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Stephen Wiggins

Publications and source records attributed to Stephen Wiggins.

At least 19 recordsLinked to original sources
arXiv2026

Rare Fluctuations from Normally Hyperbolic Invariant Manifolds

Freidlin--Wentzell theory converts weak-noise large deviations into a Hamiltonian variational problem. We study how a $k$-dimensional normally hyperbolic invariant manifold (NHIM) $N$ of the deterministic dynamics appears in this Hamiltonian system. Its zero-momentum copy $N_0=N\times\{0\}$ is invariant, but the Hamiltonian dynamics has $2k$ center directions near $N_0$: $k$ tangent to $N$ and $k$ conjugate covector directions. We construct the resulting local symplectic geometry and show that fluctuation extremals approaching $N_0$ backward in time at the strong normal rate form an $n$-dimensional exact Lagrangian invariant manifold carrying a single-valued action. A counterexample shows that a corresponding zero-energy section need not be normally hyperbolic within $H_{\mathrm{FW}}^{-1}(0)$. We then consider reaction dynamics. If a parameter moves a deterministic trajectory toward a codimension-one reactivity boundary, the minimum Freidlin--Wentzell action required to reach the boundary is quadratic in the distance from the threshold parameter. Its coefficient is determined by the relative motion of trajectory and boundary and by how effectively the available noise acts transversely. This deterministic boundary is distinct from a noise-dependent stochastic transition state or a committor surface. In a solvent--solute model, varying solvent mass moves the phase-space reactivity boundary while leaving the potential-energy surface fixed, changing the rare-event cost without changing the potential-energy barrier.

physics.chem-ph↗
arXiv2026

What solvable models reveal about Born-Oppenheimer, Born-Huang, and exact factorization

The Born-Oppenheimer (BO) and single-surface Born-Huang (BH) approximations replace the full molecular problem by nuclear motion associated with one clamped electronic state, whereas the full BH expansion and exact factorization (EF) are exact representations. We use two analytically transparent models to identify what controls one-surface error. For Fern\'andez's bilinearly coupled oscillators, the Brattsev ground-state bracket was already known; we derive closed-form differences proving strictness for every positive nuclear mass and nonzero stable coupling. Using Hunter's conditional-amplitude construction, we place the closed-form ground-state EF potential alongside BO and single-surface BH. Exact excited-state calculations show that the bracket does not extend through the spectrum. The diagonal BH correction is the quantum-metric coefficient of the retained electronic state divided by twice the nuclear mass. Fern\'andez's sixth-order expansion shows that the next omitted correction contains the same derivative couplings, an additional inverse electronic-energy separation, and a factor depending on the nuclear state. We therefore add a two-state model with a position-dependent gap. Decreasing its minimum separation makes the metric higher and narrower while leaving the total Fubini-Study length equal to pi/2. The exact EF, BO, and BH potentials can differ visibly even when their nuclear ground-state amplitudes are nearly identical. The two models show that one-surface accuracy depends on nuclear mass, electronic-state variation, energy separation from omitted states, and the nuclear wavefunction.

quant-ph↗
arXiv2026

A Three-Degree-of-Freedom Chesnavich Model for Roaming: Derivation, Phase-Space Geometry, NHIM-Anchored Dividing Surfaces, and Roaming Transport

