Browsing by Subject "Hamiltonian systems"

Now showing 1 - 15 of 15

Coherent control of cold atoms in a[n] optical lattice
(2007) Holder, Benjamin Peirce, 1976-; Reichl, L. E.
The dynamics of non-interacting, ultracold alkali atoms in the presence of counter-propagating lasers (optical lattice systems) is considered theoretically. The center of mass motion of an atom is such a system can be described by an effective Hamiltonian of a relatively simple form. Modulation of the laser fields implies a parametric variation of the effective Hamiltonian's eigenvalue spectrum, under which avoided crossings may occur. We investigate two dynamical processes arising from these near-degeneracies, which can be manipulated to coherently control atomic motion. First, we demonstrate the mechanism for the chaos-assisted, or multiple-state, tunneling observed in recent optical lattice experiments. Second, we propose a new method for the coherent acceleration of lattice atoms using the techniques of stimulated Raman adiabatic passage (STIRAP). In each case we use perturbation analysis to show the existence of a small, few level, subsystem of the full effective Schrödinger equation that determines the dynamics.
Dynamics of quantum control in cold-atom systems
(2009-05) Roy, Analabha, 1978-; Reichl, L. E.
The dynamics of mesoscopic two-boson systems that model an interacting pair of ultracold alkali atoms in the presence of electromagnetic potentials are considered. The translational degrees of freedom of such a system can be described by a simple reduced atom Hamiltonian. Introducing time modulations in the laser fields causes parametric variations of the Hamiltonian's Floquet eigenvalue spectrum. Broken symmetries cause level repulsion and avoided crossings in this spectrum that are quantum manifestations of the chaos in the underlying classical dynamics of the systems. We investigate the effects of this phenomenon in the coherent control of excitations in these systems. These systems can be coherently excited from their ground states to higher energy states via a Stimulated Raman Adiabatic Passage (STIRAP). The presence of avoided crossings alter the outcome of STIRAP. First, the classical dynamics of such two-boson systems in double wells is described and manifestations of the same to the quantum mechanical system are discussed. Second, the quantum dynamics of coherent control in the manner discussed above is detailed for a select choice(s) of system parameters. Finally, the same chaos-assisted adiabatic passage is demonstrated for optical lattice systems based on experiments on the same done with noninteracting atoms.
Hamilton's equations with Euler parameters for hybrid particle-finite element simulation of hypervelocity impact
(2002) Shivarama, Ravishankar Ajjanagadde; Fahrenthold, Eric P.
Hypervelocity impact studies (impact velocities > 1 km/sec) encompass a wide range of applications including development of anti-terrorist defense and orbital debris shield for the International Space Station (ISS). The focus of this work is on the development of a hybrid particle-finite element method for orbital debris shield simulations. The problem is characterized by finite strain kinematics, strong energy domain coupling, contact-impact, shock wave propagation and history dependent material damage effects. A novel hybrid particle finite element method based on Hamilton’s equations is presented. The model discretizes the continuum of interest simultaneously (but not redundantly) into particles and finite elements. The particles are ellipsoidal in shape and can translate and rotate in three dimensional space. Rotation is described using Euler parameters. Volumetric and contact impact effects are modeled using particles, while strength is modeled using conventional Lagrangian finite elements. The model is general enough to accommodate a wide range of material models and equations of state.
Irreversibility and extended formulation of classical and quantum nonintegrable dynamics
(1995-08) Zhang, Zili, 1964-; Not available
Numerical studies of the standard nontwist map and a renormalization group framework for breakup of invariant tori
(2004) Apte, Amit Shriram; Morrison, Philip J.
This thesis presents numerical explorations of area-preserving nontwist maps, and a renormalization group framework for the destruction of invariant tori. We study the phenomena of bifurcation and reconnection, and the emergence of meandering tori which are non-KAM invariant curves. We also study the breakup of shearless invariant tori with noble winding numbers using improved numerical techniques to implement Greene’s residue criterion. We interpret the breakup of invariant tori within a renormalization group framework by constructing renormalization group operators for the tori with winding numbers that are quadratic irrationals. We find the simple fixed points of these operators and interpret the map pairs with critical invariant tori as critical fixed points. We introduce coordinate transformations on the space of maps to relate these fixed points to each other. These transformations induce conjugacies between the corresponding operators, and provide a new perspective on the space of area-preserving maps.
