62,024 research outputs found
Uniform approximation for diffractive contributions to the trace formula in billiard systems
We derive contributions to the trace formula for the spectral density
accounting for the role of diffractive orbits in two-dimensional billiard
systems with corners. This is achieved by using the exact Sommerfeld solution
for the Green function of a wedge. We obtain a uniformly valid formula which
interpolates between formerly separate approaches (the geometrical theory of
diffraction and Gutzwiller's trace formula). It yields excellent numerical
agreement with exact quantum results, also in cases where other methods fail.Comment: LaTeX, 41 pages including 12 PostScript figures, submitted to Phys.
Rev.
Landscape Predictions for the Higgs Boson and Top Quark Masses
If the Standard Model is valid up to scales near the Planck mass, and if the
cosmological constant and Higgs mass parameters scan on a landscape of vacua,
it is well known that the observed orders of magnitude of these quantities can
be understood from environmental selection for large-scale structure and atoms.
If in addition the Higgs quartic coupling scans, with a probability
distribution peaked at low values, environmental selection for a phase having a
scale of electroweak symmetry breaking much less than the Planck scale leads to
a most probable Higgs mass of 106 GeV. While fluctuations below this are
negligible, the upward fluctuation is 25/p GeV, where p measures the strength
of the peaking of the a priori distribution of the quartic coupling. If the top
Yukawa coupling also scans, the most probable top quark mass is predicted to
lie in the range (174--178) GeV, providing the standard model is valid to at
least 10^{17} GeV. The downward fluctuation is 35 GeV/ \sqrt{p}, suggesting
that p is sufficiently large to give a very precise Higgs mass prediction.
While a high reheat temperature after inflation could raise the most probable
value of the Higgs mass to 118 GeV, maintaining the successful top prediction
suggests that reheating is limited to about 10^8 GeV, and that the most
probable value of the Higgs mass remains at 106 GeV. If all Yukawa couplings
scan, then the e,u,d and t masses are understood to be outliers having extreme
values induced by the pressures of strong environmental selection, while the s,
\mu, c, b, \tau Yukawa couplings span only two orders of magnitude, reflecting
an a priori distribution peaked around 10^{-3}. Extensions of these ideas allow
order of magnitude predictions for neutrino masses, the baryon asymmetry and
important parameters of cosmological inflation.Comment: 41 pages; v4: threshold corrrections for top Yukawa are correcte
Smooth Hamiltonian systems with soft impacts
In a Hamiltonian system with impacts (or "billiard with potential"), a point
particle moves about the interior of a bounded domain according to a background
potential, and undergoes elastic collisions at the boundaries. When the
background potential is identically zero, this is the hard-wall billiard model.
Previous results on smooth billiard models (where the hard-wall boundary is
replaced by a steep smooth billiard-like potential) have clarified how the
approximation of a smooth billiard with a hard-wall billiard may be utilized
rigorously. These results are extended here to models with smooth background
potential satisfying some natural conditions. This generalization is then
applied to geometric models of collinear triatomic chemical reactions (the
models are far from integrable -degree of freedom systems with ).
The application demonstrates that the simpler analytical calculations for the
hard-wall system may be used to obtain qualitative information with regard to
the solution structure of the smooth system and to quantitatively assist in
finding solutions of the soft impact system by continuation methods. In
particular, stable periodic triatomic configurations are easily located for the
smooth highly-nonlinear two and three degree of freedom geometric models.Comment: 33 pages, 8 figure
Isolating intrinsic noise sources in a stochastic genetic switch
The stochastic mutual repressor model is analysed using perturbation methods. This simple model of a gene circuit consists of two genes and three promotor states. Either of the two protein products can dimerize, forming a repressor molecule that binds to the promotor of the other gene. When the repressor is bound to a promotor, the corresponding gene is not transcribed and no protein is produced. Either one of the promotors can be repressed at any given time or both can be unrepressed, leaving three possible promotor states. This model is analysed in its bistable regime in which the deterministic limit exhibits two stable fixed points and an unstable saddle, and the case of small noise is considered. On small time scales, the stochastic process fluctuates near one of the stable fixed points, and on large time scales, a metastable transition can occur, where fluctuations drive the system past the unstable saddle to the other stable fixed point. To explore how different intrinsic noise sources affect these transitions, fluctuations in protein production and degradation are eliminated, leaving fluctuations in the promotor state as the only source of noise in the system. Perturbation methods are then used to compute the stability landscape and the distribution of transition times, or first exit time density. To understand how protein noise affects the system, small magnitude fluctuations are added back into the process, and the stability landscape is compared to that of the process without protein noise. It is found that significant differences in the random process emerge in the presence of protein noise
Mode structure and ray dynamics of a parabolic dome microcavity
We consider the wave and ray dynamics of the electromagnetic field in a
parabolic dome microcavity. The structure of the fundamental s-wave involves a
main lobe in which the electromagnetic field is confined around the focal point
in an effective volume of the order of a cubic wavelength, while the modes with
finite angular momentum have a structure that avoids the focal area and have
correspondingly larger effective volume. The ray dynamics indicates that the
fundamental s-wave is robust with respect to small geometrical deformations of
the cavity, while the higher order modes are associated with ray chaos and
short-lived. We discuss the incidence of these results on the modification of
the spontaneous emission dynamics of an emitter placed in such a parabolic dome
microcavity.Comment: 50 pages, 17 figure
Approximating multi-dimensional Hamiltonian flows by billiards
Consider a family of smooth potentials , which, in the limit
, become a singular hard-wall potential of a multi-dimensional
billiard. We define auxiliary billiard domains that asymptote, as
to the original billiard, and provide asymptotic expansion of
the smooth Hamiltonian solution in terms of these billiard approximations. The
asymptotic expansion includes error estimates in the norm and an
iteration scheme for improving this approximation. Applying this theory to
smooth potentials which limit to the multi-dimensional close to ellipsoidal
billiards, we predict when the separatrix splitting persists for various types
of potentials
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