In this dissertation, the critical scaling behavior of magnetic Ising models with long range interactions is studied. These long range interactions, when imposed in addition to interactions on a regular lattice, lead to small-world graphs. By using large-scale Monte Carlo simulations, together with finite-size scaling, the critical behavior of a number of different models is obtained. The Ising models studied in this dissertation include the z-model introduced by Scalettar, standard small-world bonds superimposed on a square lattice, and physical small-world bonds superimposed on a square lattice. From the scaling results of the Binder 4th order cumulant, the order parameter, and the susceptibility, the long-range interaction is found to drive the systems behavior from Ising-like to mean field, and drive the critical point to a higher temperature. It is concluded that with a large amount of strong long-range connections (compared to the interactions on regular lattices), so the long-range connection density is non-vanishing, systems have mean field behavior. With a weak interaction that vanishes for an infinite system size or for vanishing density of long-range connections the systems have Ising-like critical behavior. The crossover from Ising-like to meanield behavior due to weak long-range interactions for systems with a large amount of long-range connections is also discussed. These results provide further evidence to support the existence of physical (quasi-) small-world nanomaterials.

We study the mean-field limit of a particle-based system modeling the behavior of many indistinguishable pedestrians as their number increases. The base model is a modified version of Helbing's social force model. In the mean-field limit, the time-dependent density of two-dimensional pedestrians satisfies a four-dimensional integro-differential Fokker-Planck equation. To approximate the solution of the Fokker-Planck equation we use a time-splitting approach and solve the diffusion part using a Crank-Nicholson method. The advection part is solved using a Lax-Wendroff-Leveque method or an upwind Backward Euler method depending on the advection speed. Moreover, we use multilevel Monte Carlo to estimate observables from the particle-based system. We discuss these numerical methods, and present numerical results showing the convergence of observables that were calculated using the particle-based model as the number of pedestrians increases to those calculated using the probability density function satisfying the Fokker-Planck equation.

The mean-field game theory is the study of strategic decision making in very large populations of weakly interacting individuals. Mean-field games have been an active area of research in the last decade due to its increased significance in many scientific fields. The foundations of mean-field theory go back to the theory of statistical and quantum physics. One may describe mean-field games as a type of stochastic differential game for which the interaction between the players is of mean-field type, i.e the players are coupled via their empirical measure. It was proposed by Larsy and Lions and independently by Huang, Malhame, and Caines. Since then, the mean-field games have become a rapidly growing area of research and has been studied by many researchers. However, most of these studies were dedicated to diffusion-type games. The main purpose of this thesis is to extend the theory of mean-field games to jump case in both discrete and continuous state space. Jump processes are a very important tool in many areas of applications. Specifically, when modeling abrupt events appearing in real life. For instance, financial modeling (option pricing and risk management), networks (electricity and Banks) and statistics (for modeling and analyzing spatial data). The thesis consists of two papers and one technical report which will be submitted soon: In the first publication, we study the mean-field game in a finite state space where the dynamics of the indistinguishable agents is governed by a controlled continuous time Markov chain. We have studied the control problem for a representative agent in the linear quadratic setting. A dynamic programming approach has been used to drive the Hamilton Jacobi Bellman equation, consequently, the optimal strategy has been achieved. The main result is to show that the individual optimal strategies for the mean-field game system represent 1/N-Nash equilibrium for the approximating system of N agents. As a second article, we generalize the previous results to agents driven by a non-linear pure jump Markov processes in Euclidean space. Mathematically, this means working with linear operators in Banach spaces adapted to the integro-differential operators of jump type and with non-linear partial differential equations instead of working with linear transformations in Euclidean spaces as in the first work. As a by-product, a generalization for the Koopman operator has been presented. In this setting, we studied the control problem in a more general sense, i.e. the cost function is not necessarily of linear quadratic form. We showed that the resulting unique optimal control is of Lipschitz type. Furthermore, a fixed point argument is presented in order to construct the approximate Nash Equilibrium. In addition, we show that the rate of convergence will be of special order as a result of utilizing a non-linear pure jump Markov process. In a third paper, we develop our approach to treat a more realistic case from a modelling perspective. In this step, we assume that all players are subject to an additional common noise of Brownian type. We especially study the well-posedness and the regularity for a jump version of the stochastic kinetic equation. Finally, we show that the solution of the master equation, which is a type of second order partial differential equation in the space of probability measures, provides an approximate Nash Equilibrium. This paper, unfortunately, has not been completely finished and it is still in preprint form. Hence, we have decided not to enclose it in the thesis. However, an outlook about the paper will be included.

