157,564 research outputs found
The Born Oppenheimer wave function near level crossing
The standard Born Oppenheimer theory does not give an accurate description of
the wave function near points of level crossing. We give such a description
near an isotropic conic crossing, for energies close to the crossing energy.
This leads to the study of two coupled second order ordinary differential
equations whose solution is described in terms of the generalized
hypergeometric functions of the kind 0F3(;a,b,c;z). We find that, at low
angular momenta, the mixing due to crossing is surprisingly large, scaling like
\mu^(1/6), where \mu is the electron to nuclear mass ratio.Comment: 21 pages, 7 figure
Numerical investigations of non-uniqueness for the Navier-Stokes initial value problem in borderline spaces
We consider the Cauchy problem for the incompressible Navier-Stokes equations
in for a one-parameter family of explicit scale-invariant
axi-symmetric initial data, which is smooth away from the origin and invariant
under the reflection with respect to the -plane. Working in the class of
axi-symmetric fields, we calculate numerically scale-invariant solutions of the
Cauchy problem in terms of their profile functions, which are smooth. The
solutions are necessarily unique for small data, but for large data we observe
a breaking of the reflection symmetry of the initial data through a
pitchfork-type bifurcation. By a variation of previous results by Jia &
\v{S}ver\'ak (2013) it is known rigorously that if the behavior seen here
numerically can be proved, optimal non-uniqueness examples for the Cauchy
problem can be established, and two different solutions can exists for the same
initial datum which is divergence-free, smooth away from the origin, compactly
supported, and locally -homogeneous near the origin. In particular,
assuming our (finite-dimensional) numerics represents faithfully the behavior
of the full (infinite-dimensional) system, the problem of uniqueness of the
Leray-Hopf solutions (with non-smooth initial data) has a negative answer and,
in addition, the perturbative arguments such those by Kato (1984) and Koch &
Tataru (2001), or the weak-strong uniqueness results by Leray, Prodi, Serrin,
Ladyzhenskaya and others, already give essentially optimal results. There are
no singularities involved in the numerics, as we work only with smooth profile
functions. It is conceivable that our calculations could be upgraded to a
computer-assisted proof, although this would involve a substantial amount of
additional work and calculations, including a much more detailed analysis of
the asymptotic expansions of the solutions at large distances.Comment: 31 pages, 19 figure
Born-Oppenheimer Approximation near Level Crossing
We consider the Born-Oppenheimer problem near conical intersection in two
dimensions. For energies close to the crossing energy we describe the wave
function near an isotropic crossing and show that it is related to generalized
hypergeometric functions 0F3. This function is to a conical intersection what
the Airy function is to a classical turning point. As an application we
calculate the anomalous Zeeman shift of vibrational levels near a crossing.Comment: 8 pages, 1 figure, Lette
Spherical Universes with Anisotropic Pressure
Einstein's equations are solved for spherically symmetric universes composed
of dust with tangential pressure provided by angular momentum, L(R), which
differs from shell to shell. The metric is given in terms of the shell label,
R, and the proper time, tau, experienced by the dust particles. The general
solution contains four arbitrary functions of R - M(R), L(R), E(R) and r(0,R).
The solution is described by quadratures, which are in general elliptic
integrals. It provides a generalization of the Lemaitre-Tolman-Bondi solution.
We present a discussion of the types of solution, and some examples. The
relationship to Einstein clusters and the significance for gravitational
collapse is also discussed.Comment: 24 pages, 11 figures, accepted for publication in Classical and
Quantum Gravit
The complex life of hydrodynamic modes
We study analytic properties of the dispersion relations in classical
hydrodynamics by treating them as Puiseux series in complex momentum. The radii
of convergence of the series are determined by the critical points of the
associated complex spectral curves. For theories that admit a dual
gravitational description through holography, the critical points correspond to
level-crossings in the quasinormal spectrum of the dual black hole. We
illustrate these methods in supersymmetric Yang-Mills theory in
3+1 dimensions, in a holographic model with broken translation symmetry in 2+1
dimensions, and in conformal field theory in 1+1 dimensions. We comment on the
pole-skipping phenomenon in thermal correlation functions, and show that it is
not specific to energy density correlations.Comment: V3: 54 pages, 18 figures. Appendix added. Version to appear in JHE
Non-Filippov dynamics arising from the smoothing of nonsmooth systems, and its robustness to noise
Switch-like behaviour in dynamical systems may be modelled by highly
nonlinear functions, such as Hill functions or sigmoid functions, or
alternatively by piecewise-smooth functions, such as step functions. Consistent
modelling requires that piecewise-smooth and smooth dynamical systems have
similar dynamics, but the conditions for such similarity are not well
understood. Here we show that by smoothing out a piecewise-smooth system one
may obtain dynamics that is inconsistent with the accepted wisdom --- so-called
Filippov dynamics --- at a discontinuity, even in the piecewise-smooth limit.
By subjecting the system to white noise, we show that these discrepancies can
be understood in terms of potential wells that allow solutions to dwell at the
discontinuity for long times. Moreover we show that spurious dynamics will
revert to Filippov dynamics, with a small degree of stochasticity, when the
noise magnitude is sufficiently large compared to the order of smoothing. We
apply the results to a model of a dry-friction oscillator, where spurious
dynamics (inconsistent with Filippov's convention or with Coulomb's model of
friction) can account for different coefficients of static and kinetic
friction, but under sufficient noise the system reverts to dynamics consistent
with Filippov's convention (and with Coulomb-like friction).Comment: submitted to: Nonlinear Dynamic
A Method for the Combination of Stochastic Time Varying Load Effects
The problem of evaluating the probability that a structure becomes unsafe under a
combination of loads, over a given time period, is addressed. The loads and load effects
are modeled as either pulse (static problem) processes with random occurrence time, intensity and a specified shape or intermittent continuous (dynamic problem) processes which
are zero mean Gaussian processes superimposed 'on a pulse process. The load coincidence
method is extended to problems with both nonlinear limit states and dynamic responses,
including the case of correlated dynamic responses. The technique of linearization of a
nonlinear limit state commonly used in a time-invariant problem is investigated for timevarying
combination problems, with emphasis on selecting the linearization point. Results
are compared with other methods, namely the method based on upcrossing rate, simpler
combination rules such as Square Root of Sum of Squares and Turkstra's rule. Correlated
effects among dynamic loads are examined to see how results differ from correlated static
loads and to demonstrate which types of load dependencies are most important, i.e., affect'
the exceedance probabilities the most.
Application of the load coincidence method to code development is briefly discussed.National Science Foundation Grants CME 79-18053 and CEE 82-0759
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