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Entropy barrier, not energy, gates roaming reactions

2026-07-08 05:39 UTC pith:XOHJ24WA

load-bearing objection A clean, well-controlled model study that successfully embeds Makarov's entropic barrier into a roaming Hamiltonian. The numerics are solid and the controls are honest. The main limitation is the single-energy scope. the 2 major comments →

arxiv 2607.06437 v1 pith:XOHJ24WA submitted 2026-07-07 physics.chem-ph math.DS

An entropic bottleneck, dynamical gating, and outward redistribution of roaming in a designed Chesnavich-type model

Stephen Wiggins This is my paper
classification physics.chem-ph math.DS
keywords modelbottleneckentropicroamingangularinteriororbitsradius
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.

This paper asks a direct dynamical question: what happens to roaming reactions when the controlling bottleneck is made purely entropic? The author takes the Chesnavich model of barrierless ion-molecule dissociation and replaces its orientation-locking angular hindrance with a deliberately designed transverse-stiffness ridge, a Gaussian bump in the angular frequency peaking at an interior radius. The radial channel remains barrierless throughout, so any constriction comes from angular confinement alone. The central claim is that this ridge reshapes roaming in two ways. First, it gates entry into the inner molecular well, cutting capture from 57.3% to 14.7% of incoming trajectories and returning most flux directly to reactants. Second, it does not eliminate roaming but displaces it outward in radius, suppressing it inside the ridge and switching it on farther out. The phase-space explanation rests on three unstable periodic orbits that serve as transition states: a tight orbit at the flux-minimum radius that gates well entry, a free-rotor orbit that sorts direct from roaming trajectories, and an outer orbiting orbit on the centrifugal barrier that gates escape. The tight orbit's dividing surface coincides with the variational minimum-flux surface to within 0.15%, confirming it as the controlling bottleneck. This bottleneck carries no potential-energy barrier along the reaction coordinate; it is entirely entropic, a constriction of accessible angular phase space. The trajectories it admits roam nonstatistically, with nonexponential gap-time distributions, meaning the bottleneck governs how much is captured but not the subsequent dynamics. Strength-matched monotone controls show that gating tracks hindrance strength at the bottleneck radius; the interior maximum's contribution is placement, concentrating that strength where it gates most effectively.

Core claim

The paper's central discovery is that a localized interior maximum of transverse stiffness, placed on an otherwise barrierless radial channel, creates a deep entropic bottleneck that gates inner-well capture and redistributes roaming outward without suppressing it. The bottleneck is the tight transition state, an unstable periodic orbit whose dividing surface is the variational minimum-flux surface, and it is purely entropic: no potential barrier exists along the reaction coordinate. The ridge cuts inner capture from 57.3% to 14.7% while leaving the nonreactive roaming fraction at the classifier radius essentially unchanged (5.3% to 4.9%), but spatially displaced. This realizes Makarov's ent

What carries the argument

Chesnavich model Hamiltonian; transverse-stiffness ridge (Gaussian bump in angular frequency); unstable periodic orbits as transition states; variational minimum-flux surface; exact directional flux; gap-time survival distributions; strength-matched monotone controls

Load-bearing premise

The paper's controlled comparison attributes the gating and outward displacement to the designed ridge's localized profile, but the ridge amplitude at the bottleneck radius reaches roughly 156 kcal/mol versus about 16 kcal/mol for the original lock at the same radius. The paper's own strength-matched monotone controls, which raise the lock's amplitude to match at that radius, also reproduce both the gating (17.8% capture) and the outward displacement, meaning the central dyn

What would settle it

If a monotone angular hindrance matched in amplitude at the bottleneck radius reproduced both the gating fraction and the outward displacement exactly, with no residual difference from the ridge, then the localized interior maximum of transverse stiffness would contribute nothing beyond what any strong orientational hindrance at that radius provides.

