Non-Perturbative Methods in Quantum Field Theory and Quantum Gravity

This thesis considers non-perturbative methods in quantum field theory with applications to gravity and cosmology. In particular, there are chapters on black hole holography, inflationary model building, and the conformal bootstrap.

DIRECT NUMERICAL SIMULATION OF INCOMPRESSIBLE MULTIPHASE FLOW WITH PHASE CHANGE

Phase change problems arise in many practical applications such as air-conditioning and refrigeration, thermal energy storage systems and thermal management of electronic devices. The physical phenomenon in such applications are complex and are often difficult to be studied in detail with the help of only experimental techniques. The efforts to improve computational techniques for analyzing two-phase flow problems with phase change are therefore gaining momentum. The development of numerical methods for multiphase flow has been motivated generally by the need to account more accurately for (a) large topological changes such as phase breakup and merging, (b) sharp representation of the interface and its discontinuous properties and (c) accurate and mass conserving motion of the interface. In addition to these considerations, numerical simulation of multiphase flow with phase change introduces additional challenges related to discontinuities in the velocity and the temperature fields. Moreover, the velocity field is no longer divergence free. For phase change problems, the focus of developmental efforts has thus been on numerically attaining a proper conservation of energy across the interface in addition to the accurate treatment of fluxes of mass and momentum conservation as well as the associated interface advection. Among the initial efforts related to the simulation of bubble growth in film boiling applications the work in \cite{Welch1995} was based on the interface tracking method using a moving unstructured mesh. That study considered moderate interfacial deformations. A similar problem was subsequently studied using moving, boundary fitted grids \cite{Son1997}, again for regimes of relatively small topological changes. A hybrid interface tracking method with a moving interface grid overlapping a static Eulerian grid was developed \cite{Juric1998} for the computation of a range of phase change problems including, three-dimensional film boiling \cite{esmaeeli2004computations}, multimode two-dimensional pool boiling \cite{Esmaeeli2004} and film boiling on horizontal cylinders \cite{Esmaeeli2004a}. The handling of interface merging and pinch off however remains a challenge with methods that explicitly track the interface. As large topological changes are crucial for phase change problems, attention has turned in recent years to front capturing methods utilizing implicit interfaces that are more effective in treating complex interface deformations. The VOF (Volume of Fluid) method was adopted in \cite{Welch2000} to simulate the one-dimensional Stefan problem and the two-dimensional film boiling problem. The approach employed a specific model for mass transfer across the interface involving a mass source term within cells containing the interface. This VOF based approach was further coupled with the level set method in \cite{Son1998}, employing a smeared-out Heaviside function to avoid the numerical instability related to the source term. The coupled level set, volume of fluid method and the diffused interface approach was used for film boiling with water and R134a at the near critical pressure condition \cite{Tomar2005}. The effect of superheat and saturation pressure on the frequency of bubble formation were analyzed with this approach. The work in \cite{Gibou2007} used the ghost fluid and the level set methods for phase change simulations. A similar approach was adopted in \cite{Son2008} to study various boiling problems including three-dimensional film boiling on a horizontal cylinder, nucleate boiling in microcavity \cite{lee2010numerical} and flow boiling in a finned microchannel \cite{lee2012direct}. The work in \cite{tanguy2007level} also used the ghost fluid method and proposed an improved algorithm based on enforcing continuity and divergence-free condition for the extended velocity field. The work in \cite{sato2013sharp} employed a multiphase model based on volume fraction with interface sharpening scheme and derived a phase change model based on local interface area and mass flux. Among the front capturing methods, sharp interface methods have been found to be particularly effective both for implementing sharp jumps and for resolving the interfacial velocity field. However, sharp velocity jumps render the solution susceptible to erroneous oscillations in pressure and also lead to spurious interface velocities. To implement phase change, the work in \cite{Hardt2008} employed point mass source terms derived from a physical basis for the evaporating mass flux. To avoid numerical instability, the authors smeared the mass source by solving a pseudo time-step diffusion equation. This measure however led to mass conservation issues due to non-symmetric integration over the distributed mass source region. The problem of spurious pressure oscillations related to point mass sources was also investigated by \cite{Schlottke2008}. Although their method is based on the VOF, the large pressure peaks associated with sharp mass source was observed to be similar to that for the interface tracking method. Such spurious fluctuation in pressure are essentially undesirable because the effect is globally transmitted in incompressible flow. Hence, the pressure field formation due to phase change need to be implemented with greater accuracy than is reported in current literature. The accuracy of interface advection in the presence of interfacial mass flux (mass flux conservation) has been discussed in \cite{tanguy2007level,tanguy2014benchmarks}. The authors found that the method of extending one phase velocity to entire domain suggested by Nguyen et al. in \cite{nguyen2001boundary} suffers from a lack of mass flux conservation when the density difference is high. To improve the solution, the authors impose a divergence-free condition for the extended velocity field by solving a constant coefficient Poisson equation. The approach has shown good results with enclosed bubble or droplet but is not general for more complex flow and requires additional solution of the linear system of equations. In current thesis, an improved approach that addresses both the numerical oscillation of pressure and the spurious interface velocity field is presented by featuring (i) continuous velocity and density fields within a thin interfacial region and (ii) temporal velocity correction steps to avoid unphysical pressure source term. Also I propose a general (iii) mass flux projection correction for improved mass flux conservation. The pressure and the temperature gradient jump condition are treated sharply. A series of one-dimensional and two-dimensional problems are solved to verify the performance of the new algorithm. Two-dimensional and cylindrical film boiling problems are also demonstrated and show good qualitative agreement with the experimental observations and heat transfer correlations. Finally, a study on Taylor bubble flow with heat transfer and phase change in a small vertical tube in axisymmetric coordinates is carried out using the new multiphase, phase change method.

