53,281 research outputs found
Exact and approximate dynamics of the quantum mechanical O(N) model
We study a quantum dynamical system of N, O(N) symmetric, nonlinear
oscillators as a toy model to investigate the systematics of a 1/N expansion.
The closed time path (CTP) formalism melded with an expansion in 1/N is used to
derive time evolution equations valid to order 1/N (next-to-leading order). The
effective potential is also obtained to this order and its properties
areelucidated. In order to compare theoretical predictions against numerical
solutions of the time-dependent Schrodinger equation, we consider two initial
conditions consistent with O(N) symmetry, one of them a quantum roll, the other
a wave packet initially to one side of the potential minimum, whose center has
all coordinates equal. For the case of the quantum roll we map out the domain
of validity of the large-N expansion. We discuss unitarity violation in the 1/N
expansion; a well-known problem faced by moment truncation techniques. The 1/N
results, both static and dynamic, are also compared to those given by the
Hartree variational ansatz at given values of N. We conclude that late-time
behavior, where nonlinear effects are significant, is not well-described by
either approximation.Comment: 16 pages, 12 figrures, revte
Chaos in Time Dependent Variational Approximations to Quantum Dynamics
Dynamical chaos has recently been shown to exist in the Gaussian
approximation in quantum mechanics and in the self-consistent mean field
approach to studying the dynamics of quantum fields. In this study, we first
show that any variational approximation to the dynamics of a quantum system
based on the Dirac action principle leads to a classical Hamiltonian dynamics
for the variational parameters. Since this Hamiltonian is generically nonlinear
and nonintegrable, the dynamics thus generated can be chaotic, in distinction
to the exact quantum evolution. We then restrict attention to a system of two
biquadratically coupled quantum oscillators and study two variational schemes,
the leading order large N (four canonical variables) and Hartree (six canonical
variables) approximations. The chaos seen in the approximate dynamics is an
artifact of the approximations: this is demonstrated by the fact that its onset
occurs on the same characteristic time scale as the breakdown of the
approximations when compared to numerical solutions of the time-dependent
Schrodinger equation.Comment: 10 pages (12 figures), RevTeX (plus macro), uses epsf, minor typos
correcte
Cyclic Identities Involving Jacobi Elliptic Functions. II
Identities involving cyclic sums of terms composed from Jacobi elliptic
functions evaluated at equally shifted points on the real axis were
recently found. These identities played a crucial role in discovering linear
superposition solutions of a large number of important nonlinear equations. We
derive four master identities, from which the identities discussed earlier are
derivable as special cases. Master identities are also obtained which lead to
cyclic identities with alternating signs. We discuss an extension of our
results to pure imaginary and complex shifts as well as to the ratio of Jacobi
theta functions.Comment: 38 pages. Modified and includes more new identities. A shorter
version of this will appear in J. Math. Phys. (May 2003
Resumming the large-N approximation for time evolving quantum systems
In this paper we discuss two methods of resumming the leading and next to
leading order in 1/N diagrams for the quartic O(N) model. These two approaches
have the property that they preserve both boundedness and positivity for
expectation values of operators in our numerical simulations. These
approximations can be understood either in terms of a truncation to the
infinitely coupled Schwinger-Dyson hierarchy of equations, or by choosing a
particular two-particle irreducible vacuum energy graph in the effective action
of the Cornwall-Jackiw-Tomboulis formalism. We confine our discussion to the
case of quantum mechanics where the Lagrangian is . The
key to these approximations is to treat both the propagator and the
propagator on similar footing which leads to a theory whose graphs have the
same topology as QED with the propagator playing the role of the photon.
The bare vertex approximation is obtained by replacing the exact vertex
function by the bare one in the exact Schwinger-Dyson equations for the one and
two point functions. The second approximation, which we call the dynamic Debye
screening approximation, makes the further approximation of replacing the exact
propagator by its value at leading order in the 1/N expansion. These two
approximations are compared with exact numerical simulations for the quantum
roll problem. The bare vertex approximation captures the physics at large and
modest better than the dynamic Debye screening approximation.Comment: 30 pages, 12 figures. The color version of a few figures are
separately liste
Solitary Waves and Compactons in a class of Generalized Korteweg-DeVries Equations
We study the class of generalized Korteweg-DeVries equations derivable from
the Lagrangian: L(l,p) = \int \left( \frac{1}{2} \vp_{x} \vp_{t} - {
{(\vp_{x})^{l}} \over {l(l-1)}} + \alpha(\vp_{x})^{p} (\vp_{xx})^{2} \right)
dx, where the usual fields of the generalized KdV equation are
defined by u(x,t) = \vp_{x}(x,t). This class contains compactons, which are
solitary waves with compact support, and when , these solutions have the
feature that their width is independent of the amplitude. We consider the
Hamiltonian structure and integrability properties of this class of KdV
equations. We show that many of the properties of the solitary waves and
compactons are easily obtained using a variational method based on the
principle of least action. Using a class of trial variational functions of the
form we
find soliton-like solutions for all , moving with fixed shape and constant
velocity, . We show that the velocity, mass, and energy of the variational
travelling wave solutions are related by , where , independent of .\newline \newline PACS numbers: 03.40.Kf,
47.20.Ky, Nb, 52.35.SbComment: 16 pages. LaTeX. Figures available upon request (Postscript or hard
copy
Evaluating Descriptive Metrics of the Human Cone Mosaic
Purpose: To evaluate how metrics used to describe the cone mosaic change in response to simulated photoreceptor undersampling (i.e., cell loss or misidentification).
