvented.
  • Thermal rates exist only with a large, fast thermostat (t_th ≪ t_dec and heat capacity C ≫ E_ts²/T²); otherwise the rate drifts and decays are non-exponential.
  • The surface-flux method computes rates at arbitrarily strong exponential suppression at a cost set by t_plat, exponentially shorter than direct decay simulation.
  • The formula unifies the known perturbative prefactor and the non-perturbative weak-damping turnover as the prompt and sloshing contributions, respectively.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If Eq. (13) survives scrutiny, cosmological bubble-nucleation estimates at moderate E_ts/T may need revision, since sloshing re-crossings can contribute at a level comparable to perturbative corrections.
  • The same re-crossing logic should apply to other thermally activated processes with a dividing surface, such as sphaleron-like transitions or soliton production, where long-lived intermediates could suppress the rate.
  • A direct testable corollary: in the quartic model at T/E_ts ≈ 0.04, replacing the Gaussian initial surface ensemble with a fully non-perturbative one should alter R if the quadratic approximation misses essential sloshing states; agreement would strengthen the exactness claim.
  • The predicted exponential suppression R_non-pert ∝ exp(−α E_ts/T) with α ≈ 0.044 means the non-perturbative correction is formally exponentially small but numerically significant at moderate suppression, a window accessible to analogue condensed-matter experiments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper develops a general dynamical theory of thermal nucleation in classical systems, centered on Eq. (13): Γ = Γ_TST·(1−R), where the TST rate is corrected by a re-crossing probability R = R^(+) + R^(−). The authors give two derivations (probability-flux evolution and trajectory classification), relate them to the MRT approach, reproduce Langer's dissipative prefactor and the Mel'nikov–Meshkov weak-noise turnover, and implement the surface-flux method in 1+1-dimensional scalar field theory with quartic and Liouville potentials. They report significant non-perturbative 'sloshing' re-crossings, enhanced by oscillons, an exponential fit R_non-pert ∝ exp(−αE_ts/T) with α≈0.044, and a discussion of finite-time and quasi-stationary rates when the thermality conditions (11)–(12) are violated.

Significance. If the central formula is valid, it unifies previously conflicting expressions for the dynamical prefactor, gives a surface-independent definition of the thermal nucleation rate, and yields a numerical algorithm that is exponentially cheaper than direct decay simulations. The paper has notable strengths: two independent derivations that cross-check each other, analytic recovery of known results, public numerical codes, and direct lattice comparisons at selected parameter points. The reported agreement with direct decays at T̂=0.1 and with Langer's formula in the dissipative case is persuasive evidence that the framework is physically sound. The main weakness is that the 'exact modulo exponentially small corrections' claim rests on unproven plateau and ergodicity assumptions that are load-bearing for the definition of R.

