Exact Equal Time Statistics of Orszag-McLaughlin Dynamics By The Hopf Characteristic Functional Approach

By employing Hopf's functional method, we find the exact characteristic functional for a simple nonlinear dynamical system introduced by Orszag. Steady-state equal-time statistics thus obtained are compared to direct numerical simulation. The solution is both non-trivial and strongly non-Gaussian.Comment: 6 pages and 2 figure

Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Benard convection

An alternative computational procedure for numerically solving a class of variational problems arising from rigorous upper-bound analysis of forced-dissipative infinite-dimensional nonlinear dynamical systems, including the Navier-Stokes and Oberbeck-Boussinesq equations, is analyzed and applied to Rayleigh-Benard convection. A proof that the only steady state to which this numerical algorithm can converge is the required global optimal of the relevant variational problem is given for three canonical flow configurations. In contrast with most other numerical schemes for computing the optimal bounds on transported quantities (e.g., heat or momentum) within the "background field" variational framework, which employ variants of Newton's method and hence require very accurate initial iterates, the new computational method is easy to implement and, crucially, does not require numerical continuation. The algorithm is used to determine the optimal background-method bound on the heat transport enhancement factor, i.e., the Nusselt number (Nu), as a function of the Rayleigh number (Ra), Prandtl number (Pr), and domain aspect ratio L in two-dimensional Rayleigh-Benard convection between stress-free isothermal boundaries (Rayleigh's original 1916 model of convection). The result of the computation is significant because analyses, laboratory experiments, and numerical simulations have suggested a range of exponents alpha and beta in the presumed Nu similar to (PrRa beta)-Ra-alpha scaling relation. The computations clearly show that for Ra <= 10(10) at fixed L = 2 root 2, Nu <= 0.106Pr(0)Ra(5/12), which indicates that molecular transport cannot generally be neglected in the "ultimate" high-Ra regime.NSF DMS-0928098 DMS-1515161 DMS-0927587 PHY-1205219Simons FoundationNSFONRInstitute for Computational Engineering and Sciences (ICES

Singularities and the distribution of density in the Burgers/adhesion model

We are interested in the tail behavior of the pdf of mass density within the one and $d$-dimensional Burgers/adhesion model used, e.g., to model the formation of large-scale structures in the Universe after baryon-photon decoupling. We show that large densities are localized near ``kurtoparabolic'' singularities residing on space-time manifolds of codimension two ($d \le 2$) or higher ($d \ge 3$). For smooth initial conditions, such singularities are obtained from the convex hull of the Lagrangian potential (the initial velocity potential minus a parabolic term). The singularities contribute {\em \hbox{universal} power-law tails} to the density pdf when the initial conditions are random. In one dimension the singularities are preshocks (nascent shocks), whereas in two and three dimensions they persist in time and correspond to boundaries of shocks; in all cases the corresponding density pdf has the exponent -7/2, originally proposed by E, Khanin, Mazel and Sinai (1997 Phys. Rev. Lett. 78, 1904) for the pdf of velocity gradients in one-dimensional forced Burgers turbulence. We also briefly consider models permitting particle crossings and thus multi-stream solutions, such as the Zel'dovich approximation and the (Jeans)--Vlasov--Poisson equation with single-stream initial data: they have singularities of codimension one, yielding power-law tails with exponent -3.Comment: LATEX 11 pages, 6 figures, revised; Physica D, in pres

Single ion implantation for single donor devices using Geiger mode detectors

Electronic devices that are designed to use the properties of single atoms such as donors or defects have become a reality with recent demonstrations of donor spectroscopy, single photon emission sources, and magnetic imaging using defect centers in diamond. Improving single ion detector sensitivity is linked to improving control over the straggle of the ion as well as providing more flexibility in lay-out integration with the active region of the single donor device construction zone by allowing ion sensing at potentially greater distances. Using a remotely located passively gated single ion Geiger mode avalanche diode (SIGMA) detector we have demonstrated 100% detection efficiency at a distance of >75 um from the center of the collecting junction. This detection efficiency is achieved with sensitivity to ~600 or fewer electron-hole pairs produced by the implanted ion. Ion detectors with this sensitivity and integrated with a thin dielectric, for example 5 nm gate oxide, using low energy Sb implantation would have an end of range straggle of <2.5 nm. Significant reduction in false count probability is achieved by modifying the ion beam set-up to allow for cryogenic operation of the SIGMA detector. Using a detection window of 230 ns at 1 Hz, the probability of a false count was measured as 1E-1 and 1E-4 for operation temperatures of 300K and 77K, respectively. Low temperature operation and reduced false, dark, counts are critical to achieving high confidence in single ion arrival. For the device performance in this work, the confidence is calculated as a probability of >98% for counting one and only one ion for a false count probability of 1E-4 at an average ion number per gated window of 0.015.Comment: 10 pages, 5 figures, submitted to Nanotechnolog

