17 research outputs found
Computing Lyapunov spectra with continuous Gram-Schmidt orthonormalization
We present a straightforward and reliable continuous method for computing the
full or a partial Lyapunov spectrum associated with a dynamical system
specified by a set of differential equations. We do this by introducing a
stability parameter beta>0 and augmenting the dynamical system with an
orthonormal k-dimensional frame and a Lyapunov vector such that the frame is
continuously Gram-Schmidt orthonormalized and at most linear growth of the
dynamical variables is involved. We prove that the method is strongly stable
when beta > -lambda_k where lambda_k is the k'th Lyapunov exponent in
descending order and we show through examples how the method is implemented. It
extends many previous results.Comment: 14 pages, 10 PS figures, ioplppt.sty, iopl12.sty, epsfig.sty 44 k
Chaos and Preheating
We show evidence for a relationship between chaos and parametric resonance
both in a classical system and in the semiclassical process of particle
creation. We apply our considerations in a toy model for preheating after
inflation.Comment: 7 pages, 9 figures; uses epsfig and revtex v3.1. Matches version
accepted for publication in Phys. Rev.
Chaotic Orbits in Thermal-Equilibrium Beams: Existence and Dynamical Implications
Phase mixing of chaotic orbits exponentially distributes these orbits through
their accessible phase space. This phenomenon, commonly called ``chaotic
mixing'', stands in marked contrast to phase mixing of regular orbits which
proceeds as a power law in time. It is operationally irreversible; hence, its
associated e-folding time scale sets a condition on any process envisioned for
emittance compensation. A key question is whether beams can support chaotic
orbits, and if so, under what conditions? We numerically investigate the
parameter space of three-dimensional thermal-equilibrium beams with space
charge, confined by linear external focusing forces, to determine whether the
associated potentials support chaotic orbits. We find that a large subset of
the parameter space does support chaos and, in turn, chaotic mixing. Details
and implications are enumerated.Comment: 39 pages, including 14 figure
Chaos and the continuum limit in nonneutral plasmas and charged particle beams
This paper examines discreteness effects in nearly collisionless N-body
systems of charged particles interacting via an unscreened r^-2 force, allowing
for bulk potentials admitting both regular and chaotic orbits. Both for
ensembles and individual orbits, as N increases there is a smooth convergence
towards a continuum limit. Discreteness effects are well modeled by Gaussian
white noise with relaxation time t_R = const * (N/log L)t_D, with L the Coulomb
logarithm and t_D the dynamical time scale. Discreteness effects accelerate
emittance growth for initially localised clumps. However, even allowing for
discreteness effects one can distinguish between orbits which, in the continuum
limit, feel a regular potential, so that emittance grows as a power law in
time, and chaotic orbits, where emittance grows exponentially. For sufficiently
large N, one can distinguish two different `kinds' of chaos. Short range
microchaos, associated with close encounters between charges, is a generic
feature, yielding large positive Lyapunov exponents X_N which do not decrease
with increasing N even if the bulk potential is integrable. Alternatively,
there is the possibility of larger scale macrochaos, characterised by smaller
Lyapunov exponents X_S, which is present only if the bulk potential is chaotic.
Conventional computations of Lyapunov exponents probe X_N, leading to the
oxymoronic conclusion that N-body orbits which look nearly regular and have
sharply peaked Fourier spectra are `very chaotic.' However, the `range' of the
microchaos, set by the typical interparticle spacing, decreases as N increases,
so that, for large N, this microchaos, albeit very strong, is largely
irrelevant macroscopically. A more careful numerical analysis allows one to
estimate both X_N and X_S.Comment: 13 pages plus 17 figure
Chaos and Noise in Galactic Potentials
ABBREVIATED ABSTRACT: This paper summarises an investigation of the effects
of weak friction and noise in time-independent, nonintegrable potentials which
admit both regular and stochastic orbits. The aim is to understand the
qualitative effects of internal and external irregularities associated, e.g.,
with discreteness effects or couplings to an external environment, which stars
in any real galaxy must experience. The two principal conclusions are: (1)
These irregularities can be important on time scales much shorter than the
natural relaxation time scale t_R associated with the friction and noise. For
stochastic orbits friction and noise induce an average exponential divergence
from the unperturbed Hamiltonian trajectory at a rate set by the value of the
local Lyapunov exponent. Even weak noise can make a pointwise interpretation of
orbits suspect already on time scales much shorter than t_R. (2) The friction
and noise can also have significant effects on the statistical properties of
ensembles of stochastic orbits, these also occurring on time scales much
shorter than t_R. Potential implications for galactic dynamics are discussed,
including the problem of shadowing.Comment: 45 pages, uuencoded PostScript (figures included), LA-UR-94-282
Statistical Mechanics and the Physics of the Many-Particle Model Systems
The development of methods of quantum statistical mechanics is considered in
light of their applications to quantum solid-state theory. We discuss
fundamental problems of the physics of magnetic materials and the methods of
the quantum theory of magnetism, including the method of two-time temperature
Green's functions, which is widely used in various physical problems of
many-particle systems with interaction. Quantum cooperative effects and
quasiparticle dynamics in the basic microscopic models of quantum theory of
magnetism: the Heisenberg model, the Hubbard model, the Anderson Model, and the
spin-fermion model are considered in the framework of novel
self-consistent-field approximation. We present a comparative analysis of these
models; in particular, we compare their applicability for description of
complex magnetic materials. The concepts of broken symmetry, quantum
protectorate, and quasiaverages are analyzed in the context of quantum theory
of magnetism and theory of superconductivity. The notion of broken symmetry is
presented within the nonequilibrium statistical operator approach developed by
D.N. Zubarev. In the framework of the latter approach we discuss the derivation
of kinetic equations for a system in a thermal bath. Finally, the results of
investigation of the dynamic behavior of a particle in an environment, taking
into account dissipative effects, are presented.Comment: 77 pages, 1 figure, Refs.37