major comments (4)
  1. [Secs. 2.1–2.2, Eqs. (13), (31)] The exactness claim requires R_R^(±)(t) to stabilize to a unique plateau for t_plat ≪ t ≪ t_dec. Conditions (11)–(12) are introduced only as 'appear to be sufficient'; no proof is given that they exclude slow drift of R_R^(±)(t) on time scales up to t_dec, e.g., from long-lived sloshing trajectories or power-law return-time tails. Section 5.2 demonstrates drift when condition (12) is violated, but it does not establish a bound on drift when (12) is satisfied. Because the limit in Eq. (31) defines R, the statement that Eq. (13) is exact up to O(e^{−E_ts/T}) is unsupported. I ask the authors either to prove the plateau under (11)–(12) (or a stronger explicit mixing condition) or to state the plateau as an assumption and soften the exactness claim. A numerical plateau test over a wide time window at fixed small T/E_ts would be a useful partial check.
  2. [Sec. 2.3, Eqs. (40)–(43)] The trajectory derivation assumes that every trajectory through ∂R has ended in R_in or R_out by time t_plat, and uses this to define θ_in→out(τ_z). This is an additional ergodicity/mixing assertion that is not derived from (11)–(12). The text 'Since any trajectory through ∂R ends up either in R_in or R_out on the time scale t_plat' is asserted, not proven. If some trajectories remain in the layer or re-enter after t_plat, then N_cross(τ_z) and the probabilities in Eq. (43) are not well defined, and the claimed surface independence of Γ is not established. This assumption should be stated explicitly and justified, or the proof completed.
  3. [Sec. 4.1, App. C.1, Eq. (76), Fig. 8] The quantitative non-perturbative result R_non-pert ∝ exp(−αE_ts/T), α≈0.044, is obtained using the quadratic surface-flux ensemble around the tree-level TS and a set of ad hoc classification thresholds (decay trigger max|ϕ|>10, sloshing marker max|ϕ|<1, R_in/R_out definitions). The paper does not report the sensitivity of the fitted exponent or the perturbative/non-perturbative split to these thresholds or to ensemble preparation choices. Since the claim that sloshing/oscillon re-crossings dominate and follow an exponential law is central to Sec. 4, a threshold scan or a definition-independent check is needed. The direct-decay cross-check at T̂=0.1 is reassuring, but it covers only one parameter point and does not by itself validate Eq. (76).
  4. [Sec. 5.1, Fig. 15] The finite-time study shows that the rate approaches its late-time value as a power law, Γ_Rref(t)−Γ ∝ t^{−α} with α≈1.4–1.7, and that finite-volume effects modify the asymptotics at smaller L. This supports the concern that the plateau in R_R^(±)(t) may not be a sharp, exactly stationary regime but only an approximate one. The paper should quantify how the plateau value and the 'exact' claim depend on the measurement window and on the finite box size, especially because the algorithm is proposed as a general method for arbitrarily strong exponential suppression.
minor comments (5)
  1. [Sec. 4.2 / Fig. 11] The text says 'At T/E_ts=0.08 (left plot)' but the caption says 'T/E_ts=0.08 (right)'. Please fix the mismatch.
  2. [Sec. 2.2, Eq. (23)] The sign convention for I_R(t) at t→0± is confusing. A sentence clarifying whether I_R is the outward probability flux or the rate of change of the probability in R would help readers follow the derivation of Eq. (27).
  3. [Sec. 5.2.1, Eq. (91)] The connection between the microcanonical rate Γ_mc(E), its ensemble average in Eq. (87), and the time-dependent rate Γ_R(t) of Sec. 2.2 is asserted rather than derived. A brief derivation or explicit mapping would improve rigor.
  4. [Sec. 2.1, after Eq. (13)] The phrase 'exact modulo exponentially small corrections of order O(e^{−E_ts/T})' could be misread because R_non-pert itself is exponentially small in E_ts/T but with a much smaller coefficient (α≈0.044). It would be helpful to state explicitly that the neglected corrections are relative corrections of order e^{−E_ts/T}, not that R is negligible at that order.
  5. [Ref. [72]] Reference [72] is listed as 'To appear' with no preprint number. If available, please provide the arXiv number or update the citation.

Circularity Check

0 steps flagged

No significant circularity: Eq. (13) is derived from the dynamics of surface-flux ensembles and checked against independent analytic and direct-decay benchmarks.

full rationale

The central formula Γ = Γ_TST(1−R) is not fitted or defined into existence. It follows from the exact flux identity in Eqs. (27)/(30), where R_R^(±)(t) are defined as probabilities under surface-flux ensembles, and only the plateau limit in Eq. (31) is identified with the steady-state rate. The dynamics enters through solving the Fokker–Planck equation in Sec. 3.1 or through explicit trajectory sums in Sec. 3.2, not through an input parameter chosen to reproduce the result. The recovery of Langer's formula and of Mel'nikov–Meshkov's turnover formula is an independent check, not an input. Numerically, the re-crossing probabilities are measured from separately prepared surface-flux ensembles and then compared with direct lattice decay rates (Figs. 13–14), so the numerical claim is self-contained against an external benchmark. Some passages cite the authors' own earlier work (Refs. [18,29,73]) for numerical discrepancies, thermalization times, and direct-decay rates; these are comparisons and cross-checks, not load-bearing assumptions in the derivation of Eq. (13). The main caveat in the manuscript is that conditions (11)–(12) 'appear to be sufficient' for the stationary plateau, and Sec. 5.2 explicitly shows drift when they fail; this is a correctness/rigor limitation, not a circularity. Likewise the exponential fit (76) with α≃0.044 is an empirical fit, not a prediction derived from the formula. No circular step can be exhibited by reduction of an output equation to an input.