Monocamadas insolúveis localizadas sobre a superfície de um líquido são sistemas conhecidos e estudados há mais de 100 anos. Elas são formadas quando moléculas anfifílicas são depositadas sobre algum solvente em condições especiais. Quando sofrem compressão isotérmica, tais sistemas exibem um comportamento muito complexo podendo sofrer várias transições de fase nesse processo. Embora, com o surgimento na década de 1990 de técnicas experimentais que proporcionaram um maior ?insight? no entendimento das referidas transições, há muitas questões que permanecem em aberto, principalmente no que diz respeito à influência exercida: pelas conformações intramoleculares; pelas interações entre as moléculas anfifílicas; pelas interações entre as moléculas anfifílicas e as moléculas do solvente sobre as referidas transições. Para ajudar a preencher esta lacuna são necessários modelos moleculares que auxiliem a obtenção da resposta destas questões. É neste contexto que se insere este trabalho, onde três diferentes modelos de campo médio são empregados a fim de se descrever o comportamento das transições de fase sofridas pelas monocamadas no que se refere aos aspectos acima mencionados. Cada modelo é diferente no que diz respeito ao comportamento das caudas hidrofóbicas erguidas em direção ao ar. O emprego de tais modelos proporcionou, em linhas gerais, um melhor entendimento das transições de fase nestes sistemas. / Insoluble monolayers lying on a liquid surface are known for about one century. They are formed when amphiphilic molecules are deposited on some solvent under special conditions. Under isothermal compression, these systems may exhibit a complex behavior suffering several phase transitions. Although with recent experimental development on the area new insights on the phase transitions were obtained, many questions remain unanswered. Some of these questions are related with the influence of some variables like the intramolecular conformations and the interaction between the amphiphilic molecules and the solvent molecules. In order to fill this gap molecular models are a useful and valuable tool. So, it was employed three different mean-field models in order to describe phase transitions of the molecules. The difference between the models relies on the behavior of the hydrophobic tails lifted on the air. Such models proportioned some insight on the phase transitions of the system.

