arXiv subjects

Kevin A. Mitchell

Publications and source records attributed to Kevin A. Mitchell.

28 records · Page 2Linked to original sources

arXiv2015

Frozen reaction fronts in steady flows: a burning-invariant-manifold perspective

The dynamics of fronts, such as chemical reaction fronts, propagating in two-dimensional fluid flows can be remarkably rich and varied. For time-invariant flows, the front dynamics may simplify, settling in to a steady state in which the reacted domain is static, and the front appears "frozen". Our central result is that these frozen fronts in the two-dimensional fluid are composed of segments of burning invariant manifolds---invariant manifolds of front-element dynamics in $xyθ$-space, where $θ$ is the front orientation. Burning invariant manifolds (BIMs) have been identified previously as important local barriers to front propagation in fluid flows. The relevance of BIMs for frozen fronts rests in their ability, under appropriate conditions, to form global barriers, separating reacted domains from nonreacted domains for all time. The second main result of this paper is an understanding of bifurcations that lead from a nonfrozen state to a frozen state, as well as bifurcations that change the topological structure of the frozen front. Though the primary results of this study apply to general fluid flows, our analysis focuses on a chain of vortices in a channel flow with an imposed wind. For this system, we present both experimental and numerical studies that support the theoretical analysis developed here.

nlin.PS↗

arXiv2013

A turnstile mechanism for fronts propagating in fluid flows

We consider the propagation of fronts in a periodically driven flowing medium. It is shown that the progress of fronts in these systems may be mediated by a turnstile mechanism akin to that found in chaotic advection. We first define the modified ("active") turnstile lobes according to the evolution of point sources across a transport boundary. We then show that the lobe boundaries may be constructed from stable and unstable \emph{burning invariant manifolds}---one-way barriers to front propagation analogous to traditional invariant manifolds for passive advection. Because the burning invariant manifolds (BIMs) are one-dimensional curves in a three-dimensional ($xyθ$) phase space, their projection into $xy$-space exhibits several key differences from their advective counterparts: (lobe) areas are not preserved, BIMs may self-intersect, and an intersection between stable and unstable BIMs does not map to another such intersection. These differences must be accommodated in the correct construction of the new turnstile. As an application, we consider a lobe-based treatment protocol for protecting an ocean bay from an invading algae bloom.

nlin.CD↗

arXiv2012

Invariant manifolds and the geometry of front propagation in fluid flows

Recent theoretical and experimental work has demonstrated the existence of one-sided, invariant barriers to the propagation of reaction-diffusion fronts in quasi-two-dimensional periodically-driven fluid flows. These barriers were called burning invariant manifolds (BIMs). We provide a detailed theoretical analysis of BIMs, providing criteria for their existence, a classification of their stability, a formalization of their barrier property, and mechanisms by which the barriers can be circumvented. This analysis assumes the sharp front limit and negligible feedback of the front on the fluid velocity. A low-dimensional dynamical systems analysis provides the core of our results.

nlin.CD↗

arXiv2012

Scattering by an oscillating barrier: quantum, classical, and semiclassical comparison

We present a detailed study of scattering by an amplitude-modulated potential barrier using three distinct physical frameworks: quantum, classical, and semiclassical. Classical physics gives bounds on the energy and momentum of the scattered particle, while also providing the foundation for semiclassical theory. We use the semiclassical approach to selectively add quantum-mechanical effects such as interference and diffraction. We find good agreement between the quantum and semiclassical momentum distributions. Our methods and results can be used to understand quantum and classical aspects of transport mechanisms involving time-varying potentials, such as quantum pumping.