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

If this is right

  • If entropic bottlenecks can be placed at chosen radii by designing transverse stiffness profiles, then molecular systems with tunable angular hindrance, such as those involving steric or solvent effects, could exhibit controllable gating of reactive capture without any energy barrier.
  • The separation between the entropic bottleneck radius (2.46 Å), the roaming classifier radius (3.5 Å), and the centrifugal escape barrier (7.48 Å) implies that no single dividing surface captures the full dynamics of barrierless reactions, challenging standard single-transition-state rate models.
  • The nonexponential gap-time distributions, decomposing into well-separated direct and roaming populations, suggest that experimental measurements of transit times could distinguish entropic gating from energetic gating in real molecular systems.
  • The finding that strength-matched monotone controls reproduce both gating and outward displacement implies that the causal agent is hindrance amplitude at the bottleneck radius, with the localized profile providing efficiency and economy rather than a qualitatively distinct mechanism.

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

2 major / 8 minor

Summary. The manuscript constructs a designed Chesnavich-type roaming model in which the orientation-locking angular hindrance is replaced by a localized Gaussian maximum of the transverse stiffness (angular frequency) at an interior radius. The central dynamical result is that this ridge gates inner-well capture (57.3%→14.7%) and displaces roaming outward without suppressing it, while the nonreactive roaming fraction at the classifier radius is essentially unchanged (5.3%→4.9%). The phase-space explanation identifies three unstable periodic orbits (tight, free-rotor, orbiting), with the tight orbit's dividing surface coinciding with the variational minimum-flux surface (action 2.902 vs. flux 2.907, 0.15% agreement). Strength-matched monotone controls separate amplitude from localization effects. The gap-time distributions are non-exponential for both models, indicating nonstatistical dynamics. The model is a designed theoretical construct, not fitted to a specific molecule, and the paper is transparent about this throughout.

Significance. The paper makes a clean contribution to the phase-space theory of roaming by embedding Makarov's transverse-stiffening entropic barrier mechanism into a Hamiltonian roaming model and tracing its dynamical consequences. The strengths are substantial: (1) the controlled comparison isolates the angular interaction while holding the radial channel, energy, and initial conditions fixed, and the LJ(8,4) substitution confirms insensitivity to the radial potential shape; (2) the tight-orbit action (2.902) agreeing with the fixed-radius flux minimum (2.907) to 0.15% is a non-trivial numerical verification that the variational bottleneck is the periodic-orbit dividing surface—a computed result, not an assumption; (3) energy conservation to |H−E|≲2×10⁻⁷ kcal/mol and grid convergence (57.11%/14.68% on 120×120 vs. 57.27%/14.75% on 240×240) are documented; (4) the strength-matched monotone controls are an honest and informative decomposition of amplitude vs. localization effects; (5) the gap-time analysis directly tests statisticality on the same ensemble used for the dynamical characterization. The paper is transparent about the designed nature of the model and about the qualifications on its中央