本論文包括對 Jaynes-Cummings-Hubbard (JCH) 系統的研究。在這個系統中，每個耦合光學腔之內都放置了一顆雙態原子，偶極相互作用導致系統有激發光子和原子的自由度。對量子電動力學系統的研究，使我們對光子與原子間的相互作用以及量子相變有更深的認識。 / 我們研究了一維JCH系統中有兩個激發子的本徵態。我們發現當真空拉比頻率與腔間穿隧率之比超過某一臨界值，兩個激發子的束縛態就會出現。 / 我們還為兩個腔的JCH系統之演化作出研究，並指出系統的量子態在一定條件下可演變成薛丁格貓態。從相干態演化到薛丁格貓態所需的時間亦被估計。 / 最後，我們使用主方程來探討驅動JCH系統的去相干。在這篇論文中，我們提出了一些低激發量的穩態的例子。許多不同的穩態系統的光子統計將被討論。 / This thesis comprises of an investigation of the Jaynes-Cummings-Hubbard (JCH) system. In such a system, single two-level atoms are embedded in each coupled optical cavity, and the dipole interaction leads to dynamics involving photonic and atomic degrees of freedom. The investigation of quantum electrodynamics in the system provides insight about the behaviour of strongly interacting photons and atoms and quantum phase transition. / We examine the eigenstates of the one-dimensional JCH system in the two-excitation subspace. We discover that two-excitation bound states emerge when the ratio of vacuum Rabi frequency to the tunnelling rate between cavities exceeds a critical value. / We also study for the time evolution of a two-cavity JCH system, and indicate that the evolved state can be a Schrödinger's cat state under certain conditions. The time required for evolving a coherent state into a Schrödinger's cat state is also estimated. / Finally, we investigate the decoherence of a driven JCH system by using the master equation. In this thesis we present several examples of steady state when the total number of excitation is low. The photon statistics of the steady state of the system will be discussed. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Wong, Tsz Ching Max = Jaynes-Cummings-Hubbard模型的課題 / 黃子澄. / "November 2012." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 79-84). / Abstracts also in Chinese. / Wong, Tsz Ching Max = Jaynes-Cummings-Hubbard mo xing de ke ti / Huang Zicheng. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Basic Description of the Jaynes-Cummings-Hubbard Model --- p.4 / Chapter 2.1 --- Jaynes-Cummings Model --- p.4 / Chapter 2.1.1 --- Hamiltonian --- p.5 / Chapter 2.1.2 --- Energy Eigenstates --- p.6 / Chapter 2.2 --- Coupled-cavity System without Atoms --- p.8 / Chapter 2.2.1 --- A One-dimensional Coupled-cavity Chain --- p.8 / Chapter 2.2.2 --- Normal Modes --- p.9 / Chapter 2.3 --- Jaynes-Cummings-Hubbard Model --- p.10 / Chapter 2.3.1 --- Eigenstates of Single Excitation --- p.11 / Chapter 2.3.2 --- Dynamics of Single Excitation --- p.12 / Chapter 2.3.3 --- Single Excitation in the Weak Coupling Limit --- p.15 / Chapter 2.3.4 --- Single Excitation in the Strong Coupling Limit --- p.16 / Chapter 3 --- Solution of the JCH Model with Two Excitations --- p.17 / Chapter 3.1 --- Two-particle Bound States in the Bose-Hubbard Model --- p.17 / Chapter 3.2 --- Two-polariton Bound States in the JCH Model --- p.19 / Chapter 3.2.1 --- Bound State Eigenvectors --- p.24 / Chapter 3.2.2 --- Bound State Eigenvalues --- p.26 / Chapter 3.2.3 --- Critical Coupling --- p.28 / Chapter 3.2.4 --- Analytic Approximations in Strong Coupling Regime n≫k --- p.30 / Chapter 3.3 --- Dynamics of Two Excitations in Strong Coupling Regime --- p.33 / Chapter 3.3.1 --- Initial Condition: --- p.33 / Chapter 3.3.2 --- Initial Condition: Superposition of j2; gin in Gaussian Distribution --- p.34 / Chapter 3.3.3 --- Initial Condition: Superposition of j1; gin j1; gim in Gaussian Distribution --- p.39 / Chapter 3.3.4 --- Initial Condition: Superposition of j1;+in j1; ¡im in Gaussian Distribution --- p.39 / Chapter 4 --- JCH Model in a Two-cavity Con¯guration --- p.41 / Chapter 4.1 --- Generation of SchrÄodinger's Cat State: Numerical Simulation --- p.41 / Chapter 4.2 --- Generation of SchrÄodinger's Cat States: Analytic Solution --- p.45 / Chapter 4.2.1 --- Estimation of Optimum Parameters --- p.53 / Chapter 4.3 --- Coherent States as Initial Conditions --- p.55 / Chapter 5 --- Decoherence of a Weakly Driven JCH Model --- p.60 / Chapter 5.1 --- Master Equation --- p.60 / Chapter 5.2 --- Driven JCH Model: N-cavity Configuration --- p.61 / Chapter 5.2.1 --- One-excitation Approximation --- p.63 / Chapter 5.2.2 --- Simple Harmonic Oscillator Limit --- p.66 / Chapter 5.3 --- Driven JCH Model: Two-cavity Configuration --- p.69 / Chapter 6 --- Conclusion --- p.76 / Bibliography --- p.79