Axiom & Free-Parameter Ledger

5 free parameters · 8 axioms · 0 invented entities

All free parameters are empirical characterizations of simulation output, not inputs to Eq. (13). The core formula itself uses no fitted constants: it rests on the classical Hamiltonian/Langevin setup, the Fokker–Planck description, and the assumed existence of a stationary plateau. The most substantial ad hoc assumptions are the sufficiency of conditions (11),(12) and the claim that the quadratic surface-flux ensemble sees non-perturbative sloshing; both are partially validated by comparison with direct-decay simulations.

free parameters (5)
  • perturbative re-crossing slope (quartic) = ≈1.0 T̂
    Linear fit to R_pert in Fig. 8; used to characterize prompt re-crossings in the ϕ⁴ model.
  • perturbative re-crossing slope (Liouville) = ≈3.5 T/E_ts
    Linear fit to R_pert in Fig. 10; no-oscillon model.
  • non-perturbative sloshing exponent α = ≈0.044
    Exponential fit R_non-pert ∝ exp(−α E_ts/T), Eq. (76); coefficient is empirical, not derived.
  • classification thresholds (decay trigger, sloshing marker, R_in/R_out) = max|ϕ|>10, max|ϕ|<1, R_in: max|ϕ|<1, R_out: max|ϕ|>3√2, C=3.1√T̂
    Hand-chosen to define trajectory classes and the non-perturbative boundary in Appendix C.2; affect measured R but direct-simulation cross-checks support the extracted values.
  • plateau time t_plat = ≈50–200/m in the simulations
    Time at which R(t) stabilizes; extracted by inspection and used in Eq. (16).
axioms (8)
  • domain assumption Classical regime T_q ≪ T ≪ E_ts; Hamiltonian at most quadratic in momenta (Eq. 5)
    Limits applicability to classical thermal nucleation; stated in Sec. 2.1.
  • domain assumption Thermostat described by linear damping plus additive white noise with FDT (Eqs. 6–7)
    Underpins the Fokker–Planck equation (8); other dissipation kernels are deferred to Sec. 6.
  • ad hoc to paper Thermality conditions t_th ≪ t_dec and C ≫ E_ts²/T² are sufficient for a stationary-flux plateau
    Eqs. (11),(12); text says 'appear to be sufficient'—not proven; Sec. 5.2 explores violations.
  • ad hoc to paper Every trajectory through ∂R ends in R_in or R_out within t_plat (Sec. 2.3, Eqs. (40)–(43))
    Needed for the trajectory indicator and time-independent re-crossing probabilities.
  • ad hoc to paper Quadratic surface-flux ensemble near the transition state captures both prompt and non-perturbative sloshing re-crossings (Sec. 4.1, App. C.1)
    Asserted to be sufficient for leading contributions; cross-checked against direct decays at T̂=0.1 but not proven at arbitrary coupling.
  • domain assumption Lattice discretization and symplectic/pseudo-spectral integrators approximate the continuum field theory; finite-size effects are small
    Simulations use a=0.024/m, L=100/m and checked insensitivity; periodic-box effects appear at late times (Sec. 5.1).
  • domain assumption Small-noise expansion in 1d mechanics: deviations from the deterministic trajectory are small enough for linearization (Sec. 3.2)
    Needed for the Mel'nikov–Meshkov reproduction; condition expressed around Eq. (73).
  • standard math Liouville theorem and conservation of energy along Hamiltonian flow (Sec. 2.2)
    Used to change variables from final to initial phase-space points in Eq. (20).

pith-pipeline@v1.3.0-alltime-deepseek · 41060 in / 20580 out tokens · 223074 ms · 2026-08-04T01:07:13.112165+00:00 · methodology