Roaming reactions, in which a dissociating fragment moves through a flat region of the potential surface rather than down the minimum-energy path, lie outside the assumptions of conventional transition state theory. The phase-space theory of roaming -- unstable periodic orbits and their invariant manifolds organizing transport -- has been developed for the Chesnavich model of $\mathrm{CH_4^+}\to\mathrm{CH_3^+}+\mathrm{H}$, which is cylindrically symmetric and reduces to two degrees of freedom (2-DoF). We construct and analyze a three-degree-of-freedom (3-DoF) extension. From the rigid-body formulation of Ezra and Wiggins, we break the symmetry with an azimuthal coupling respecting the three-fold ($C_3$) symmetry of the methyl fragment, obtaining a family $H_b$ whose planar reduction at $b=0$ is the 2-DoF model exactly and which is genuinely 3-DoF for $b>0$. This activates the out-of-plane degree of freedom at once: with the physical planar-top inertia ratio $I_z=2I_x$, arbitrarily weak coupling makes the periodic orbit on the roaming shelf transversely unstable, opening an escape route out of the reaction plane. Apart from a narrow elliptic window $0.58\lesssim b\lesssim0.63$, the instability persists across the range studied, changing type through a period-doubling at $b_c\approx0.63$. Because a periodic orbit cannot anchor a dividing surface in three degrees of freedom, we construct the objects that do -- three three-dimensional normally hyperbolic invariant manifolds, one per transition state -- at $b=0$, and prove that every compact interior piece of each persists for sufficiently small $b>0$. At $E=0.5\ \mathrm{kcal\,mol^{-1}}$ the coupling lowers the direct non-reactive fraction of a microcanonical ensemble of incoming trajectories by $0.032$ and raises the two roaming fractions by $0.040$; the effect decreases as the energy increases.

physics.chem-ph↗
arXiv2026

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

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.

physics.chem-ph↗
arXiv2026

Preparation-Space Diagnostics and Logical Information Loss in a Driven Kerr-Cat Qubit

A Kerr-cat qubit encodes a logical bit in the two wells of a parametrically driven nonlinear oscillator, and a logic gate is a transient change of the drive. In the phase plane the gate deforms the double well and can split its separatrix into a turnstile that carries trajectories across the dividing surface between the wells; the same pulse, acting on the quantum oscillator, can corrupt the encoded bit. We study this process over a disk of coherent-state preparations, comparing classical phase-space transport diagnostics with the open-system quantum outcome on a common domain so that the two can be compared point by point. The central finding is that the corruption depends on the full temporal protocol, not on pulse strength alone: a sudden quench erases the bit, whereas a smooth ramp of the same peak amplitude largely preserves it. A finite-time sensitivity field locates the classical transport boundary, and a Loschmidt echo evaluated near the end of the gate predicts the much later quantum outcome. Sweeps of pulse amplitude and width, of cat size, and of engineered two-photon dissipation map where the classical transport picture predicts the quantum loss of the bit and where it does not.

quant-ph↗
arXiv2026

A Phase Space Signature of Quantum Roaming in Chesnavich's Model

Roaming reactions occur when a molecule enters a near-dissociation region, avoids immediate separation, and later forms products by a pathway not controlled by the conventional tight transition-state bottleneck. Classical studies have shown that roaming is best understood in phase space: inner and outer transition-state structures, together with their invariant manifolds, organize trapping, return, and dissociation. The corresponding quantum question is less settled. Can a single quantum resonance carry a recognizable signature of the classical roaming region? We address this question in Chesnavich's two-degree-of-freedom model for the ion--molecule reaction $\mathrm{CH}_4^+\rightarrow\mathrm{CH}_3^+ + \mathrm{H}$. Resonance states are computed with a complex absorbing potential and analyzed using diagnostics designed to mirror the classical phase-space picture: radial probability weights derived from the tight and outer transition-state structures, radial Husimi projections, angular-momentum channel weights, and coherent-state probes of the classical periodic orbits. One resonance is distinguished from the rest of the computed resonance ensemble. Its wavefunction is concentrated in the projected region between the inner and outer transition-state structures, its radial phase-space distribution is centered at intermediate radius with nearly zero radial momentum, and its angular structure is consistent with a standing rather than a directed rotating component. We interpret this state as a phase-space-localized quantum analogue of classical roaming. The result provides a controlled example in which quantum roaming is identified directly from a resonance wavefunction and its phase-space diagnostics, rather than only from product-state or scattering signatures.

physics.chem-ph↗
arXiv2026

Phase Space Bottlenecks in an Adiabatic Marcus Hamiltonian: Cusp Geometry, NHIMs, and Mixed Valence Electron Transfer

