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Jeremy Marzuola

Publications and source records attributed to Jeremy Marzuola.

16 recordsLinked to original sources
arXiv2021

Dispersion control in pressure-driven flow through bowed rectangular microchannels

In fully-developed pressure-driven flow, the spreading of a dissolved solute is enhanced in the flow direction due to transverse velocity variations in a phenomenon now commonly referred to as Taylor-Aris dispersion. It is well understood that the characteristics of the dispersion are sensitive to the channel's cross-sectional geometry. Here we demonstrate a method for manipulation of dispersion in a single rectangular microchannel via controlled deformation of its upper wall. Using a rapidly prototyped multi-layer microchip, the channel wall is deformed by a controlled pressure source allowing us to characterize the dependence of the dispersion on the deflection of the channel wall and overall channel aspect ratio. For a given channel aspect ratio, an optimal deformation to minimize dispersion is found, consistent with prior numerical and theoretical predictions. Our experimental measurements are also compared directly to numerical predictions using an idealized geometry.

physics.flu-dyn↗
arXiv2020

Limiting Behaviors of High Dimensional Stochastic Spin Ensembles

Lattice spin models in statistical physics are used to understand magnetism. Their Hamiltonians are a discrete form of a version of a Dirichlet energy, signifying a relationship to the Harmonic map heat flow equation. The Gibbs distribution, defined with this Hamiltonian, is used in the Metropolis-Hastings (M-H) algorithm to generate dynamics tending towards an equilibrium state. In the limiting situation when the inverse temperature is large, we establish the relationship between the discrete M-H dynamics and the continuous Harmonic map heat flow associated with the Hamiltonian. We show the convergence of the M-H dynamics to the Harmonic map heat flow equation in two steps: First, with fixed lattice size and proper choice of proposal size in one M-H step, the M-H dynamics acts as gradient descent and will be shown to converge to a system of Langevin stochastic differential equations (SDE). Second, with proper scaling of the inverse temperature in the Gibbs distribution and taking the lattice size to infinity, it will be shown that this SDE system converges to the deterministic Harmonic map heat flow equation. Our results are not unexpected, but show remarkable connections between the M-H steps and the SDE Stratonovich formulation, as well as reveal trajectory-wise out of equilibrium dynamics to be related to a canonical PDE system with geometric constraints.

math.PR↗
arXiv2018

Existence and uniqueness of solutions for a quasilinear KdV equation with degenerate dispersion

We consider a quasilinear KdV equation that admits compactly supported traveling wave solutions (compactons). This model is one of the most straightforward instances of degenerate dispersion, a phenomenon that appears in a variety of physical settings as diverse as sedimentation, magma dynamics and shallow water waves. We prove the existence and uniqueness of solutions with sufficiently smooth, spatially localized initial data.

math.AP↗
arXiv2012

Nonlinear bound states on weakly homogeneous spaces

We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schrödinger equations. We examine two constrained minimization problems, which give rise to such solutions. One yields what we call $F_λ$-minimizers, the other energy minimizers. We produce such ground state solutions on a class of Riemannian manifolds called weakly homogeneous spaces, and establish smoothness, positivity, and decay properties. We also identify classes of Riemannian manifolds with no such minimizers, and classes for which essential uniqueness of positive solutions to the associated elliptic PDE fails.

math.AP↗
arXiv2011

Bose-Einstein condensation transition studies for atoms confined in Laguerre-Gaussian laser modes

Multiply-connected traps for cold, neutral atoms fix vortex cores of quantum gases. Laguerre-Gaussian laser modes are ideal for such traps due to their phase stability. We report theoretical calculations of the Bose-Einstein condensation transition properties and thermal characteristics of neutral atoms trapped in multiply connected geometries formed by Laguerre-Gaussian LG{p}{l} beams. Specifically, we consider atoms confined to the anti-node of a LG{0}{1} laser mode detuned to the red of an atomic resonance frequency, and those confined in the node of a blue-detuned LG{1}{1} beam. We compare the results of using the full potential to those approximating the potential minimum with a simple harmonic oscillator potential. We find that deviations between calculations of the full potential and the simple harmonic oscillator can be up to 3%-8% for trap parameters consistent with typical experiments.

physics.atom-ph↗
arXiv2010

Wave packet parametrices for evolutions governed by PDO's with rough symbols

In this note, we consider wave packet parametrices for Schrodinger-like evolution equations. Under an integrability condition along the flow, we prove that the flow is then globally well-defined and bilipschitz. Under an additional smallness assumption, we prove that the kernel of the phase space operator decays rapidly away from the graph of the Hamilton flow. This can be used to prove that smoothness is preserved for finite times by the flow of the equation.

math.AP↗
arXiv2009

Strichartz estimates on Schwarzschild black hole backgrounds

We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of earlier work of Metcalfe-Tataru, in order to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere.

math.AP↗
arXiv2009

A class of stable perturbations for a minimal mass soliton in three dimensional saturated nonlinear Schrödinger equations

