Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results

We review some recent attempts to extract information about the nature of quantum gravity, with and without matter, by quantum field theoretical methods. More specifically, we work within a covariant lattice approach where the individual space-time geometries are constructed from fundamental simplicial building blocks, and the path integral over geometries is approximated by summing over a class of piece-wise linear geometries. This method of ``dynamical triangulations'' is very powerful in 2d, where the regularized theory can be solved explicitly, and gives us more insights into the quantum nature of 2d space-time than continuum methods are presently able to provide. It also allows us to establish an explicit relation between the Lorentzian- and Euclidean-signature quantum theories. Analogous regularized gravitational models can be set up in higher dimensions. Some analytic tools exist to study their state sums, but, unlike in 2d, no complete analytic solutions have yet been constructed. However, a great advantage of our approach is the fact that it is well-suited for numerical simulations. In the second part of this review we describe the relevant Monte Carlo techniques, as well as some of the physical results that have been obtained from the simulations of Euclidean gravity. We also explain why the Lorentzian version of dynamical triangulations is a promising candidate for a non-perturbative theory of quantum gravity.Comment: 69 pages, 16 figures, references adde

Novel simulation methods for coulomb and hydrodynamic interactions

Novel simulation methods for Coulomb and hydrodynamic interactions This thesis presents new methods to simulate systems with hydrodynamic and electrostatic interactions. Part 1 is devoted to computer simulations of Brownian particles with hydrodynamic interactions. The main in uence of the solvent on the dynamics of Brownian particles is that it mediates hydrodynamic interactions. In the method, this is simulated by numerical solution of the Navier{Stokes equation on a lattice. To this end, the Lattice{Boltzmann method is used, namely its D3Q19 version. This model is capable to simulate compressible ow. It gives us the advantage to treat dense systems, in particular away from thermal equilibrium. The Lattice{Boltzmann equation is coupled to the particles via a friction force. In addition to this force, acting on point particles, we construct another coupling force, which comes from the pressure tensor. The coupling is purely local, i. e. the algorithm scales linearly with the total number of particles. In order to be able to map the physical properties of the Lattice{Boltzmann uid onto a Molecular Dynamics (MD) uid, the case of an almost incompressible ow is considered. The Fluctuation{Dissipation theorem for the hybrid coupling is analyzed, and a geometric interpretation of the friction coe cient in terms of a Stokes radius is given. Part 2 is devoted to the simulation of charged particles. We present a novel method for obtaining Coulomb interactions as the potential of mean force between charges which are dynamically coupled to a local electromagnetic eld. This algorithm scales linearly, too. We focus on the Molecular Dynamics version of the method and show that it is intimately related to the Car{Parrinello approach, while being equivalent to solving Maxwell's equations with freely adjustable speed of light. The Lagrangian formulation of the coupled particles{ elds system is derived. The quasi{Hamiltonian dynamics of the system is studied in great detail. For implementation on the computer, the equations of motion are discretized with respect to both space and time. The discretization of the electromagnetic elds on a lattice, as well as the interpolation of the particle charges on the lattice is given. The algorithm is as local as possible: Only nearest neighbors sites of the lattice are interacting with a charged particle. Unphysical self{energies arise as a result of the lattice interpolation of charges, and are corrected by a subtraction scheme based on the exact lattice Green's function. The method allows easy parallelization using standard domain decomposition. Some benchmarking results of the algorithm are presented and discussed. Supervisor PD Dr. Burkhard Dunweg May 17, 2004 Neue Methoden zur Simulation von Systemen mit elektrostatischer und hydrodynamischer Wechselwirkung Die vorliegende Dissertation stellt neue Methoden zur Simulation von Systemen mit hydrodynamischer und elektrostatischer Wechselwirkung vor. Teil 1 widmet sich der Computersimulation von Brown'schen Teilchen mit hydrodynamischer Wechselwirkung. Der wichtigste Ein u des Losungsmittels auf die Dynamik der Brown'schen Teilchen besteht darin, da es hydrodynamische Wechselwirkungen vermittelt. In der vorgestellten Methode wird dies simuliert durch numerische Losung der Navier{Stokes{Gleichung auf einem Gitter. Hierzu wird die \Lattice Boltzmann"{Methode benutzt, und zwar in ihrer sogenannten \D3Q19"{Version. Dieses Modell ist imstande, kompressible Stromungen zu simulieren. Dies hat den Vorteil, da dichte Systeme studiert werden konnen, insbesondere auch unter Nichtgleichgewichtsbedingungen. Die \Lattice Boltzmann"{Gleichung wird mit den Teilchen uber eine Reibungskraft gekoppelt. Zusatzlich zu dieser Kraft, die auf Punktteilchen wirkt, konstruieren wir eine weitere Kraft, die vom Drucktensor herruhrt. Diese Kopplung ist streng lokal, d. h. der Algorithmus skaliert linear mit der Gesamtzahl der Teilchen. Um imstande zu sein, die physikalischen Eigenschaften der \Lattice Boltzmann"{Flussigkeit auf diejenigen einer Molekulardynamik{Flussigkeit abzubilden, wird der Fall einer fast inkompressiblen Stromung betrachtet. Die Analyse des Fluktuations{Dissipations{Theorems fur die Hybridkopplung fuhrt auf eine geometrische Interpretation des Reibungskoe zienten im Sinne eines Stokes{Radius. Teil 2 widmet sich der Simulation geladener Teilchen. Wir prasentieren eine neue Methode, um Coulomb{Wechselwirkungen als das \potential of mean force" zwischen Ladungen zu erhalten, die dynamisch an ein lokales elektromagnetisches Feld angekoppelt werden. Dieser Algorithmus skaliert ebenfalls linear. Wir konzentrieren uns auf die Molekulardynamik{Version der Methode, und zeigen, da ein enger Zusammenhang zum Car{Parrinello{Verfahren besteht. Au erdem wird gezeigt, da die Methode auf die Losung der Maxwell{Gleichungen mit frei anpa barer Lichtgeschwindigkeit hinauslauft. Die Lagrange'sche Formulierung des gekoppelten Systems Teilchen{Felder wird hergeleitet. Die quasi{Hamilton'sche Dynamik des Systems wird im Detail studiert. Zur Implementation auf dem Computer werden die Bewegungsgleichungen sowohl raumlich als auch zeitlich diskretisiert. Die Diskretisierung der elektromagnetischen Felder auf dem Gitter sowie die Interpolation der Teilchenladungen auf das Gitter werden angegeben. Der Algorithmus ist so lokal wie nur moglich: Nur die nachsten Nachbarn des Gitters wechselwirken mit einem geladenen Teilchen. Die Gitter{Interpolation der Ladungen fuhrt zu unphysikalischen Selbstenergien; diese werden durch ein Subtraktionsverfahren korrigiert, welches auf der exakten Gitter{Greensfunktion beruht. Die Methode la t sich mit Standard{ Gebietszerlegung leicht parallelisieren. Einige \Benchmark"{Testergebnisse des Algorithmus werden vorgestellt und diskutiert. Betreuer PD Dr. Burkhard Dunweg 17. Mai 200