Old and new perspectives on effective equations : a study of quantum many-body systems
(2020-05-09) Rosenzweig, Matthew Harry; Pavlović, Nataša; Staffilani, Gigliola; Chen, Thomas; Caffarelli, Luis; Gamba, Irene M
This dissertation focuses on the study of nonlinear-Schrodinger-type equations as partial differentiation equations (PDEs) arising as effective descriptions of systems of finitely many interacting bosons. We approach this topic from two perspectives. The old perspective consists of proving quantitative convergence in an appropriate function space of solutions to the finite problem to a solution of an effective, limiting PDE, as the number of particles tends to infinity. The new perspective consists of proving qualitative convergence of geometric structure, such as the properties of being an integrable and Hamiltonian system. Through these two complementary perspectives, focusing on both quantitative and qualitative convergence, we gain a deeper understanding of how field theories, both classical and quantum, may be deformed to a new field theory, and of how this deformation may be reversed.
On the Hamiltonian structure of the linearized Maxwell-Vlasov system
(1995-05) Shadwick, Bradley Allan, 1964-; Not available
Plasma turbulence in the equatorial electrojet observations, theories, models, and simulations
(2015-12) Hassan, Ehab Mohamed Ali Hussein; Morrison, Philip J.; Horton, Wendell; Fitzpatrick, Richard; Bengtson, Roger; Humphreys, Todd
The plasma turbulence in the equatorial electrojet due to the presence of two different plasma instability mechanisms has been observed and studied for more than seven decades. The sharp density-gradient and large conductivity give rise to gradient-drift and Farley-Buneman instabilities, respectively, of different scale-lengths. A new 2-D fluid model is derived by modifying the standard two-stream fluid model with the ion viscosity tensor and electron polarization drift, and is capable of describing both instabilities in a unified system. Numerical solution of the model in the linear regime demonstrates the capacity of the model to capture the salient characteristics of the two instabilities. Nonlinear simulations of the unified model of the equatorial electrojet instabilities reproduce many of the features that are found in radar observations and sounding rocket measurements under multiple solar and ionospheric conditions. The linear and nonlinear numerical results of the 2-D unified fluid model are found to be comparable to the fully kinetic and hybrid models which have high computational cost and small coverage area of the ionosphere. This gives the unified fluid model a superiority over those models. The distribution of the energy content in the system is studied and the rate of change of the energy content in the evolving fields obeys the law of energy conservation. The dynamics of the ions were found to have the largest portion of energy in their kinetic and internal thermal energy components. The redistribution of energy is characterized by a forward cascade generating small-scale structures. The bracket of the system dynamics in the nonlinear partial differential equation was proved to be a non-canonical Hamiltonian system as that bracket satisfies the Jacobi identity. The penetration of the variations in the interplanetary magnetic and electric fields in the solar winds to the dip equator is observed as a perfect match with the variations in the horizontal components of the geomagnetic and electric fields at the magnetic equator. Three years of concurrent measurements of the solar wind parameters at Advanced Composition Explorer (ACE) and Interplanetary Monitoring Platform (IMP) space missions used to establish a Kernel Density Estimation (KDE) functions for these parameters at the IMP-8 location. The KDE functions can be used to generate an ensemble of the solar wind parameters which has many applications in space weather forecasting and data-driven simulations. Also, categorized KDE functions ware established for the solar wind categories that have different origin from the Sun.
Renormalization of continuous-time dynamical systems with KAM applications
(2006) Kocić, Saša; Koch, Hans
In this dissertation, we construct a sequence of renormalization group transformations on a space of analytic vector fields. We apply these transformations to study the persistence of quasiperiodic motion (invariant tori) with sufficiently incommensurate frequency vectors w in near-integrable systems. The renormalization transformations preserve geometrical “classes” of the vector fields, such as Hamiltonian, divergence-free, time-reversible, and symmetric with respect to an involution. Two different approaches have been developed. One approach makes use of a recent multidimensional generalization of the continued fraction algorithm and applies to Diophantine frequency vectors w. The other approach applies to the larger set of Brjuno frequency vectors. We prove the existence of an integrable limit set of the renormalization and show that there exists a finite-codimension stable manifold W for the sequence of renormalization maps, associated to this set. We show that every vector field on W has an analytic elliptic invariant torus on which the flow is conjugate to a rotation with a Diophantine or, more generally, Brjuno frequency vector w. Consequently, every family of vector fields that intersects W has a member which has an analytic invariant torus with frequency vector w. We show that the number of parameters of a family can be reduced if a non-degeneracy condition is satisfied. In certain classes of vector fields, e.g. Hamiltonian vector fields, the number of parameters can be reduced to zero, and analogous statements are true for individual vector fields. In the special case of two degree of freedom Hamiltonian vector fields we also construct a sequence of renormalization group transformations with an attracting integrable limit set, directly on a space of Hamiltonian functions. As an application of the scheme we give a proof of KAM theorem for Hamiltonians satisfying a nondegeneracy condition. On a numerical level, the scheme can be applied to obtain the critical function of one-parameter families of two-degree of freedom Hamiltonian systems.