Spectral graph theory with applications to quantum adiabatic optimization

In this dissertation I draw a connection between quantum adiabatic optimization, spectral graph theory, heat-diffusion, and sub-stochastic processes through the operators that govern these processes and their associated spectra. In particular, we study Hamiltonians which have recently become known as ``stoquastic'' or, equivalently, the generators of sub-stochastic processes. The operators corresponding to these Hamiltonians are of interest in all of the settings mentioned above. I predominantly explore the connection between the spectral gap of an operator, or the difference between the two lowest energies of that operator, and certain equilibrium behavior. In the context of adiabatic optimization, this corresponds to the likelihood of solving the optimization problem of interest. I will provide an instance of an optimization problem that is easy to solve classically, but leaves open the possibility to being difficult adiabatically. Aside from this concrete example, the work in this dissertation is predominantly mathematical and we focus on bounding the spectral gap. Our primary tool for doing this is spectral graph theory, which provides the most natural approach to this task by simply considering Dirichlet eigenvalues of subgraphs of host graphs. I will derive tight bounds for the gap of one-dimensional, hypercube, and general convex subgraphs. The techniques used will also adapt methods recently used by Andrews and Clutterbuck to prove the long-standing ``Fundamental Gap Conjecture''.

Parsing, Generation and Grammar

Humans use their grammatical knowledge in more than one way. On one hand, they use it to understand what others say. On the other hand, they use it to say what they want to convey to others (or to themselves). In either case, they need to assemble the structure of sentences in a systematic fashion, in accordance with the grammar of their language. Despite the fact that the structures that comprehenders and speakers assemble are systematic in an identical fashion (i.e., obey the same grammatical constraints), the two ‘modes’ of assembling sentence structures might or might not be performed by the same cognitive mechanisms. Currently, the field of psycholinguistics implicitly adopts the position that they are supported by different cognitive mechanisms, as evident from the fact that most psycholinguistic models seek to explain either comprehension or production phenomena. The potential existence of two independent cognitive systems underlying linguistic performance doubles the problem of linking the theory of linguistic knowledge and the theory of linguistic performance, making the integration of linguistics and psycholinguistic harder. This thesis thus aims to unify the structure building system in comprehension, i.e., parser, and the structure building system in production, i.e., generator, into one, so that the linking theory between knowledge and performance can also be unified into one. I will discuss and unify both existing and new data pertaining to how structures are assembled in understanding and speaking, and attempt to show that the unification between parsing and generation is at least a plausible research enterprise. In Chapter 1, I will discuss the previous and current views on how parsing and generation are related to each other. I will outline the challenges for the current view that the parser and the generator are the same cognitive mechanism. This single system view is discussed and evaluated in the rest of the chapters. In Chapter 2, I will present new experimental evidence suggesting that the grain size of the pre-compiled structural units (henceforth simply structural units) is rather small, contrary to some models of sentence production. In particular, I will show that the internal structure of the verb phrase in a ditransitive sentence (e.g., The chef is donating the book to the monk) is not specified at the onset of speech, but is specified before the first internal argument (the book) needs to be uttered. I will also show that this timing of structural processes with respect to the verb phrase structure is earlier than the lexical processes of verb internal arguments. These two results in concert show that the size of structure building units in sentence production is rather small, contrary to some models of sentence production, yet structural processes still precede lexical processes. I argue that this view of generation resembles the widely accepted model of parsing that utilizes both top-down and bottom-up structure building procedures. In Chapter 3, I will present new experimental evidence suggesting that the structural representation strongly constrains the subsequent lexical processes. In particular, I will show that conceptually similar lexical items interfere with each other only when they share the same syntactic category in sentence production. The mechanism that I call syntactic gating, will be proposed, and this mechanism characterizes how the structural and lexical processes interact in generation. I will present two Event Related Potential (ERP) experiments that show that the lexical retrieval in (predictive) comprehension is also constrained by syntactic categories. I will argue that the syntactic gating mechanism is operative both in parsing and generation, and that the interaction between structural and lexical processes in both parsing and generation can be characterized in the same fashion. In Chapter 4, I will present a series of experiments examining the timing at which verbs’ lexical representations are planned in sentence production. It will be shown that verbs are planned before the articulation of their internal arguments, regardless of the target language (Japanese or English) and regardless of the sentence type (active object-initial sentence in Japanese, passive sentences in English, and unaccusative sentences in English). I will discuss how this result sheds light on the notion of incrementality in generation. In Chapter 5, I will synthesize the experimental findings presented in this thesis and in previous research to address the challenges to the single system view I outlined in Chapter 1. I will then conclude by presenting a preliminary single system model that can potentially capture both the key sentence comprehension and sentence production data without assuming distinct mechanisms for each.