Arkadas Ozakin

A geometric theory of thermal stresses

In this paper we formulate a geometric theory of thermal stresses. Given a temperature distribution, we associate a Riemannian material manifold to the body, with a metric that explicitly depends on the temperature distribution. A change in temperature corresponds to a change in the material metric. In this sense, a temperature change is a concrete example of the so-called referential evolutions. We also make a concrete connection between our geometric point of view and the multiplicative decomposition of deformation gradient into thermal and elastic parts. We study the stress-free temperature distributions of the finite-deformation theory using curvature tensor of the material manifold. We find the zero-stress temperature distributions in nonlinear elasticity. Given an equilibrium configuration, we show that a change in the material manifold, i.e., a change in the material metric will change the equilibrium configuration. In the case of a temperature change, this means that given an equilibrium configura...

Controlling trapping potentials and stray electric fields in a microfabricated ion trap through design and compensation

Recent advances in quantum information processing with trapped ions have demonstrated the need for new ion trap architectures capable of holding and manipulating chains of many (>10) ions. Here we present the design and detailed characterization of a new linear trap, microfabricated with scalable complementary metal-oxide-semiconductor (CMOS) techniques, that is well-suited to this challenge. Forty-four individually controlled dc electrodes provide the many degrees of freedom required to construct anharmonic potential wells, shuttle ions, merge and split ion chains, precisely tune secular mode frequencies, and adjust the orientation of trap axes. Microfabricated capacitors on dc electrodes suppress radio-frequency pickup and excess micromotion, while a top-level ground layer simplifies modeling of electric fields and protects trap structures underneath. A localized aperture in the substrate provides access to the trapping region from an oven below, permitting deterministic loading of particular isotopic/elemental sequences via species-selective photoionization. The shapes of the aperture and radio-frequency electrodes are optimized to minimize perturbation of the trapping pseudopotential. Laboratory experiments verify simulated potentials and characterize trapping lifetimes, stray electric fields, and ion heating rates, while measurement and cancellation of spatially-varying stray electric fields permits the formation of nearly-equally spaced ion chains.

Affine development of closed curves in Weitzenböck manifolds and the Burgers vector of dislocation mechanics

In the theory of dislocations, the Burgers vector is usually defined by referring to a crystal structure. Using the notion of affine development of curves on a differential manifold with a connection, we give a differential geometric definition of the Burgers vector directly in the continuum setting, without making use of an underlying crystal structure. As opposed to some other approaches to the continuum definition of the Burgers vector, our definition is completely geometric, in the sense that it involves no ambiguous operations such as the integration of a vector field: when we integrate a vector field, it is a vector field living in the tangent space at a given point in the manifold. For a body with distributed dislocations, the material manifold, which describes the geometry of the stress-free state of the body, is commonly taken to be a Weitzenböck manifold, i.e. a manifold with a metric-compatible, flat connection with torsion. We show that for such a manifold, the density of the Burgers vector calculated according to our definition reproduces the commonly stated relation between the density of dislocations and the torsion tensor.

Stability analysis of ion motion in asymmetric planar ion traps

Motivated by recent developments in ion trap design and fabrication, we investigate the stability of ion motion in asymmetrical, planar versions of the classic Paul trap. The equations of motion of an ion in such a trap are generally coupled due to a nonzero relative angle θ between the principal axes of RF and DC fields, invalidating the assumptions behind the standard stability analysis for symmetric Paul traps. We obtain stability diagrams for the coupled system for various values of θ, generalizing the standard q-a stability diagrams. We use multi-scale perturbation theory to obtain approximate formulas for the boundaries of the primary stability region and obtain some of the stability boundaries independently by using the method of infinite determinants. We cross-check the consistency of the results of these methods. Our results show that while the primary stability region is quite robust to changes in θ, a secondary stability region is highly variable, joining the primary stability region at the special case of θ = 45◦, which results in a significantly enlarged stability region for this particular angle. We conclude that while the stability diagrams for classical, symmetric Paul traps are not entirely accurate for asymmetric surface traps (or for other types of traps with a relative angle between the RF and DC axes), they are “safe” in the sense that operating conditions deemed stable according to standard stability plots are in fact stable for asymmetric traps, as well. By ignoring the coupling in the equations, one only underestimates the size of the primary stability region.Motivated by recent developments in ion trap design and fabrication, we investigate the stability of the motion of an ion in asymmetrical, planar versions of the linear Paul trap. The equations of motion of an ion in such a trap are generally coupled due to a nonzero relative angle θ between the principal axes of RF and DC fields, invalidating the assumptions behind the standard stability analysis for symmetric Paul traps. Using numerical methods, we obtain stability diagrams for the coupled system for various values of θ, generalizing the standard q-a stability diagrams. We then use multi-scale perturbation theory to obtain approximate formulas for the boundaries of the primary stability region and check our formulas against results from numerical analysis. Our results show that while the primary stability region is quite robust to changes in θ, a secondary stability region is highly variable, joining the primary stability region at the special case of θ=45°, which results in a significantly enlarged sta...

Nonlinear Elasticity in a Deforming Ambient Space

In this paper, we formulate a nonlinear elasticity theory in which the ambient space is evolving. For a continuum moving in an evolving ambient space, we model time dependency of the metric by a time-dependent embedding of the ambient space in a larger manifold with a fixed background metric. We derive both the tangential and the normal governing equations. We then reduce the standard energy balance written in the larger ambient space to that in the evolving ambient space. We consider quasi-static deformations of the ambient space and show that a quasi-static deformation of the ambient space results in stresses, in general. We linearize the nonlinear theory about a reference motion and show that variation of the spatial metric corresponds to an effective field of body forces.

Multibody multipole methods

A three-body potential function can account for interactions among triples of particles which are uncaptured by pairwise interaction functions such as Coulombic or Lennard-Jones potentials. Likewise, a multibody potential of order n can account for interactions among n-tuples of particles uncaptured by interaction functions of lower orders. To date, the computation of multibody potential functions for a large number of particles has not been possible due to its O ( N n ) scaling cost. In this paper we describe a fast tree-code for efficiently approximating multibody potentials that can be factorized as products of functions of pairwise distances. For the first time, we show how to derive a Barnes-Hut type algorithm for handling interactions among more than two particles. Our algorithm uses two approximation schemes: (1) a deterministic series expansion-based method; (2) a Monte Carlo-based approximation based on the central limit theorem. Our approach guarantees a user-specified bound on the absolute or relative error in the computed potential with an asymptotic probability guarantee. We provide speedup results on a three-body dispersion potential, the Axilrod-Teller potential.

Learning Protein Folding Energy Functions

A critical open problem in \emph{ab initio} protein folding is protein energy function design, which pertains to defining the energy of protein conformations in a way that makes folding most efficient and reliable. In this paper, we address this issue as a weight optimization problem and utilize a machine learning approach, learning-to-rank, to solve this problem. We investigate the ranking-via-classification approach, especially the Ranking SVM method and compare it with the state-of-the-art approach to the problem using the MINUIT optimization package. To maintain the physicality of the results, we impose non-negativity constraints on the weights. For this we develop two efficient non-negative support vector machine (NNSVM) methods, derived from L2-norm SVM and L1-norm SVMs, respectively. We demonstrate an energy function which maintains the correct ordering with respect to structure dissimilarity to the native state more often, is more efficient and reliable for learning on large protein sets, and is qualitatively superior to the current state-of-the-art energy function.