One of the extraordinary properties of the Sun is the observed range of motion scales from the convection granules to the cyclic variation of magnetic activity. The Sun's magnetic field exhibits coherence in space and time on much larger scales than the turbulent convection that ultimately powers the dynamo. Motivated by the scale separation considerations, in this thesis we study the parametric scale selection of dynamo action. Although helioseismology has made a lot of progress in the study of the solar interior, the precise motions of plasma are still unknown. In this work, we assume that the model flow is forced with helical viscous body forces acting on different characteristic scales and weak and strong large-scale shear flows that are believed to be present near the base of the convection zone. In this thesis, we look for numerical evidence of a large-scale magnetic field relative to the characteristic scale of the model flow. The investigation is based on the simulations of incompressible MHD equations in elongated triply-periodic domains. To commence the investigation, a linear stability analysis of the coarsening instability in a one-dimensional periodic system is performed to study the stability threshold in the mean-field limit that assumes large scale separation in the system. The simulations are used to discriminate between different forms of the mean-field α -effect and domain aspect ratio. The notion of scale selection refers to methods for estimating characteristic scales. We define the dynamo scale through the characteristic scales of the underlying model flow, forcing and the realised magnetic field. The aspect ratio of the elongated domains plays a crucial role in all considered cases. In Part II, we examine the dynamo generated by the imposed model flows. The transition from large-scale dynamo at the onset to small-scale dynamo as we increase Rm is smooth and takes place in two stages: a fast transition into a predominantly small-scale magnetic energy state and a slower transition into even smaller scales. The long wavelength perturbation imposed on the ABC flow in the modulated case is not preserved in the eigenmodes of the magnetic field. In the presence of the linear (semi-linear shearing-box approximation) and the sinusoidal shearing motions, the field again undergoes a smooth transition at the slow non-sheared rate, which is associated with the balance of the advection and diffusion terms in the induction equation. Part III considers the nonlinear extension of the analysis in Part II, where the incompressible cellular and sheared flows interact with the exponentially growing magnetic field via the Lorentz force in the dynamical regime. Both sheared and non-sheared helical cellular flows become unstable to large-scale perturbations even in the limit of high viscosity. Due to the helical properties of the imposed forcing, the inverse cascade of helicity leads to energy accumulation in the largest scales of the domain, albeit the characteristic lengthscale exhibits the transitional nature at a highly reduced rate in the mean-field limit. As Rm is increased, the transition resembles that of the kinematic regime. The unique properties of the anisotropic shear reduce the componentality of the system, which in turn is able to half the rate of transition from the large-scale dynamo at the onset to a small-scale one.