Methods: Using an adaptive optics ophthalmoscope, we acquired images of the cone mosaic from the center of fixation to 10° along the temporal, superior, inferior, and nasal meridians in 20 healthy subjects. Regions of interest (n = 1780) were extracted at regular intervals along each meridian. Cone mosaic geometry was assessed using a variety of metrics â density, density recovery profile distance (DRPD), nearest neighbor distance (NND), intercell distance (ICD), farthest neighbor distance (FND), percentage of six-sided Voronoi cells, nearest neighbor regularity (NNR), number of neighbors regularity (NoNR), and Voronoi cell area regularity (VCAR). The âperformanceâ of each metric was evaluated by determining the level of simulated loss necessary to obtain 80% statistical power.
Results: Of the metrics assessed, NND and DRPD were the least sensitive to undersampling, classifying mosaics that lost 50% of their coordinates as indistinguishable from normal. The NoNR was the most sensitive, detecting a significant deviation from normal with only a 10% cell loss.
Conclusions: The robustness of cone spacing metrics makes them unsuitable for reliably detecting small deviations from normal or for tracking small changes in the mosaic over time. In contrast, regularity metrics are more sensitive to diffuse loss and, therefore, better suited for detecting such changes, provided the fraction of misidentified cells is minimal. Combining metrics with a variety of sensitivities may provide a more complete picture of the integrity of the photoreceptor mosaic
Time evolution of the chiral phase transition during a spherical expansion
We examine the non-equilibrium time evolution of the hadronic plasma produced
in a relativistic heavy ion collision, assuming a spherical expansion into the
vacuum. We study the linear sigma model to leading order in a large-
expansion. Starting at a temperature above the phase transition, the system
expands and cools, finally settling into the broken symmetry vacuum state. We
consider the proper time evolution of the effective pion mass, the order
parameter , and the particle number distribution. We
examine several different initial conditions and look for instabilities
(exponentially growing long wavelength modes) which can lead to the formation
of disoriented chiral condensates (DCCs). We find that instabilities exist for
proper times which are less than 3 fm/c. We also show that an experimental
signature of domain growth is an increase in the low momentum spectrum of
outgoing pions when compared to an expansion in thermal equilibrium. In
comparison to particle production during a longitudinal expansion, we find that
in a spherical expansion the system reaches the ``out'' regime much faster and
more particles get produced. However the size of the unstable region, which is
related to the domain size of DCCs, is not enhanced.Comment: REVTex, 20 pages, 8 postscript figures embedded with eps
Chaos in effective classical and quantum dynamics
We investigate the dynamics of classical and quantum N-component phi^4
oscillators in the presence of an external field. In the large N limit the
effective dynamics is described by two-degree-of-freedom classical Hamiltonian
systems. In the classical model we observe chaotic orbits for any value of the
external field, while in the quantum case chaos is strongly suppressed. A
simple explanation of this behaviour is found in the change in the structure of
the orbits induced by quantum corrections. Consistently with Heisenberg's
principle, quantum fluctuations are forced away from zero, removing in the
effective quantum dynamics a hyperbolic fixed point that is a major source of
chaos in the classical model.Comment: 6 pages, RevTeX, 5 figures, uses psfig, changed indroduction and
conclusions, added reference
Formulas for Continued Fractions. An Automated Guess and Prove Approach
We describe a simple method that produces automatically closed forms for the
coefficients of continued fractions expansions of a large number of special
functions. The function is specified by a non-linear differential equation and
initial conditions. This is used to generate the first few coefficients and
from there a conjectured formula. This formula is then proved automatically
thanks to a linear recurrence satisfied by some remainder terms. Extensive
experiments show that this simple approach and its straightforward
generalization to difference and -difference equations capture a large part
of the formulas in the literature on continued fractions.Comment: Maple worksheet attache
Pauli equation and the method of supersymmetric factorization
We consider different variants of factorization of a 2x2 matrix
Schroedinger/Pauli operator in two spatial dimensions. They allow to relate its
spectrum to the sum of spectra of two scalar Schroedinger operators, in a
manner similar to one-dimensional Darboux transformations. We consider both the
case when such factorization is reduced to the ordinary 2-dimensional SUSY QM
quasifactorization and a more general case which involves covariant
derivatives. The admissible classes of electromagnetic fields are described and
some illustrative examples are given.Comment: 18 pages, Late
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