Renormalization of isoenergetically degenerate Hamiltonian flows, and instability of solitons in shear hydrodynamic flows
(2003) Gaidashev, Denis Gennad'yevich; Koch, Hans A.; Bona, J. L.
Part I of this Thesis presents a study of the renormalization group transformation acting on an appropriate space of Hamiltonian functions in two angle and two action variables. In particular, we study the existence of real invariant tori, on which the flow is conjugate to a rotation with the rotation number equal to the golden mean (ω-tori). We demonstrate that the stable manifold of the renormalization operator at the “simple” fixed point contains isoenergetically degenerate Hamiltonians possessing shearless ω-tori. We also show that one-parameter families of Hamiltonians transverse to the stable manifold undergo a bifurcation: for a certain range of the parameter values the members of these families posses two distinct ω-tori, the members of such families lying on the stable manifold posses one shearless ω-torus, while the members corresponding to other parameter values do not posses any. We also present some numerical evidence for universality associated with the breakup of shearless invariant tori, and compute the relevant critical renormalization and scaling eigenvalues. Part II of the Thesis presents a stability analysis of plane solitonsin hydrodynamic shear flows obeying a (2+1) analogue of the Benjamin–Ono equation. The instability region and the short-wave instability threshold for plane solitons are found numerically. We also determine the dependence of the growth rate on the propagation angle in the longwave limit and demonstrate the existence of a critical angle which separates two types of behaviour of the growth rate.
Renormalization, invariant tori, and periodic orbits for Hamiltonian flows
(2001-05) Abad, Juan José, 1967-; Koch, Hans A.
Consideration is given to a family of renormalization transformations developed to study the existence of invariant tori in Hamiltonian systems. These transformations are used to construct invariant tori with self–similar frequency vectors, as well as sequences of periodic orbits approximating them, for near– integrable Hamiltonians in two or more degrees of freedom. Results on the location of the periodic orbits, and accumulation rates of these orbits to the invariant tori are presented. A numerical implementation of one of these transformations is used to to search for non–trivial fixed points in a reduced space of Hamiltonians with two degrees of freedom. Evidence supporting the existence of a fixed point that seems to be related to the break–up of golden invariant tori is provided. The critical indexand scaling found for this point are in good agreement with previous numerical experiments carried out for area preserving maps. Finally, a simple space of Hamiltonians in two degrees of freedom, all having a common invariant torus, is analyzed to gain insight vii into the dynamics of the flow on the invariant torus as the conjugacy to linear motion breaks down. By studying a particular one-parameter family of Hamiltonians, it is observed how the flow on the torus changes from being dense quasi–periodic to having an invariant quasi–periodic Cantor set. This transition is triggered by the appearance of a fixed point for the flow on the torus. Accumulation ratios of periodic orbits approaching the torus with this particular flow are found numerically, and shown to be different from those associated with known renormalization fixed points
Resonance overlap, secular effects and non-integrability: an approach from ensemble theory
(2003) Li, Chun Biu; Prigogine, I. (Ilya); Petrosky, Tomio Y.
The role of the Van Hove singularity in the time evolution of electronic states in a low-dimensional superlattice semiconductor
(2007-05) Garmon, Kenneth Sterling, 1978-; Petrosky, Tomio Y.; Reichl, L. E.
