809 research outputs found
Periodic orbit quantization of the Sinai billiard in the small scatterer limit
We consider the semiclassical quantization of the Sinai billiard for disk
radii R small compared to the wave length 2 pi/k. Via the application of the
periodic orbit theory of diffraction we derive the semiclassical spectral
determinant. The limitations of the derived determinant are studied by
comparing it to the exact KKR determinant, which we generalize here for the A_1
subspace. With the help of the Ewald resummation method developed for the full
KKR determinant we transfer the complex diffractive determinant to a real form.
The real zeros of the determinant are the quantum eigenvalues in semiclassical
approximation. The essential parameter is the strength of the scatterer
c=J_0(kR)/Y_0(kR). Surprisingly, this can take any value between plus and minus
infinity within the range of validity of the diffractive approximation kR <<4.
We study the statistics exhibited by spectra for fixed values of c. It is
Poissonian for |c|=infinity, provided the disk is placed inside a rectangle
whose sides obeys some constraints. For c=0 we find a good agreement of the
level spacing distribution with GOE, whereas the form factor and two-point
correlation function are similar but exhibit larger deviations. By varying the
parameter c from 0 to infinity the level statistics interpolates smoothly
between these limiting cases.Comment: 17 pages LaTeX, 5 postscript figures, submitted to J. Phys. A: Math.
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Escape from noisy intermittent repellers
Intermittent or marginally-stable repellers are commonly associated with a
power law decay in the survival fraction. We show here that the presence of
weak additive noise alters the spectrum of the Perron - Frobenius operator
significantly giving rise to exponential decays even in systems that are
otherwise regular. Implications for ballistic transport in marginally stable
miscrostructures are briefly discussed.Comment: 3 ps figures include
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Yang-Mills Measure and the Master Field on the Sphere
AbstractWe study the Yang–Mills measure on the sphere with unitary structure group. In the limit where the structure group has high dimension, we show that the traces of loop holonomies converge in probability to a deterministic limit, which is known as the master field on the sphere. The values of the master field on simple loops are expressed in terms of the solution of a variational problem. We show that, given its values on simple loops, the master field is characterized on all loops of finite length by a system of differential equations, known as the Makeenko–Migdal equations. We obtain a number of further properties of the master field. On specializing to families of simple loops, our results identify the high-dimensional limit, in non-commutative distribution, of the Brownian bridge in the group of unitary matrices starting and ending at the identity.</jats:p
On the duality between periodic orbit statistics and quantum level statistics
We discuss consequences of a recent observation that the sequence of periodic
orbits in a chaotic billiard behaves like a poissonian stochastic process on
small scales. This enables the semiclassical form factor to
agree with predictions of random matrix theories for other than infinitesimal
in the semiclassical limit.Comment: 8 pages LaTe
Lyapunov exponents and anomalous diffusion of a Lorentz gas with infinite horizon using approximate zeta functions
We compute the Lyapunov exponent, generalized Lyapunov exponents and the
diffusion constant for a Lorentz gas on a square lattice, thus having infinite
horizon. Approximate zeta functions, written in terms of probabilities rather
than periodic orbits, a re used in order to avoid the convergence problems of
cycle expansions. The emphasis is on the relation between the analytic
structure of the zeta function, where a branch cut plays an important role, and
the asymptotic dynamics of the system. We find a diverging diffusion constant
and a phase transition for the generalized Lyapunov
exponents.Comment: 14 pages LaTeX, figs 2-3 on .uu file, fig 1 available from autho
Stability ordering of cycle expansions
We propose that cycle expansions be ordered with respect to stability rather
than orbit length for many chaotic systems, particularly those exhibiting
crises. This is illustrated with the strong field Lorentz gas, where we obtain
significant improvements over traditional approaches.Comment: Revtex, 5 incorporated figures, total size 200
Pseudoacromegaly: A Differential Diagnostic Problem for Acromegaly With a Genetic Solution.
Acromegaly is usually not a difficult condition to diagnose once the possibility of this disease has been raised. However, a few conditions present with some aspects of acromegaly or gigantism but without growth hormone (GH) excess. Such cases are described as "pseudoacromegaly" or "acromegaloidism". Here we describe a female patient investigated for GH excess at 10 years of age for tall stature since infancy (height and weight > +3 standard deviations) and typical acromegalic features, including large hands/feet, large jaw, tongue, hoarse deep voice, and headache. Results of radiography of the sella turcica and GH response at an oral glucose tolerance test and insulin-arginine- thyrotrophin-luteinizing hormone-releasing hormone test were normal. Ethinylestradiol and medroxyprogesterone were given for 2 years; this successfully stopped further height increase. Although the patient's growth rate plateaued, coarsening of the facial features and acral enlargement also led to investigations for suspicion of acromegaly at 23 and 36 years of age, both with negative results. On referral at the age of 49 years, she had weight gain, sweating, sleep apnea, headaches, joint pain, and enlarged tongue. Endocrine assessment again showing normal GH axis was followed by genetic testing with a macrocephaly/overgrowth syndrome panel. A denovo mutation in the NSD1 gene (c.6605G>C; p.Cys2202Ser) was demonstrated. Mutations affecting the same cysteine residue have been identified in patients with Sotos syndrome. In summary, Sotos syndrome and other overgrowth syndromes can mimic the clinical manifestations of acromegaly or gigantism. Genetic assessment could be helpful in these cases
Accelerating cycle expansions by dynamical conjugacy
Periodic orbit theory provides two important functions---the dynamical zeta
function and the spectral determinant for the calculation of dynamical averages
in a nonlinear system. Their cycle expansions converge rapidly when the system
is uniformly hyperbolic but greatly slowed down in the presence of
non-hyperbolicity. We find that the slow convergence can be associated with
singularities in the natural measure. A properly designed coordinate
transformation may remove these singularities and results in a dynamically
conjugate system where fast convergence is restored. The technique is
successfully demonstrated on several examples of one-dimensional maps and some
remaining challenges are discussed
Anomalous Diffusion in Infinite Horizon Billiards
We consider the long time dependence for the moments of displacement < |r|^q
> of infinite horizon billiards, given a bounded initial distribution of
particles. For a variety of billiard models we find ~ t^g(q) (up to
factors of log t). The time exponent, g(q), is piecewise linear and equal to
q/2 for q2. We discuss the lack of dependence of this result
on the initial distribution of particles and resolve apparent discrepancies
between this time dependence and a prior result. The lack of dependence on
initial distribution follows from a remarkable scaling result that we obtain
for the time evolution of the distribution function of the angle of a
particle's velocity vector.Comment: 11 pages, 7 figures Submitted to Physical Review
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