From B Modes to Quantum Gravity and Unification of Forces

It is commonly anticipated that gravity is subject to the standard principles of quantum mechanics. Yet some (including Einstein) have questioned that presumption, whose empirical basis is weak. Indeed, recently Freeman Dyson has emphasized that no conventional experiment is capable of detecting individual gravitons. However, as we describe, if inflation occurred, the Universe, by acting as an ideal graviton amplifier, affords such access. It produces a classical signal, in the form of macroscopic gravitational waves, in response to spontaneous (not induced) emission of gravitons. Thus recent BICEP2 observations of polarization in the cosmic microwave background will, if confirmed, provide empirical evidence for the quantization of gravity. Their details also support quantitative ideas concerning the unification of strong, electromagnetic, and weak forces, and of all these with gravity.Comment: 4 pages, no figures. v2: minor typos corrected, reference added. v3: very minor typo corrected. Winning entry in Gravity Research Foundation essay competitio

Using Cosmology to Establish the Quantization of Gravity

While many aspects of general relativity have been tested, and general principles of quantum dynamics demand its quantization, there is no direct evidence for that. It has been argued that development of detectors sensitive to individual gravitons is unlikely, and perhaps impossible. We argue here, however, that measurement of polarization of the Cosmic Microwave Background due to a long wavelength stochastic background of gravitational waves from Inflation in the Early Universe would firmly establish the quantization of gravity.Comment: 4 pages, no figures, revised in response to referee's reports. Accepted for publication in PR

Lagrangian Particle Statistics in Turbulent Flows from a Simple Vortex Model

The statistics of Lagrangian particles in turbulent flows is considered in the framework of a simple vortex model. Here, the turbulent velocity field is represented by a temporal sequence of Burgers vortices of different circulation, strain, and orientation. Based on suitable assumptions about the vortices' statistical properties, the statistics of the velocity increments is derived. In particular, the origin and nature of small-scale intermittency in this model is investigated both numerically and analytically

Vortices, zero modes and fractionalization in bilayer-graphene exciton condensate

A real-space formulation is given for the recently discussed exciton condensate in a symmetrically biased graphene bilayer. We show that in the continuum limit an oddly-quantized vortex in this condensate binds exactly one zero mode per valley index of the bilayer. In the full lattice model the zero modes are split slightly due to intervalley mixing. We support these results by an exact numerical diagonalization of the lattice Hamiltonian. We also discuss the effect of the zero modes on the charge content of these vortices and deduce some of their interesting properties.Comment: (v2) A typo in Fig. 1 and a slight error in Eq. (4) corrected; all the main results and conclusions remain unchange

Statistical and numerical investigations of fluid turbulence

The statistical description of fully developed turbulence up to today remains a central open issue of classical physics. Apart from the fact that turbulence plays a key role in many natural and engineering environments, the solution of the problem is also of interest on a conceptual level. Hydrodynamical turbulence may be regarded as a paradigmatic example for a strongly interacting system with a high number of degrees of freedom out of equilibrium, for which a comprehensive statistical mechanics is yet to be formulated. The statistical formulation of turbulent flows can either be approached from a phenomenological side or by deriving statistical relations right from the basic equations of motion. While phenomenological theories often lead to a good description of a variety of statistical quantities, the amount of physical insights to be possibly gained depends heavily on the validity of the assumptions made. On the contrary, statistical theories based on first principles have to face the famous closure problem of turbulence, which prevents a straightforward solution of the statistical problem. The present thesis aims at the investigation of a statistical theory of turbulence in terms of probability density functions (PDFs) based on first principles. To this end we make use of the statistical framework of the Lundgren-Monin-Novikov hierarchy, which allows to derive evolution equations for probability density functions right from the equations of fluid motion. The arising unclosed terms are estimated from highly resolved direct numerical simulations of fully developed turbulence, which allows to make a connection between basic dynamical features of turbulence and the observed statistics. As a technical prerequisite, a parallel pseudospectral code for the direct numerical simulation (DNS) of fully developed turbulence has been developed and tested within this thesis. A number of standard statistical evaluations are presented with the purpose both to benchmark the numerical results as well as to characterize the statistical features of turbulence. Studying the PDF equations, a comprehensive treatment of the single-point velocity and vorticity statistics is achieved within the current work. By making use of statistical symmetries present in the case of homogeneous isotropic turbulence, exact expressions for, e.g., the stationary PDF are derived in terms of correlations between the turbulent field and various quantities determining the dynamics of the field. The joint numerical and analytical investigations eventually lead to an explanation of the slightly subGaussian tails for the velocity statistics and the highly non-Gaussian vorticity statistics with pronounced tails. To contribute to the characterization of the multi-point statistics of turbulence, the two-point enstrophy statistics is investigated. The results quantify the interaction of different spatial scales and give insights into the spatial structure of the vorticity field. Along the lines of preceding works in this context the local conditional structure of the vorticity field and its relation to the multi-point statistics of the vorticity field is discussed and applied to the two-point enstrophy statistics. Finally, the closure problem of turbulence is treated on a more conceptual level by pursuing the question how to establish a model for the two-point PDF which is consistent with the single-point evolution equation and a number of statistical constraints to be imposed on probability density functions. A simple analytical model for the joint PDF is developed and improvements in the context of maximum entropy methods are discussed. Both models are compared to results from DNS. Altogether, the results of the current thesis help to establish a connection between the flow topology, dynamical quantities that determine the temporal evolution of the turbulent fields and the resulting statistics. Beyond the characterization and explanation of these statistical quantities this provides new insights for future modeling and closure strategies