Strongly correlated materials are an important class of materials for research in condensed matter physics. Other than ordinary solid-state physical systems, which can be well described and analyzed by the energy band theory, the electron-electron correlation effects in strongly correlated materials are far more significant. So it is necessary to develop theories and methods that are beyond the energy band theory to describe their rich and varied behaviors. Not only are there electron-electron correlations, typically the multiple degrees of freedom in strongly correlated materials, such as charge distribution, orbital occupancies, spin orientations, and lattice structure exhibit cooperative or competitive behaviors, giving rise to rich phase diagrams and sensitive or non-perturbative responses to changes in external parameters such as temperature, strain, electromagnetic fields, etc. This thesis is divided into two parts. In the first part, we use the density functional theory (DFT) plus U correction, i.e., the DFT+U method, to calculate the equilibrium and nonequilibrium phase transitions of LuNiO3 and VO2. The effect of adding U is manifested in both materials as the change of band structure in response to the change of orbital occupancies of electrons, i.e., the soft band effect. This effect bring about competitions of electrons between different orbitals by lowering the occupied orbitals and raising the empty orbitals in energy, giving rise to multiple metastable states. In the second part, we study the dynamic mean field theory (DMFT) as a beyond band-theory method. This is a Green's function based theory for open quantum systems. By selecting one lattice site of an interacting lattice model as an open system, the other lattice sites as the environment are equivalently replaced by a set of non-interaction orbitals according to the hybridization function, so the whole system is transformed into an Anderson impurity model. We studied how to use the density matrix renormalization group (DMRG) method to perform real-time evolutions of the Anderson impurity model to study the non-equilibrium dynamics of a strongly correlated lattice system. We begin in Chapter 1 with an introduction to strongly correlated materials, density functional theory (DFT) and dynamical mean-field theory (DMFT). The Kohn-Sham density functional theory and its plus U correction are discussed in detail. We also demonstrate how the DMFT reduces the lattice sites other than the impurity site as a set of non-interacting bath orbitals. Then in Chapters 2 and 3, we show material-related studies of LuNiO3 as an example of rare-earth nickelates under substrate strain, and VO2 as an example of a narrow-gap Mott insulator in a pump-probe experiment. These are two types of strongly correlated materials with localized 3d orbitals (for Ni and V). We use the DFT+U method to calculate their band structures and study the structural phase transitions in LuNiO3 and metal-insulator transitions in both materials. The competition between the charge-ordered and Jahn-Teller distorted phases of LuNiO3 is studied at various substrate lattice constants within DFT+U. A Landau energy function is constructed based on group theory to understand the competition of various distortion modes of the NiO6 octahedra. VO2 is known for its metal-insulator transition at 68 degree C, above which temperature it's a metal and below which it's an insulator with a doubled unit cell. For VO2 in a pump-probe experiment, a metastable metal phase was found to exist in the crystal structure of the equilibrium insulating phase. Our work is to understand this novel metastable phase from a soft-band picture. We also use quantum Boltzmann equation to justify the prethermalization of electrons over the lifetime of the metastable metal, so that the photoinduced transition can be understood in a hot electron picture. Finally, in Chapters 4 and 5, we show a focused study of building a real-time solver for the Anderson impurity model out of equilibrium using the density matrix renormalization group (DMRG) method, towards the goal of building an impurity solver for nonequilibrium dynamical mean-field theory (DMFT). We study both the quenched and driven single-impurity Anderson models (SIAM) in real time, evolving the wave function written in a form with 4 matrix product states (MPS) in DMRG. For the quenched model, we find that the computational cost is polynomial time if the bath orbitals in the MPSs are ordered in energy. The same energy-ordering scheme works for the driven model in the short driving period regime in which the Floquet-Magnus expansion converges. In the long-period regime, we find that the computational time grows exponentially with the physical time, or the number of periods reached. The computational cost reduces in the long run when the bath orbitals are quasi-energy ordered, which is discussed in further detail in the thesis.

Abstract Detailed in this manuscript is a methodology to model ground state properties of 4He droplets at zero pressure and zero Kelvin using a density functional theory of liquid helium. The density functional approach examined here consists of two noted functionals from the literature and corresponding mean field definitions. A mean field and trial density are defined for each system and optimized to self-consistency using a matrix diagonalization technique. Initial calculations of planar slabs are performed and demonstrate reasonable agreement with experiment and with prior studies using density functional theory. Quantum properties of droplets and droplets containing atomic dopants are calculated. Three different He-dopant potentials are examined to test the limits of the functional methods. For each impurity interaction, an average of 12 atoms were found to reside in the first solvation shell with an atomic dopant placed at the droplet center. Maximum densities in the first solvation shell reached those of solid helium as predicted by DF methods.

No description available.

Thesis (Ph. D.)--University of Florida, 2000. / Title from first page of PDF file. Document formatted into pages; contains ix, 242 p.; also contains graphics. Vita. Includes bibliographical references (p. 232-241).