major comments (2)
  1. The entire analysis—trajectory ensembles, flux calculations, periodic orbit locations, and gap-time distributions—is conducted at a single energy E=2 kcal/mol. The entropic bottleneck's effectiveness depends on the ratio of angular confinement to available channel energy ε(R)=E−V_rad(R); at E=2 the confinement factor C≈0.10, but at higher excess energies C would increase and the gating could weaken substantially. The central claim that the ridge 'gates inner capture and displaces roaming outward' is demonstrated only at one point in energy space. A second energy (e.g., E=4 or 6 kcal/mol) showing whether the gating persists, weakens, or qualitatively changes would substantially strengthen the robustness of the central claim. If this is not feasible in revision, the authors should at minimum discuss the expected energy dependence qualitatively, since the confinement factor C(R,E) is energy
  2. Sec. 4, qualification (ii): the ridge's B(r_c)≈156 kcal/mol is ~10× the Chesnavich lock's ~16 kcal/mol at the same radius. The r_c-matched monotone control (V_0(r_c)=156) gives 17.8% capture vs. the ridge's 14.7%, a 3.1-percentage-point difference. The authors attribute this residual to the localized profile shape. While this is statistically significant given the 57,600-trajectory ensemble, the framing in the abstract and Sec. 1 ('a localized interior maximum of the transverse stiffness reshapes roaming') could overstate the causal role of the interior maximum per se, since the dominant gating effect (57.3%→17.8%) follows from amplitude at the bottleneck radius. The qualification in Sec. 4 is honest and well-placed, but the abstract's one-sentence summary ('a localized interior maximum of the transverse stiffness reshapes roaming, gating inner capture and displacing roaming outward') el
minor comments (8)
  1. Sec. 3, Eq. (7): the factor a(r)^{-1} in B(r) is chosen to cancel the r-dependence of the angular mass so that Ω(r) is an exact Gaussian. This is clearly stated, but the resulting B(r) values (~156 kcal/mol at r_c) are large compared to typical chemical energy scales. A brief comment on whether this amplitude is physically realizable or whether the model should be understood as purely illustrative would help readers calibrate expectations.
  2. Table 1: the parameter Ω_0^2=200 (kcal/mol) is listed without units on Ω_0 itself. Given that Ω(r) has dimensions of frequency, the units should be stated explicitly.
  3. Sec. 7: the classifier radius r_class=3.5 Å is described as having a 'broad maximum' in roaming fraction near 3.4–3.5 Å, but no quantitative sensitivity analysis (e.g., how the four-class partition changes as r_class varies over 3.0–4.0 Å) is provided. A small table or inset showing this sensitivity would strengthen the claim that the choice of 3.5 Å is representative.
  4. Fig. 2b caption: the variational bottleneck r*=2.47 Å is marked, but the figure uses the LJ(8,4) channel while the text in Sec. 4 primarily discusses the V_CH channel (r*=2.46 Å). The caption should clarify which channel is plotted and whether the values are harmonized.
  5. Appendix A: the time limit t_max=320 model units (≈15.6 ps) leaves 0.085% of trajectories (49 of 57,600) unclassified. The authors should briefly state whether these unclassified trajectories are concentrated in any particular region of the launch grid (e.g., near the roaming/ direct boundary) or are approximately uniformly distributed.
  6. Sec. 6: the statement that B(r_OTS)~10^{-47} makes the ridge 'negligible' at the orbiting radius is correct, but the exponent should be checked for consistency with the Gaussian width σ=0.5 Å and the center r_c=2.2 Å. At r=7.48 Å, the Gaussian factor exp(-(7.48-2.2)^2/(2×0.5^2))≈exp(-55.5)≈10^{-24}, so the stated 10^{-47} may include the a(r)^{-1} prefactor. The basis for the exponent should be stated.
  7. The reference list is thorough but the self-citation pattern is heavy: references [22]–[32] are all co-authored by the author or close collaborators. This is appropriate given the lineage of the work, but the authors should ensure that the contributions of independent groups (e.g., Suits, Bowman) are cited where relevant to the dynamical questions, not only to the phase-space framework.
  8. Sec. 1, paragraph on entropy barriers: the connection to the 'entropic intermediate' of nonstatistical organic dynamics [7] is mentioned as an 'adjacent phenomenon' but the distinction is not elaborated. A sentence clarifying whether the entropic bottleneck here could manifest as an entropic intermediate in a higher-dimensional system would help readers working in that area.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful and constructive report. Both major comments are well-taken. We address them in turn.

read point-by-point responses

  1. Referee: The entire analysis is conducted at a single energy E=2 kcal/mol. The entropic bottleneck's effectiveness depends on the ratio of angular confinement to available channel energy. A second energy or at least a qualitative discussion of energy dependence is requested.

    Authors: The referee is correct that the central dynamical claim is established at a single energy, and that the confinement factor C(R,E) is energy-dependent: as E increases, the available channel energy ε(R)=E−V_rad(R) grows, the accessible transverse phase space widens, and the bottleneck must weaken. We will address this in two ways in the revision. First, we will add a qualitative discussion of the expected energy dependence, using the confinement factor C(R,E) and the flux formula (Eq. 11) to explain the trend: at higher E the same transverse stiffness Ω(r) constricts a smaller fraction of the (now larger) transverse phase space, so C rises toward unity and the gating softens. Second, we will add flux calculations at E=4 and E=6 kcal/mol (the flux integral of Eq. 11 is inexpensive and does not require new trajectory ensembles) to show quantitatively how the bottleneck depth and location shift with energy. We expect this to confirm the referee's intuition that the gating weakens but does not vanish at moderate excess energies, given that Ω₀²=200 kcal/mol is large. We note that repeating the full trajectory ensemble, periodic-orbit search, and gap-time analysis at a second energy is a substantial computation that we would undertake in a follow-up, but the flux-level calculation at additional energies directly tests the robustness of the statistical bottleneck that the dynamics track. revision: partial

  2. Referee: The abstract's framing ('a localized interior maximum of the transverse stiffness reshapes roaming') could overstate the causal role of the interior maximum per se, since the dominant gating effect (57.3%→17.8%) follows from amplitude at the bottleneck radius, with the localized profile contributing the residual (17.8%→14.7%).