Model study of the sign problem in the mean-field approximation

We argue the sign problem of the fermion determinant at finite density. It is unavoidable not only in Monte-Carlo simulations on the lattice but in the mean-field approximation as well. A simple model deriving from Quantum Chromodynamics (QCD) in the double limit of large quark mass and large quark chemical potential exemplifies how the sign problem arises in the Polyakov loop dynamics at finite temperature and density. In the color SU(2) case our mean-field estimate is in excellent agreement with the lattice simulation. We combine the mean-field approximation with a simple phase reweighting technique to circumvent the complex action encountered in the color SU(3) case. We also investigate the mean-field free energy, from the saddle-point of which we can estimate the expectation value of the Polyakov loop.Comment: 14 page, 18 figures, typos corrected, references added, some clarification in sec.I

The Lundgren-Monin-Novikov Hierarchy: Kinetic Equations for Turbulence

We present an overview of recent works on the statistical description of turbulent flows in terms of probability density functions (PDFs) in the framework of the Lundgren-Monin-Novikov (LMN) hierarchy. Within this framework, evolution equations for the PDFs are derived from the basic equations of fluid motion. The closure problem arises either in terms of a coupling to multi-point PDFs or in terms of conditional averages entering the evolution equations as unknown functions. We mainly focus on the latter case and use data from direct numerical simulations (DNS) to specify the unclosed terms. Apart from giving an introduction into the basic analytical techniques, applications to two-dimensional vorticity statistics, to the single-point velocity and vorticity statistics of three-dimensional turbulence, to the temperature statistics of Rayleigh-B\'enard convection and to Burgers turbulence are discussed.Comment: Accepted for publication in C. R. Acad. Sc

Bias in particle tracking acceleration measurement

We investigate sources of error in acceleration statistics from Lagrangian Particle Tracking (LPT) data and demonstrate techniques to eliminate or minimise bias errors introduced during processing. Numerical simulations of particle tracking experiments in isotropic turbulence show that the main sources of bias error arise from noise due to position uncertainty and selection biases introduced during numerical differentiation. We outline the use of independent measurements and filtering schemes to eliminate these biases. Moreover, we test the validity of our approach in estimating the statistical moments and probability densities of the Lagrangian acceleration. Finally, we apply these techniques to experimental particle tracking data and demonstrate their validity in practice with comparisons to available data from literature. The general approach, which is not limited to acceleration statistics, can be applied with as few as two cameras and permits a substantial reduction in the spatial resolution and sampling rate required to adequately measure statistics of Lagrangian acceleration

Some Calculable Contributions to Entanglement Entropy

Entanglement entropy appears as a central property of quantum systems in broad areas of physics. However, its precise value is often sensitive to unknown microphysics, rendering it incalculable. By considering parametric dependence on correlation length, we extract finite, calculable contributions to the entanglement entropy for a scalar field between the interior and exterior of a spatial domain of arbitrary shape. The leading term is proportional to the area of the dividing boundary; we also extract finite subleading contributions for a field defined in the bulk interior of a waveguide in 3+1 dimensions, including terms proportional to the waveguide's cross-sectional geometry; its area, perimeter length, and integrated curvature. We also consider related quantities at criticality and suggest a class of systems for which these contributions might be measurable.Comment: 4+ pages, 1 figure. v2: Some clarifications and more references; updated to resemble version published in PR