Marcus--Hush theory explains electron transfer in terms of reorganization energies, driving forces, electronic couplings, and reduced free-energy or energy-gap descriptions. These descriptions do not by themselves determine when the underlying adiabatic dynamics possesses a genuine phase space transition state. We address this question for a minimal asymmetric two-degree-of-freedom adiabatic Marcus Hamiltonian obtained from two coupled diabatic harmonic surfaces. Passing to the lower adiabatic sheet gives a classical Hamiltonian with one electron-transfer coordinate and one transverse mode. We derive an explicit cusp condition in the plane of dimensionless asymmetry and coupling parameters that is necessary and sufficient for the lower sheet to possess an index-one saddle. This cusp criterion is the Marcus-specific result of the paper: it identifies when the lower adiabatic surface supports a local Hamiltonian bottleneck rather than only an energetic barrier in a reduced-coordinate picture. Inside the cusp, the corresponding Hamiltonian equilibrium is of saddle-centre type, and the standard local phase-space transition-state structures follow: in two degrees of freedom the normally hyperbolic invariant manifold is an unstable periodic orbit, with stable and unstable manifolds and an attached no-recrossing dividing surface. Outside the cusp, this lower-sheet local transition-state structure is absent. The construction provides a Hamiltonian complement to standard adiabatic Marcus theory, clarifies the role of the lower-sheet bottleneck in a minimal mixed valence setting, and separates the conservative adiabatic problem from dissipative solvent theories and nonadiabatic mixed quantum-classical formulations.

physics.chem-ph↗
arXiv2026

Lobe Dynamics, Phase-Space Transport, and Non-Adiabatic Leakage Thresholds in the Nonautonomous Kerr-Cat Qubit

The Kerr-nonlinear parametric oscillator (KPO) provides a foundational semiclassical model for cat-state quantum hardware. Standard analyses of the KPO typically rely on autonomous, frozen-time approximations to describe the stabilization of macroscopic coherent states. However, state preparation and gate manipulation are driven by explicitly time-dependent microwave pulses, so the operational dynamics are inherently nonautonomous. In this paper, we show that static algebraic equilibrium pictures are incomplete for describing both state formation and gate-induced transport in the Kerr-cat qubit. For nonautonomous state preparation, we analyze the ramped resonant model by combining a linear nonautonomous stability analysis with a local invariant-graph reduction near the vacuum trajectory. This yields a quintic reduced normal form in the critical direction and identifies two symmetric post-threshold moving branches that organize the local state-formation dynamics. The associated diagnostics separate the reduced branch dynamics from the full two-dimensional phase-twist relaxation observed in the hardware coordinates. For gate execution, we model a fast pulse as a weak aperiodic perturbation of the conservative resonant figure-eight separatrix and apply Melnikov's method to derive a leading-order transport criterion. In this framework, transient lobe dynamics emerge as a semiclassical mechanism for non-adiabatic leakage, and the resulting amplitude-width threshold curve provides a leading-order geometric indicator for the onset of gate-pulse-induced transport.

quant-ph↗
arXiv2026

A Periodic Orbit Trace Formula for Quantum Scrambling: The Role of the Normally Hyperbolic Invariant Manifold

Out-of-Time-Order Correlators (OTOCs) quantify quantum information scrambling, but their connection to localized phase-space structures, such as chemical transition states, requires formal development. We derive a leading-order semiclassical expansion for the local microcanonical OTOC in systems with an index-1 saddle point, expressing the scrambling rate as a coherent sum over unstable periodic orbits on the Normally Hyperbolic Invariant Manifold (NHIM). Valid in the semiclassical limit and the intermediate-time regime before the Ehrenfest time, our derivation utilizes the Normal Form theory of the transition state, which transforms the Hamiltonian near the saddle into an integrable (though generally non-separable) form dependent on conserved actions. We outline the derivation of the microcanonical trace, the semiclassical propagator for integrable systems, the factorization of the stability matrix, and the Schur complement reduction of the stationary phase approximation. Our result extends periodic-orbit trace methods to scrambling observables, yielding a local instability exponent {\Lambda}(J) governing the leading semiclassical growth window. As a special case, when the observation time coincides with the intrinsic periods of the contributing orbits, the trace sum reduces to an effective 1.5{\Lambda} scaling, resulting from the competition between local hyperbolic growth and wavepacket dilution. This simplified form is conditional; the full expansion retains a coherent sum over orbit periods. Finally, we discuss how the dependence of the instability on transverse actions establishes a theoretical mechanism for mode-selective control of scrambling, and outline a numerical evaluation strategy to test these predictions.