    Authors: The referee is right that the abstract's one-sentence summary, read in isolation, could attribute more of the gating to the interior maximum than the controls justify. The body of the paper is already precise on this point: Sec. 4, qualification (ii), states that 'both headline effects follow from strong orientational hindrance in the region of the bottleneck radius and are not unique to an interior maximum,' and the abstract's penultimate sentence says that 'the gating tracks the hindrance strength at the bottleneck radius; what the interior maximum supplies is placement.' Nevertheless, the opening framing could be tightened. We will revise the abstract so that the causal claim is stated more carefully: the ridge (as a whole, including its amplitude) gates inner capture and displaces roaming, and the interior maximum's specific contribution is placement and economy—concentrating the hindrance at the gating radius—rather than the bulk of the gating itself. This makes the abstract consistent with the controlled decomposition already in Sec. 4 without diminishing the paper's central finding. revision: yes

Circularity Check

2 steps flagged

No significant circularity. The paper's central dynamical results are computed from trajectory integration of a designed Hamiltonian, not restatements of cited prior work or fitted inputs.

specific steps
  1. self citation load bearing [Sec. 7, first paragraph; Sec. 8, final paragraph; Appendix B, first paragraph]
    "Mauguière, Collins, Ezra, Farantos and Wiggins constructed phase-space dividing surfaces from the periodic orbits of the Chesnavich model and classified trajectories by how many times they cross them [22]. Krajňák and Waalkens computed the stable and unstable manifolds of those orbits and described the transport through their lobes [27]; Krajňák and Wiggins obtained the same structures with Lagrangian descriptors... We adopt their distinction between direct and roaming trajectories."

    The paper's classification scheme (direct vs. roaming via periodic-orbit dividing surfaces) and the nonstatistical gap-time diagnostic are adopted from prior work by the author and collaborators [22, 27, 28, 16]. However, this is a methodological borrowing, not a circular derivation: the cited work establishes the phase-space framework and the nonstatistical finding for the original Chesnavich model, while the present paper applies these tools to a new, designed Hamiltonian with a different angular interaction. The central dynamical results (gating from 57% to 15%, outward displacement of roaming, the tight-orbit/flux-minimum coincidence at r*≈2.46 Å) are computed from trajectory integration and flux evaluation on the new model, not restated from the citations. The self-citation provides a

  2. fitted input called prediction [Sec. 3, Eqs. (7)–(8)]
    "B(r) = 1/2 a(r)^{-1} Ω_0^2 e^{-(r-r_c)^2/σ^2} ... Ω(r) = sqrt(2B(r)a(r)) = Ω_0 e^{-(r-r_c)^2/(2σ^2)}. The factor a(r)^{-1} in B(r) is chosen precisely to cancel the r-dependence of the angular mass, so that Ω(r) is an exact Gaussian with a single smooth interior maximum at r=r_c"

    The transverse frequency Ω(r) is an exact Gaussian by construction: B(r) is defined with a factor a(r)^{-1} so that the mass cancels in Ω = sqrt(2Ba), yielding Ω(r) = Ω_0 exp(...). The paper then uses the interior maximum of Ω(r) to place the entropic bottleneck. This is transparent design, not circularity: the paper explicitly states the model is 'construct[ed] by hand to have a prescribed property rather than fit to a real molecule.' The dynamical consequences (gating, outward displacement) are computed from trajectory integration, not assumed from the Gaussian form. The construction is an input, and the output is an independent computation.