Over the last three decades, the rapid development of efficient synthetic routes for the preparation of expanded porphyrin macrocycles has allowed the exploration of a new frontier involving “porphyrin-like” coordination chemistry. This doctoral dissertation describes the author’s exploratory journey into the area of transition metal cation complexation using oligopyrrolic macrocycles. The reported synthetic findings were used to gain new insights into the factors affecting the observed coordination modes and to probe the emerging roles of counter-anion effects, tautomeric equilibria and hydrogenbonding interactions in regulating the metalation chemistry of expanded porphyrins. The first chapter provides an updated overview of this relatively young coordination chemistry subfield and introduces the idea of expanded porphyrins as a diverse family of ligands for metalation studies. Chapter 2 details the synthesis of a series of binuclear complexes and illustrates the importance of metal oxidation state, macrocycle protonation and counter-anion effects in terms of defining the final structure of the observed metal complexes. The binding study reported in Chapter 3 demonstrates a strong positive allosteric effect for the coordination of silver(I) cations in a Schiff base expanded porphyrin. Chapter 4 introduces the use of oligopyrrolic macrocycles for the stabilization of early transition metal cations. Specifically, the preparation of a series of vanadium complexes illustrates the bimodal (i.e., covalent and noncovalent) recognition of the non-spherical dioxovanadium(V) species within the macrocyclic cavities. Experimental procedures and characterization data are reported in Chapter 5.
Spatially-homogeneous Vlasov-Einstein dynamics
(2008-12) Okabe, Takahide; Morrison, Philip J.
The influence of matter described by the Vlasov equation, on the evolution of anisotropy in the spatially-homogeneous universes, called the Bianchi cosmologies, is studied. Due to the spatial-homogeneity, the Einstein equations for each Bianchi Type are reduced to a set of coupled ordinary differential equations, which has Hamiltonian form with the metric components being the canonical coordinates. In the vacuum Bianchi cosmologies, it is known that a curvature potential, which comes from the symmetries of the three-dimensional Lie groups, determines the basic properties of the evolution of anisotropy. In this work, matter potentials are constructed for Vlasov matter. They are obtained by first introducing a new matter action principle for the Vlasov equation, in terms of a conjugate pair of functions, and then enforcing the symmetry to obtain a reduction. This yields an expression for the matter potential in terms of the phase space density, which is further reduced by assuming cold streaming matter. Some vacuum Bianchi cosmologies and Type I with Vlasov matter are compared. It is shown that the Vlasov-matter potential for cold streaming matter results in qualitatively distinct dynamics from the well-known vacuum Bianchi cosmologies.
Topics in Lagrangian and Hamiltonian fluid dynamics : relabeling symmetry and ion-acoustic wave stability
(1998) Padhye, Nikhil Subhash, 1970-; Morrison, Philip J.
Relabeling symmetries of the Lagrangian action are found for the ideal, compressible fluid and magnetohydrodynamics (MHD). These give rise to conservation laws of potential vorticity (Ertel's theorem) and helicity in the ideal fluid, cross helicity in MHD, and a conservation law for an ideal fluid with three thermodynamic variables. The symmetry that gives rise to Ertel's theorem is generated by an infinite parameter group, and leads to a generalized Bianchi identity. The existence of a more general symmetry is also shown, with dependence on time and space derivatives of the fields, and corresponds to a family of conservation laws associated with the potential vorticity. In the Hamiltonian formalism, Casimir invariants of the noncanonical formulation are directly constructed from the symmetries of the reduction map from Lagrangian to Eulerian variables. Casimir invariants of MHD include a gauge-dependent family of invariants that incorporates magnetic helicity as a special case. Novel examples of finite dimensional, noncanonical Hamiltonian dynamics are also presented: the equations for a magnetic field line flow with a symmetry direction, and Frenet formulas that describe a curve in 3-space. In the study of Lyapunov stability of ion-acoustic waves, existence of negative energy perturbations is found at short wavelengths. The effect of adiabatic, ionic pressure on ion-acoustic waves is investigated, leading to explicit solitary and nonlinear periodic wave solutions for the adiabatic exponent r = 3. In particular, solitary waves are found to exist at any wave speed above Mach number one, without an upper cutoff speed. Negative energy perturbations are found to exist despite the addition of pressure, which prevents the establishment of Lyapunov stability; however the stability of ion-acoustic waves is established in the KdV limit, in a manner far simpler than the proof of KdV soliton stability. It is also shown that the KdV free energy (Benjamin, 1972) is recovered upon evaluating (the negative of) the ion-acoustic free energy at the critical point, in the KdV approximation. Numerical study of an ion-acoustic solitary wave with a negative energy perturbation shows transients with increased perturbation amplitude. The localized perturbation moves to the left in the wave-frame, leaving the solitary wave peak intact, thus indicating that the wave may be stable.