quant-ph↗
arXiv2026

Symplectic Constraints in Classical Reaction Dynamics: From Gromov's Camel to Reaction Rates

We investigate whether ideas from symplectic topology, in particular Gromov's non-squeezing theorem and symplectic capacity, can provide useful geometric insight into classical reaction dynamics near an index-1 saddle. Using Poincar\'e-Birkhoff normal form theory, we describe the phase-space structures that organize transport through the transition-state region, including dividing surfaces, normally hyperbolic invariant manifolds (NHIMs), and the associated bath-action geometry. For quadratic saddle-center and saddle-center-center models, the normal-form geometry identifies natural bath-action area scales associated with the reactive bottleneck. For anharmonic systems (Eckart-Morse and Eckart-Morse-Morse), we formulate corresponding candidate symplectic width scales -- based on transverse bath actions -- using high-order normal forms for bounded local neighborhoods associated with the reaction bottleneck near the saddle. We then present two numerical illustrations: the backward propagation of a locally coupled phase-space ball to examine linear non-squeezing behavior, and a bath-localized ensemble calculation in an anharmonic normal-form model. These computations are consistent with the idea that heavily biasing the initial phase-space distribution of an ensemble toward the high-action boundaries of the bath modes can induce a severe finite-time dynamical delay, influencing reactivity in ways not captured by total phase-space volume or flux alone. The results suggest a new geometric perspective on mode selectivity and reaction bottlenecks, while highlighting open mathematical questions concerning the precise relation between these candidate width scales and genuine symplectic capacities of suitably defined reactive neighborhoods.

math.DS↗
arXiv2026

Symplectic Constraints in Quantum Reaction Dynamics: Squeezed-State Suppression and Candidate Width Scales

Classical reaction dynamics suggests transport through an index-1 saddle is organized not just by flux, but by local symplectic width scales of bounded proxy neighborhoods near the bottleneck. We investigate if a related geometric effect appears in the quantum regime for highly squeezed Gaussian wavepackets. Building on de Gosson's symplectic approach, we analyze how transverse bath-mode squeezing modifies transmission across a quantum normal-form (QNF) bottleneck. To avoid the instability of propagating states with extreme phase-space eccentricity, we use the Weyl-symbol formulation of the QNF. For the quadratic saddle-center model, we derive an exact baseline transmission formula by convolving the bath's squeezed-state number distribution with the 1D Kemble transmission factor. For anharmonic truncated QNF models, we enforce strict algebraic energy conservation and evaluate exact Gaussian expectation-value diagnostics of the Weyl symbol via Wick-Isserlis moment formulas. Results reveal a pronounced squeeze-induced suppression of transmission. As the squeezed state's bath-plane geometric scale grows relative to the classical candidate width, the expected bath action grows rapidly. Consequently, effective reactive energy is strongly depleted, driving transmission into a severely suppressed regime. We interpret this as evidence of a quantum geometric suppression mechanism consistent with the classical candidate symplectic-width picture. While not yet a rigorous quantum non-squeezing theorem, this work provides a concrete framework linking squeezed-state covariance geometry, normal-form action scales, and mode-specific quantum reactivity near an index-1 saddle.

quant-ph↗
arXiv2026

Geometric Diagnostics of Scrambling-Related Sensitivity in a Bohmian Preparation Space