full rationale

The paper is largely self-contained. The designed Hamiltonian (Sec. 3) is an explicit construction, not a fit to data, and the paper is transparent that Ω(r) is Gaussian by design. The central dynamical results—inner-capture gating (57.3%→14.7%), outward roaming displacement, the tight-orbit/flux-minimum coincidence (verified numerically to 0.15%, Sec. 8), and non-exponential gap times—are all computed from trajectory integration (57,600 trajectories, Appendix A) and flux evaluation (Eq. 11, Sec. 5), not restated from cited work. The self-citations to Mauguière et al. [22], Krajňák and Waalkens [27, 28], and Ezra, Waalkens, and Wiggins [16] adopt the phase-space classification framework and gap-time diagnostic, but these are methodological tools applied to a new model, not load-bearing premises whose validity is assumed. The strength-matched monotone controls (Sec. 4, qualification ii) are an honest internal check showing that amplitude alone accounts for most of the gating, with the localized profile contributing a residual 3.1 pp—this is the opposite of circular, as it explicitly separates what the design contributes from what mere amplitude does. The only minor concern is that the classification scheme and nonstatisticality finding are borrowed from the author's own prior work, but this is standard methodological continuity, not a circular derivation. Score 2: one minor self-citation that is not load-bearing for the central new results.

Axiom & Free-Parameter Ledger

6 free parameters · 4 axioms · 1 invented entities

The free parameters are design choices for a model Hamiltonian, not fits to experimental data. The paper is transparent about this: the model is 'designed' to isolate the effect of a transverse-stiffness maximum. The axioms are standard results from the transition-state theory literature. The single invented entity (the ridge) is a model construction whose consequences are computed, not postulated.

free parameters (6)
  • Ω₀² = 200 kcal²/(mol²·u·Å²)
    Ridge peak frequency squared; chosen by hand to create a deep bottleneck. Not fitted to data but designed to produce a prescribed property.
  • r_c = 2.2 Å
    Ridge center; chosen by hand to place the bottleneck at an interior radius.
  • σ = 0.5 Å
    Ridge width; chosen by hand to localize the transverse stiffness.
  • E = 2 kcal/mol
    Roaming energy; fixed just above dissociation threshold.
  • r_class = 3.5 Å
    Classifier radius for roaming; chosen as representative of a broad maximum in the roaming fraction vs radius.
  • D_e, r_e, c_1, c_2, V_e, α, μ, I_CH3 = see Table 1
    Chesnavich model parameters; phenomenological, not fitted to data, inherited from prior literature [20, 22].
axioms (4)
  • standard math Periodic orbits in 2-DoF Hamiltonian systems are the transition states; their dividing surfaces are nonrecrossing locally.
    Invoked in Sec. 1 and Secs. 5-8; standard result from Pechukas-Pollak and Waalkens-Schubert-Wiggins [13, 19].
  • standard math The variational minimum-flux dividing surface is bounded by an unstable periodic orbit.
    Invoked in Sec. 8 to identify the tight orbit with the flux minimum; standard result from Pollak and Pechukas [13].
  • domain assumption A localized maximum of transverse frequency generates an entropic free-energy barrier via F(x) = V(x) + k_BT ln(ω⊥/ω₀).
    Invoked in Sec. 3 via Eq. (1); from Makarov [9]. The paper notes this is a canonical heuristic and computes the microcanonical counterpart (exact transverse flux) instead.
  • domain assumption The angular motion is fast compared to the radial approach, so averaging over the transverse mode is valid.
    Invoked in Sec. 3 to justify the adiabatic elimination leading to Eq. (9). The paper uses this heuristically and computes the exact flux separately.
invented entities (1)
  • Transverse-stiffness ridge (Gaussian bump in B(r)) independent evidence
    purpose: To create a localized interior maximum of the transverse frequency, producing a deep entropic bottleneck.
    The ridge is a designed model feature, not a postulated physical object. Its effects are verified by trajectory integration and flux computation. The paper explicitly states it is 'constructed by hand to have a prescribed property rather than fit to a real molecule.'

pith-pipeline@v1.1.0-glm · 20621 in / 3229 out tokens · 321516 ms · 2026-07-08T05:39:32.381843+00:00 · methodology