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read the original abstract

We study dynamical effects during nucleation in thermal first-order phase transitions in field theory. Focusing on the classical regime of the decay of a metastable state, we present the general formula for the thermal decay rate including the dynamical prefactor and give a recipe for its systematic evaluation. We describe the physical mechanism which reduces the actual thermal decay rate with respect to the statistical rate obtained in equilibrium theory. We also discuss the thermality conditions ensuring the existence of a steady-state thermal rate, in which case our formula is exact up to exponentially small corrections. We show that it reproduces the known results for the nucleation rate in stochastic mechanics and field theory, and allows us to unify and go beyond them. We illustrate this in real-time numerical simulations of simple field theory models. We observe significant non-perturbative contributions which can dominate the dynamical prefactor in weakly-coupled field theories at moderate exponential suppression of the decay rate. We explore the connection of these non-perturbative effects to oscillons. Notably, our numerical method requires exponentially less computing time than direct simulations of decays and is thus applicable to systems with arbitrarily strong exponential suppression. Finally, we discuss small or poorly thermalized systems when the thermality conditions are violated and the steady-state rate does not exist.

Figures

Figures reproduced from arXiv: 2608.00169 by Andrey Shkerin, Joonas Hirvonen, Oliver Gould, Sergey Sibiryakov.

Figure 1
Figure 1. Figure 1: Phase-space trajectories and the metastable region [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Potential with a metastable minimum at q− = q0, and TS at q− = 0. The assumption that the TS is unique does not hold in general. In fact, it is commonly violated in field theory where there is a continuous family of critical bubbles differing by spatial translations. Since translations do not cost any energy, they are associated with zero-frequency collective perturbations (zero modes). Further zero modes … view at source ↗
Figure 3
Figure 3. Figure 3: Different types of re-crossing motion. The particle starts from the top of the [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Three regions of phase space, here shown on a one-dimensional potential: [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Different types of trajectories crossing the boundary of the metastable region [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Sloshing re-crossing trajectories that contribute into the MRT rate ( [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Scalar field potentials for the numerical investigation of re-crossing probability. [PITH_FULL_IMAGE:figures/full_fig_p032_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The re-crossing probability R measured in simulations in the model with the quartic potential (75a) (black dots). Perturbative (blue triangles) and non-perturbative (red squares) contributions to R are also shown. The blue dashed line is the linear fit to the perturbative contribution, the red dash-dotted line is the exponential fit to the non￾perturbative contribution, and the black solid line is the sum … view at source ↗
Figure 9
Figure 9. Figure 9: Left: Number density of re-crossings as a function of time. Perturbative prompt re-crossings are shown in blue and non-perturbative sloshing re-crossings are shown in red. We take Tˆ = 0.05 corresponding to T/Ets = 0.0375. The gap at small times is due to the decay trigger requiring max |ϕ(x)| > 10. Right: Distribution of re-crossings in number of oscillations across the false vacuum, at the same temperatu… view at source ↗
Figure 10
Figure 10. Figure 10: The re-crossing probability for the model with the Liouville potential ( [PITH_FULL_IMAGE:figures/full_fig_p037_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Number density of re-crossings as a function of time, in the theory with the [PITH_FULL_IMAGE:figures/full_fig_p038_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Re-crossing probabilities R(±) in the dissipative case in the theory with the potential (75a), as functions of temperature and at two values of the dissipation coefficient: η/m = 10−2 (left) and η/m = 10−1 (right). The black dashed line is the leading-order perturbative prediction (54). Filled and empty dots show R(−) and R(+), respectively. Blue triangles (red squares) stand for perturbative (non-perturb… view at source ↗
Figure 13
Figure 13. Figure 13: Dynamical prefactor in the dissipative case, in the theory with the potential ( [PITH_FULL_IMAGE:figures/full_fig_p040_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The early-time behavior of the rate (82) in the theory (74) with the quartic potential (75a), at Tˆ = 0.1 and Lm = 50. The rate is measured in direct simulations (blue bars) and calculated using the surface-flux ensemble (82) (red solid line). The tips of the bars of different color and the width of the line show the statistical uncertainty. The dash-dotted blue line is the analytical 1-loop TST predictio… view at source ↗
Figure 15
Figure 15. Figure 15: Late-time behavior of the rate (82), for different lattice size L. We take Tˆ = 0.0125 and t∗ = 50/m. The wiggles on the lines correspond to statistical fluctuations. Γ, Eq. (13). The re-crossing probability in Γ is found from simulations described in Sec. 4, using the same lattice parameters as for calculating ΓRref(t). Note the uncertainty band for Γ, reflecting the residual time-dependence of R at t ∼ … view at source ↗

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