The Out-of-Time-Order Correlator (OTOC) is a standard algebraic diagnostic of quantum information scrambling, but it offers limited direct geometric intuition. In this note, we propose a Bohmian, trajectory-based framework for constructing a geometric diagnostic of scrambling-related sensitivity using Lagrangian Descriptors (LDs). To avoid the uncertainty-principle obstruction to assigning independent initial position and momentum within a single wave function, we evaluate Bohmian dynamics over a two-dimensional preparation space of localized Gaussian wavepackets labeled by their initial center and momentum kick. For the inverted harmonic oscillator, this construction is analytically tractable: the wavepacket-center dynamics and their dependence on preparation parameters can be written explicitly. In particular, away from the equilibrium origin, the exponential growth of the associated preparation-space stability matrix yields an $\mathcal{O}(e^{\omega T})$ bound on the sensitivity of the wavepacket-center LDs, motivating a semiclassical comparison with sensitivity structures associated with OTOC growth. In this sense, the LD provides a geometric indicator of scrambling-related sensitivity. We conclude by discussing how this preparation-space picture suggests a program for future work regarding the distinct microcanonical regimes previously reported for the inverted harmonic oscillator.

quant-ph↗
arXiv2026

Bridging Classical Sensitivity and Quantum Scrambling: A Tutorial on Out-of-Time-Ordered Correlators

In classical dynamical systems, chaotic behavior is often associated with exponential sensitivity to initial conditions together with global phase-space structure. Translating this geometric concept to the strictly linear framework of quantum mechanics presents a conceptual puzzle. The out-of-time-ordered correlator (OTOC) is often motivated as the quantum analogue of the classical butterfly effect, but this slogan can hide important mathematical distinctions. This tutorial bridges the gap between applied mathematics and quantum information by detailing the mathematical machinery of the OTOC. We explore how classical sensitivity translates to operator non-commutativity, why standard two-point correlation functions fail to cleanly detect this sensitivity, and how the delocalization of quantum observables relates to classical notions of mixing. Crucially, we outline what the OTOC can and cannot diagnose, distinguishing between local instability and global chaos. Ultimately, we provide a precise and usable conceptual map, exploring how the Koopman-von Neumann formalism offers a framework to view classical and quantum dynamics through a shared linear perspective.

quant-ph↗
arXiv2025

A Mathematical Framework for Quantifying Nonlinear Uncertainty Propagation in Eddy Identification Criteria

Ocean eddies are swirling mesoscale features that play a fundamental role in oceanic transport and mixing. Eddy identification relies on diagnostic criteria that are inherently nonlinear functions of the flow variables. However, estimating the ocean flow field is subject to uncertainty due to its turbulent nature and the use of sparse and noisy observations. This uncertainty interacts with nonlinear diagnostics, complicating its quantification and limiting the accuracy of eddy identification. In this paper, an analytically tractable mathematical and computational framework for studying eddy identification is developed. It aims to address how uncertainty interacts with the nonlinearity in the eddy diagnostics and how the uncertainty in the eddy diagnostics is reduced when additional information from observations is incorporated. The framework employs a simple stochastic model for the flow field that mimics turbulent dynamics, allowing closed-form solutions for assessing uncertainty in eddy statistics. It also leverages a nonlinear, yet analytically tractable, data assimilation scheme to incorporate observations, facilitating the study of uncertainty reduction in eddy identification, which is quantified rigorously using information theory. Applied to the Okubo-Weiss (OW) parameter, a widely used eddy diagnostic criterion, the framework leads to three key results. First, closed formulae reveal inhomogeneous spatial patterns in the OW uncertainty despite homogeneous flow field uncertainty. Second, it shows a close link between local minima of the OW expectation (eddy centers) and local maxima of its uncertainty. Third, it reveals a practical information barrier: the reduction in uncertainty in diagnostics asymptotically saturates, limiting the benefit of additional observations.

physics.geo-ph↗
arXiv2024

Structured pathways in the turbulence organizing recent oil spill events in the Eastern Mediterranean