0 comments
read the original abstract

Roaming reactions are organized not by potential-energy saddles but by transition states that are unstable invariant objects in phase space, periodic orbits in the two degrees of freedom studied here. To ask what controls roaming, we modify the Chesnavich model of a barrierless ion--molecule dissociation: its orientation-dependent angular hindrance is replaced by a transverse-stiffness ridge whose angular frequency peaks at an interior radius, and the classical dynamics are studied at a fixed energy just above the dissociation threshold. Comparing two ensembles that differ only in this angular interaction (same radial channel, energy, and inward initial conditions) isolates its effect. The ridge gates entry into the inner well, cutting inner capture from $57\%$ to $15\%$ and returning most of the incoming flux directly to reactants; it does not eliminate roaming but relocates it outward, suppressing it inside the ridge and switching it on farther out. The model retains analogues of the original model's three transition states (tight, free-rotor, and outer orbiting orbits), which we locate as unstable periodic orbits. The tight orbit spans a dividing surface that coincides, within numerical accuracy, with the variational minimum-flux surface, and it carries no barrier along the reaction coordinate: a deep entropic bottleneck placed at an interior radius by the stiffness maximum. Its entropic character is shared with the original model. Strength-matched monotone controls show that the gating tracks the hindrance strength at the bottleneck radius; what the interior maximum supplies is placement, concentrating that strength where it gates most effectively. The trajectories it admits roam nonstatistically, with nonexponential gap-time distributions: the entropic bottleneck governs how much is captured, not the dynamics that follow.

Figures

Figures reproduced from arXiv: 2607.06437 by Stephen Wiggins.

Figure 1
Figure 1. Figure 1: Construction of the designed model. (a) The two barrierless radial channels, VCH(r) and the LJ(8,4) form; neither has a radial barrier along the approach, and both decay as −1/r4 at long range. (b) Angular hindrance amplitude versus r: the monotone Chesnavich lock V0(r) (α = 1) and the entropic ridge B(r), taller and concentrated near rc = 2.2 ˚A. (c) Harmonic transverse frequency: ΩCh(r) decreases monoton… view at source ↗
Figure 2
Figure 2. Figure 2: The controlled comparison at E = 2. The radial channel VCH(r), the energy, and the inward fixed-energy ensemble are identical in the two runs; only the angular term differs, set to the Chesnavich lock (the orientation-locking term of the original model) in one and to the entropic ridge (the localized transverse-stiffness maximum at rc) in the other. (a) Inner-capture outcome: the ensemble partitioned into … view at source ↗
Figure 3
Figure 3. Figure 3: The variational flux bottleneck at E = 2. (a) Exact directional flux ΦR/Φ∞, Eq. (11), for the designed (LJ+ridge) and Chesnavich surfaces; the designed model has a deep, narrow minimum where the Chesnavich surface has only a shallow one. (b) Angular confinement C(R, E), Eq. (12): the designed constriction (C ≃ 0.10) is reproduced by the matched control (ridge on the Chesnavich channel, dashed), well below … view at source ↗
Figure 4
Figure 4. Figure 4: The three transition states of the designed model at E = 2, all unstable periodic orbits, in the configuration plane (x, y) = (r cos θ, r sin θ). (a) All three: the orbiting orbit (circle, rOTS = 7.48 ˚A), the free-rotor orbit (loop, r ∈ [3.5, 3.7] ˚A), and the tight orbit (short arc near the aligned axis, r ∗ ≃ 2.46 ˚A; its θ = π mirror is shown faintly). (b) The inner region: the tight orbit is a librati… view at source ↗
Figure 5
Figure 5. Figure 5: The outer orbiting transition state of the designed model (LJ(8,4) channel). Outer-region effective radial potential Veff(r;L) at E = 2: at L = 3.047 the orbiting transition state is the unstable centrifugal barrier maximum at rOTS = 7.48 ˚A. The ridge is negligible in this region, so the orbiting transition state is the standard centrifugal (Langevin) barrier. with Lagrangian descriptors, computationally … view at source ↗
Figure 6
Figure 6. Figure 6: Roaming on the designed surface at E = 2 (LJ(8,4) channel; 240 × 240 inward grid at r0 = 6 ˚A). (a) Launch grid (θ0, pθ,0) colored by trajectory class: reactive trajectories form centrifugal-trapping bands, roaming sets are thin, and most trajectories are direct and nonreactive. The classification is that of the non-absorbing run, in which a small roaming-reactive class appears. (b) Roaming fraction versus… view at source ↗
Figure 7
Figure 7. Figure 7: Nonreactive gap-time survival (r0 = 6 ˚A launch, conditional on inward entry). (a) Chesnavich lock and entropic ridge; dotted curves are matched single exponentials. Both distributions are narrower than exponential, with a delayed onset set by the minimum transit time. (b) The ridge ensemble resolved into direct and roaming-classified returns: the two populations are individually narrow (coefficient of var… view at source ↗

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Reference graph

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