The Gap-Tooth Method in Particle Simulations

We explore the gap-tooth method for multiscale modeling of systems represented by microscopic physics-based simulators, when coarse-grained evolution equations are not available in closed form. A biased random walk particle simulation, motivated by the viscous Burgers equation, serves as an example. We construct macro-to-micro (lifting) and micro-to-macro (restriction) operators, and drive the coarse time-evolution by particle simulations in appropriately coupled microdomains (teeth) separated by large spatial gaps. A macroscopically interpolative mechanism for communication between the teeth at the particle level is introduced. The results demonstrate the feasibility of a closure-on-demand approach to solving hydrodynamics problems

Insecurity for compact surfaces of positive genus

A pair of points in a riemannian manifold $M$ is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in $M$ are secure. A manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. We prove this for surfaces of genus greater than zero. We also prove that a closed surface of genus greater than one with any riemannian metric and a closed surface of genus one with generic metric are totally insecure.Comment: 37 pages, 11 figure

Aspects of the Noisy Burgers Equation

The noisy Burgers equation describing for example the growth of an interface subject to noise is one of the simplest model governing an intrinsically nonequilibrium problem. In one dimension this equation is analyzed by means of the Martin-Siggia-Rose technique. In a canonical formulation the morphology and scaling behavior are accessed by a principle of least action in the weak noise limit. The growth morphology is characterized by a dilute gas of nonlinear soliton modes with gapless dispersion law with exponent z=3/2 and a superposed gas of diffusive modes with a gap. The scaling exponents and a heuristic expression for the scaling function follow from a spectral representation.Comment: 23 pages,LAMUPHYS LaTeX-file (Springer), 13 figures, and 1 table, to appear in the Proceedings of the XI Max Born Symposium on "Anomalous Diffusion: From Basics to Applications", May 20-24, 1998, Ladek Zdroj, Polan

Cohomogeneity one manifolds and selfmaps of nontrivial degree

We construct natural selfmaps of compact cohomgeneity one manifolds with finite Weyl group and compute their degrees and Lefschetz numbers. On manifolds with simple cohomology rings this yields in certain cases relations between the order of the Weyl group and the Euler characteristic of a principal orbit. We apply our construction to the compact Lie group SU(3) where we extend identity and transposition to an infinite family of selfmaps of every odd degree. The compositions of these selfmaps with the power maps realize all possible degrees of selfmaps of SU(3).Comment: v2, v3: minor improvement

Classification of integrable Weingarten surfaces possessing an sl(2)-valued zero curvature representation

In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an sl(2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary differential equation to govern the functional dependence of the principal curvatures. Employing the scaling and translation (offsetting) symmetry, we give a general solution of the governing equation in terms of elliptic integrals. We show that the instances when the elliptic integrals degenerate to elementary functions were known to nineteenth century geometers. Finally, we characterize the associated normal congruences

A remark on an overdetermined problem in Riemannian Geometry

Let $(M,g)$ be a Riemannian manifold with a distinguished point $O$ and assume that the geodesic distance $d$ from $O$ is an isoparametric function. Let $\Omega\subset M$ be a bounded domain, with $O \in \Omega$, and consider the problem $\Delta_p u = -1$ in $\Omega$ with $u=0$ on $\partial \Omega$, where $\Delta_p$ is the $p$-Laplacian of $g$. We prove that if the normal derivative $\partial_{\nu}u$ of $u$ along the boundary of $\Omega$ is a function of $d$ satisfying suitable conditions, then $\Omega$ must be a geodesic ball. In particular, our result applies to open balls of $\mathbb{R}^n$ equipped with a rotationally symmetric metric of the form $g=dt^2+\rho^2(t)\,g_S$, where $g_S$ is the standard metric of the sphere.Comment: 8 pages. This paper has been written for possible publication in a special volume dedicated to the conference "Geometric Properties for Parabolic and Elliptic PDE's. 4th Italian-Japanese Workshop", organized in Palinuro in May 201