In this result, we develop the techniques of \cite{KS1} and \cite{BW} in order to determine a class of stable perturbations for a minimal mass soliton solution of a saturated, focusing nonlinear Schrödinger equation {c} i u_t + Δu + β(|u|^2) u = 0 u(0,x) = u_0 (x), in $\reals^3$. By projecting into a subspace of the continuous spectrum of $\mathcal{H}$ as in \cite{S1}, \cite{KS1}, we are able to use a contraction mapping similar to that from \cite{BW} in order to show that there exist solutions of the form e^{i λ_{\min} t} (R_{min} + e^{i \mathcal{H} t} ϕ+ w(x,t)), where $e^{i \mathcal{H} t} ϕ+ w(x,t)$ disperses as $t \to \infty$. Hence, we have long time persistance of a soliton of minimal mass despite the fact that these solutions are shown to be nonlinearly unstable in \cite{CP1}.

math.AP↗
arXiv2009

Dispersive estimates using scattering theory for matrix Hamiltonian equations

We develop the techniques of \cite{KS1} and \cite{ES1} in order to derive dispersive estimates for a matrix Hamiltonian equation defined by linearizing about a minimal mass soliton solution of a saturated, focussing nonlinear Schrödinger equation {c} i u_t + Δu + β(|u|^2) u = 0 u(0,x) = u_0 (x), in $\reals^3$. These results have been seen before, though we present a new approach using scattering theory techniques. In further works, we will numerically and analytically study the existence of a minimal mass soliton, as well as the spectral assumptions made in the analysis presented here.

math.AP↗
arXiv2008

Eigenfunction concentration for polygonal billiards

In this note, we extend the results on eigenfunction concentration in billiards as proved by the third author in \cite{M1}. There, the methods developed in Burq-Zworski \cite{BZ3} to study eigenfunctions for billiards which have rectangular components were applied. Here we take an arbitrary polygonal billiard $B$ and show that eigenfunction mass cannot concentrate away from the vertices; in other words, given any neighbourhood $U$ of the vertices, there is a lower bound $$ \int_U |u|^2 \geq c \int_B |u|^2 $$ for some $c = c(U) > 0$ and any eigenfunction $u$.

math.AP↗
arXiv2008

Counting numerical sets with no small atoms

A numerical set $S$ with Frobenius number $g$ is a set of integers with $\min(S) = 0$ and $\max(\Zbb - S)=g$, and its atom monoid is $A(S) = \setpres{n \in \Zbb}{$n+s \in S$ for all $s \in S$}$. Let $γ_g$ be the number of numerical sets $S$ having $A(S) = \set{0} \cup (g,\infty)$ divided by the total number of numerical sets with Frobenius number $g$. We show that the sequence $\set{γ_g}$ is decreasing and converges to a number $γ_\infty \approx .4844$ (with accuracy to within $.0050$). We also examine the singularities of the generating function for $\set{γ_g}$. Parallel results are obtained for the ratio $\gsymm{g}$ of the number of symmetric numerical sets $S$ with $A(S) = \set{0} \cup (g,\infty)$ by the number of symmetric numerical sets with Frobenius number $g$. These results yield information regarding the asymptotic behavior of the number of finite additive 2-bases.

math.CO↗
arXiv2007

Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations

In this article we study global-in-time Strichartz estimates for the Schrödinger evolution corresponding to long-range perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. In the aforementioned paper, the third author proved the local smoothing estimates for small perturbations of the Laplacian. Here we consider the case of large perturbations in three increasingly favorable scenarios: (i) without non-trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term, (ii) with non-trapping assumptions we prove global estimates modulo a lower order spatially localized error term, and (iii) for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum.

math.AP↗
arXiv2006

Soliton splitting by external delta potentials

We show that a soliton scattered by an external delta potential splits into two solitons and a radiation term. Theoretical analysis gives the amplitudes and phases of the reflected and transmitted solitons with errors going to zero as the velocity of the incoming soliton tends to infinity. Numerical analysis shows that this asymptotic relation is valid for all but very slow solitons. We also show that the total transmitted mass, that is the square of the $L^2$ norm of the solution restricted on the transmitted side of the delta potential is in good agreement with the quantum transmission rate of the delta potential. This paper is a numerical companion to our analytical paper on the same topic, "Fast soliton scattering by delta impurities," math.AP/0602187.

math.AP↗
arXiv2006

Fast soliton scattering by delta impurities

We study the Gross-Pitaevskii equation (nonlinear Schroedinger equation) with a repulsive delta function potential. We show that a high velocity incoming soliton is split into a transmitted component and a reflected component. The transmitted mass (L^2 norm squared) is shown to be in good agreement with the quantum transmission rate of the delta function potential. We further show that the transmitted and reflected components resolve into solitons plus dispersive radiation, and quantify the mass and phase of these solitons.

math.AP↗
arXiv2005

Nonconcentration of eigenfunctions for partially rectangular billiards

In this paper, the results of Burq and Zworski are further developed to study nonconcentration of eigenfunctions for billiards which have rectangular components: these include the Buminovich billiard, the Sinai billiard, and certain pseudointegrable billiards. The results presented are an application of using a "black box" point of view as presented by the same authors.

math.AP↗