Computationally Efficient Steady--State Simulation Algorithms for Finite-Element Models of Electric Machines.

The finite element method is a powerful tool for analyzing the magnetic characteristics of electric machines, taking account of both complex geometry and nonlinear material properties. When efficiency is the main quantity of interest, loss calculations can be affected significantly due to the development of eddy currents as a result of Faraday’s law. These effects are captured by the periodic steady-state solution of the magnetic diffusion equation. A typical strategy for calculating this solution is to analyze an initial value problem over a time window of sufficient length so that the transient part of the solution becomes negligible. Unfortunately, because the time constants of electric machines are much smaller than their excitation period at peak power, the transient analysis strategy requires simulating the device over many periods to obtain an accurate steady-state solution. Two other categories of algorithms exist for directly calculating the steady-state solution of the magnetic diffusion equation; shooting methods and the harmonic balance method. Shooting methods search for the steady-state solution by solving a periodic boundary value problem. These methods have only been investigated using first order numerical integration techniques. The harmonic balance method is a Fourier spectral method applied in the time dimension. The standard iterative procedures used for the harmonic balance method do not work well for electric machine simulations due to the rotational motion of the rotor. This dissertation proposes several modifications of these steady-state algorithms which improve their overall performance. First, we demonstrate how shooting methods may be implemented efficiently using Runge-Kutta numerical integration methods with mild coefficient restrictions. Second, we develop a preconditioning strategy for the harmonic balance equations which is robust against large time constants, strong nonlinearities, and rotational motion. Third, we present an adaptive framework for refining the solutions based on a local error criterion which further reduces simulation time. Finally, we compare the performance of the algorithms on a practical model problem. This comparison demonstrates the superiority of the improved steady-state analysis methods, and the harmonic balance method in particular, over transient analysis.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113322/1/pries_1.pd

Quantum Loewner Evolution

What is the scaling limit of diffusion limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the {\em dielectric breakdown model} $\eta$-DBM, a generalization of DLA in which particle locations are sampled from the $\eta$-th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider $\eta$-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter $\gamma \in [0,2]$. In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE$(\gamma^2, \eta)$. QLE is defined in terms of the radial Loewner equation like radial SLE, except that it is driven by a measure valued diffusion $\nu_t$ derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of $\nu_t$ using an SPDE. For each $\gamma \in (0,2]$, there are two or three special values of $\eta$ for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of $\nu_t$. We also explain discrete versions of our construction that relate DLA to loop-erased random walk and the Eden model to percolation. A certain "reshuffling" trick (in which concentric annular regions are rotated randomly, like slot machine reels) facilitates explicit calculation. We propose QLE$(2,1)$ as a scaling limit for DLA on a random spanning-tree-decorated planar map, and QLE$(8/3,0)$ as a scaling limit for the Eden model on a random triangulation. We propose using QLE$(8/3,0)$ to endow pure LQG with a distance function, by interpreting the region explored by a branching variant of QLE$(8/3,0)$, up to a fixed time, as a metric ball in a random metric space.Comment: 132 pages, approximately 100 figures and computer simulation