cond-mat.quant-gas↗

arXiv2011

Demonstration of Turnstiles as a Chaotic Ionization Mechanism in Rydberg Atoms

We present an experimental and theoretical study of the chaotic ionization of quasi-one-dimensional potassium Rydberg wavepackets via a phase-space turnstile mechanism. Turnstiles form a general transport mechanism for numerous chaotic systems, and this study explicitly illuminates their relevance to atomic ionization. We create time-dependent Rydberg wavepackets, subject them to alternating applied electric-field "kicks", and measure the electron survival probability. Ionization depends not only on the initial electron energy, but also on the phase-space position of the electron with respect to the turnstile --- that part of the electron packet inside the turnstile ionizes after the applied ionization sequence, while that part outside the turnstile does not. The survival data thus encode information on the geometry and location of the turnstile, and are in good agreement with theoretical predictions.

physics.atom-ph↗

arXiv2006

Fractal templates in the escape dynamics of trapped ultracold atoms

We consider the dynamic escape of a small packet of ultracold atoms launched from within an optical dipole trap. Based on a theoretical analysis of the underlying nonlinear dynamics, we predict that fractal behavior can be seen in the escape data. This data would be collected by measuring the time-dependent escape rate for packets launched over a range of angles. This fractal pattern is particularly well resolved below the Bose-Einstein transition temperature--a direct result of the extreme phase space localization of the condensate. We predict that several self-similar layers of this novel fractal should be measurable and we explain how this fractal pattern can be predicted and analyzed with recently developed techniques in symbolic dynamics.

physics.atom-ph↗

arXiv2000

Geometric Phase, Curvature, and Extrapotentials in Constrained Quantum Systems

We derive an effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) by an infinite restoring force. We pay special attention to how this Hamiltonian depends on quantities which are external to the constraint manifold, such as the external curvature of the constraint manifold, the (Riemannian) curvature of the ambient space, and the constraining potential. In particular, we find the remarkable fact that the twisting of the constraining potential appears as a gauge potential in the constrained Hamiltonian. This gauge potential is an example of geometric phase, closely related to that originally discussed by Berry. The constrained Hamiltonian also contains an effective potential depending on the external curvature of the constraint manifold, the curvature of the ambient space, and the twisting of the constraining potential. The general nature of our analysis allows applications to a wide variety of problems, such as rigid molecules, the evolution of molecular systems along reaction paths, and quantum strip waveguides.

quant-ph↗

arXiv1999

Kinematic Orbits and the Structure of the Internal Space for Systems of Five or More Bodies

The internal space for a molecule, atom, or other n-body system can be conveniently parameterised by 3n-9 kinematic angles and three kinematic invariants. For a fixed set of kinematic invariants, the kinematic angles parameterise a subspace, called a kinematic orbit, of the n-body internal space. Building on an earlier analysis of the three- and four-body problems, we derive the form of these kinematic orbits (that is, their topology) for the general n-body problem. The case n=5 is studied in detail, along with the previously studied cases n=3,4.

physics.chem-ph↗

arXiv1999

Boundary Conditions on Internal Three-Body Wave Functions

For a three-body system, a quantum wave function $Ψ^\ell_m$ with definite $\ell$ and $m$ quantum numbers may be expressed in terms of an internal wave function $χ^\ell_k$ which is a function of three internal coordinates. This article provides necessary and sufficient constraints on $χ^\ell_k$ to ensure that the external wave function $Ψ^\ell_m$ is analytic. These constraints effectively amount to boundary conditions on $χ^\ell_k$ and its derivatives at the boundary of the internal space. Such conditions find similarities in the (planar) two-body problem where the wave function (to lowest order) has the form $r^{|m|}$ at the origin. We expect the boundary conditions to prove useful for constructing singularity free three-body basis sets for the case of nonvanishing angular momentum.

physics.chem-ph↗

arXiv1998

The rovibrational kinetic energy for complexes of rigid molecules

The rovibrational kinetic energy for an arbitrary number of rigid molecules is computed. The result has the same general form as the kinetic energy in the molecular rovibrational Hamiltonian, although certain quantities are augmented to account for the rotational energy of the monomers. No specific choices of internal coordinates or body frame are made in order to accommodate the large variety of such conventions. However, special attention is paid to how key quantities transform when these conventions are changed. An example system is explicitly analysed as an illustration of the formalism.

physics.atm-clus↗