The chaotic nature of ocean motion is a major challenge that hinders the discovery of spatio-temporal current routes that govern the transport of material. Certain material, such as oil spills, pose significant environmental threats and these are enhanced by the fact that they evolve in a chaotic sea, in a way which still nowadays is far from being systematically anticipated. Recently such an oil spill event has affected the Mediterranean coast of several Middle Eastern countries. No accidents were reported for these spills previous to their arrival at the coast, and therefore there was no hint of their origin. Modelling such an event, in which uncertainties are increased due to the lack of information on where and when the spills was produced, stretches available technologies to their limits, and requires the use of novel ideas that help to understand the essential features of oil and tar transport by ocean currents. In this regard Lagrangian Coherent Structures enable us to find order within ocean chaos and provide powerful insights into chaotic events and their relationships over different locations and times like the one addressed. Using the observed locations of the oil impacting the coast at specific times, we seek to determine its original location and the time it was released in the open ocean. We have determined both using a combination of earlier satellite observations and computational modelling of the time evolution. The observed agreement between modeled cases and satellite observations highlights the power of these ideas.

physics.ao-ph↗
arXiv2024

Data-Driven Model Identification Using Time Delayed Nonlinear Maps for Systems with Multiple Attractors

This study presents a method, along with its algorithmic and computational framework implementation, and performance verification for dynamical system identification. The approach incorporates insights from phase space structures, such as attractors and their basins. By understanding these structures, we have improved training and testing strategies for operator learning and system identification. Our method uses time delay and non-linear maps rather than embeddings, enabling the assessment of algorithmic accuracy and expressibility, particularly in systems exhibiting multiple attractors. This method, along with its associated algorithm and computational framework, offers broad applicability across various scientific and engineering domains, providing a useful tool for data-driven characterization of systems with complex nonlinear system dynamics.

math.DS↗
arXiv2024

Taming Uncertainty in a Complex World: The Rise of Uncertainty Quantification -- A Tutorial for Beginners

This paper provides a tutorial about uncertainty quantification (UQ) for those who have no background but are interested in learning more in this area. It exploits many very simple examples, which are understandable to undergraduates, to present the ideas of UQ. Topics include characterizing uncertainties using information theory, UQ in linear and nonlinear dynamical systems, UQ via data assimilation, the role of uncertainty in diagnostics, and UQ in advancing efficient modeling. The surprisingly simple examples in each topic explain why and how UQ is essential. Both MATLAB and Python codes are made available for these simple examples.

math.DS↗
arXiv2024

Probabilistic Eddy Identification with Uncertainty Quantification

Mesoscale eddies are critical in ocean circulation and the global climate system. Standard eddy identification methods are usually based on deterministic optimal point estimates of the ocean flow field. However, uncertainty exists in estimating the flow field due to noisy, sparse, and indirect observations and turbulent flow models. Because of the intrinsic strong nonlinearity in the eddy identification diagnostics, even a small uncertainty in estimating the flow field can cause a significant error in the identified eddies. This paper presents a general probabilistic eddy identification framework that adapts existing identification methods to incorporate uncertainty into the diagnostic, emphasizing the interaction between the uncertainty in state estimation and the nonlinearity in diagnostics for affecting the identification results. The probabilistic eddy identification framework starts by sampling an ensemble of flow realizations from the probabilistic state estimation, followed by applying traditional nonlinear eddy diagnostics to individual realizations. The corresponding eddy statistics are then aggregated from the diagnostic results based on these realizations. The framework is applied to a scenario mimicking the Beaufort Gyre marginal ice zone, where large uncertainty appears in estimating the ocean field using Lagrangian data assimilation with sparse ice floe trajectories. The skills in counting the number of eddies and computing the probability of each eddy event are significantly improved under the probabilistic framework. Notably, incorporating the nonlinear propagation of uncertainty in diagnostics provides a more accurate mean estimate than standard deterministic methods in estimating eddy lifetime. It also facilitates uncertainty quantification in inferring such a crucial dynamical quantity.

math.DS↗