Simulating Radiating and Magnetized Flows in Multi-Dimensions with ZEUS-MP

This paper describes ZEUS-MP, a multi-physics, massively parallel, message- passing implementation of the ZEUS code. ZEUS-MP differs significantly from the ZEUS-2D code, the ZEUS-3D code, and an early "version 1" of ZEUS-MP distributed publicly in 1999. ZEUS-MP offers an MHD algorithm better suited for multidimensional flows than the ZEUS-2D module by virtue of modifications to the Method of Characteristics scheme first suggested by Hawley and Stone (1995), and is shown to compare quite favorably to the TVD scheme described by Ryu et. al (1998). ZEUS-MP is the first publicly-available ZEUS code to allow the advection of multiple chemical (or nuclear) species. Radiation hydrodynamic simulations are enabled via an implicit flux-limited radiation diffusion (FLD) module. The hydrodynamic, MHD, and FLD modules may be used in one, two, or three space dimensions. Self gravity may be included either through the assumption of a GM/r potential or a solution of Poisson's equation using one of three linear solver packages (conjugate-gradient, multigrid, and FFT) provided for that purpose. Point-mass potentials are also supported. Because ZEUS-MP is designed for simulations on parallel computing platforms, considerable attention is paid to the parallel performance characteristics of each module. Strong-scaling tests involving pure hydrodynamics (with and without self-gravity), MHD, and RHD are performed in which large problems (256^3 zones) are distributed among as many as 1024 processors of an IBM SP3. Parallel efficiency is a strong function of the amount of communication required between processors in a given algorithm, but all modules are shown to scale well on up to 1024 processors for the chosen fixed problem size.Comment: Accepted for publication in the ApJ Supplement. 42 pages with 29 inlined figures; uses emulateapj.sty. Discussions in sections 2 - 4 improved per referee comments; several figures modified to illustrate grid resolution. ZEUS-MP source code and documentation available from the Laboratory for Computational Astrophysics at http://lca.ucsd.edu/codes/currentcodes/zeusmp2

Continuum Mechanics and Thermodynamics in the Hamilton and the Godunov-type Formulations

Continuum mechanics with dislocations, with the Cattaneo type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov type system of the first order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov type formulation brings the mathematical rigor (the well-posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization)

Capacity Fade Analysis and Model Based Optimization of Lithium-ion Batteries

Electrochemical power sources have had significant improvements in design, economy, and operating range and are expected to play a vital role in the future in a wide range of applications. The lithium-ion battery is an ideal candidate for a wide variety of applications due to its high energy/power density and operating voltage. Some limitations of existing lithium-ion battery technology include underutilization, stress-induced material damage, capacity fade, and the potential for thermal runaway. This dissertation contributes to the efforts in the modeling, simulation and optimization of lithium-ion batteries and their use in the design of better batteries for the future. While physics-based models have been widely developed and studied for these systems, the rigorous models have not been employed for parameter estimation or dynamic optimization of operating conditions. The first chapter discusses a systems engineering based approach to illustrate different critical issues possible ways to overcome them using modeling, simulation and optimization of lithium-ion batteries. The chapters 2-5, explain some of these ways to facilitate: i) capacity fade analysis of Li-ion batteries using different approaches for modeling capacity fade in lithium-ion batteries,: ii) model based optimal design in Li-ion batteries and: iii) optimum operating conditions: current profile) for lithium-ion batteries based on dynamic optimization techniques. The major outcomes of this thesis will be,: i) comparison of different types of modeling efforts that will help predict and understand capacity fade in lithium-ion batteries that will help design better batteries for the future,: ii) a methodology for the optimal design of next-generation porous electrodes for lithium-ion batteries, with spatially graded porosity distributions with improved energy efficiency and battery lifetime and: iii) optimized operating conditions of batteries for high energy and utilization efficiency, safer operation without thermal runaway and longer life

Heavy quarks and jets as probes of the QGP

Quark-Gluon Plasma (QGP), a QCD state of matter created in ultra-relativistic heavy-ion collisions, has remarkable properties, including, for example, a low shear viscosity over entropy ratio. By detecting the collection of low-momentum particles that arise from the collision, it is possible to gain quantitative insight into the created matter. However, its fast evolution and thermalization properties remain elusive. Only using high momentum objects as probes of QGP can unveil its constituents at different wavelengths. In this review, we attempt to provide a comprehensive picture of what was, so far, possible to infer about QGP given our current theoretical understanding of jets, heavy-flavor, and quarkonia. We will bridge the resulting qualitative picture to the experimental observations done at the LHC and RHIC. We will focus on the phenomenological description of experimental observations, provide a brief analytical summary of the description of hard probes, and an outlook on the main difficulties we will need to surpass in the following years. To benchmark QGP-related effects, we will also address nuclear modifications to the initial state and hadronization effects