422 research outputs found
Exact equations for smoothed Wigner transforms and homogenization of wave propagation
The Wigner Transform (WT) has been extensively used in the formulation of
phase-space models for a variety of wave propagation problems including
high-frequency limits, nonlinear and random waves. It is well known that the WT
features counterintuitive 'interference terms', which often make computation
impractical. In this connection, we propose the use of the smoothed Wigner
Transform (SWT), and derive new, exact equations for it, covering a broad class
of wave propagation problems. Equations for spectrograms are included as a
special case. The 'taming' of the interference terms by the SWT is illustrated,
and an asymptotic model for the Schroedinger equation is constructed and
numerically verified.Comment: 16 pages, 8 figure
Semiclassical Completely Integrable Systems : Long-Time Dynamics And Observability Via Two-Microlocal Wigner Measures
We look at the long-time behaviour of solutions to a semi-classical
Schr\"odinger equation on the torus. We consider time scales which go to
infinity when the semi-classical parameter goes to zero and we associate with
each time-scale the set of semi-classical measures associated with all possible
choices of initial data. On each classical invariant torus, the structure of
semi-classical measures is described in terms of two-microlocal measures,
obeying explicit propagation laws. We apply this construction in two
directions. We first analyse the regularity of semi-classical measures, and we
emphasize the existence of a threshold : for time-scales below this threshold,
the set of semi-classical measures contains measures which are singular with
respect to Lebesgue measure in the "position" variable, while at (and beyond)
the threshold, all the semi-classical measures are absolutely continuous in the
"position" variable, reflecting the dispersive properties of the equation.
Second, the techniques of two- microlocal analysis introduced in the paper are
used to prove semiclassical observability estimates. The results apply as well
to general quantum completely integrable systems.Comment: This article contains and develops the results of hal-00765928. arXiv
admin note: substantial text overlap with arXiv:1211.151
Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation
We consider the dispersive logarithmic Schrödinger equation in a semi-classical scaling. We extend the results about the large time behaviour of the solution (dispersion faster than usual with an additional logarithmic factor, convergence of the rescaled modulus of the solution to a universal Gaussian profile) to the case with semi-classical constant. We also provide a sharp convergence rate to the Gaussian profile in Kantorovich-Rubinstein metric through a detailed analysis of the Fokker-Planck equation satisfied by this modulus. Moreover, we perform the semiclassical limit of this equation thanks to the Wigner Transform in order to get a (Wigner) measure. We show that those two features are compatible and the density of a Wigner Measure has the same large time behaviour as the modulus of the solution of the logarithmic Schrödinger equation. Lastly, we discuss about the related kinetic equation (which is the Kinetic Isothermal Euler System) and its formal properties, enlightened by the previous results and a new class of explicit solutions
Long-time dynamics of completely integrable Schr\"odinger flows on the torus
In this article, we are concerned with long-time behaviour of solutions to a
semi-classical Schr\"odinger-type equation on the torus. We consider time
scales which go to infinity when the semi-classical parameter goes to zero and
we associate with each time-scale the set of semi-classical measures associated
with all possible choices of initial data. We emphasize the existence of a
threshold: for time-scales below this threshold, the set of semi-classical
measures contains measures which are singular with respect to Lebesgue measure
in the "position" variable, while at (and beyond) the threshold, all the
semi-classical measures are absolutely continuous in the "position" variable.Comment: 41 page
Measuring nonadiabaticity of molecular quantum dynamics with quantum fidelity and with its efficient semiclassical approximation
We propose to measure nonadiabaticity of molecular quantum dynamics
rigorously with the quantum fidelity between the Born-Oppenheimer and fully
nonadiabatic dynamics. It is shown that this measure of nonadiabaticity applies
in situations where other criteria, such as the energy gap criterion or the
extent of population transfer, fail. We further propose to estimate this
quantum fidelity efficiently with a generalization of the dephasing
representation to multiple surfaces. Two variants of the multiple-surface
dephasing representation (MSDR) are introduced, in which the nuclei are
propagated either with the fewest-switches surface hopping (FSSH) or with the
locally mean field dynamics (LMFD). The LMFD can be interpreted as the
Ehrenfest dynamics of an ensemble of nuclear trajectories, and has been used
previously in the nonadiabatic semiclassical initial value representation. In
addition to propagating an ensemble of classical trajectories, the MSDR
requires evaluating nonadiabatic couplings and solving the Schr\"{o}dinger (or
more generally, the quantum Liouville-von Neumann) equation for a single
discrete degree of freedom. The MSDR can be also used to measure the importance
of other terms present in the molecular Hamiltonian, such as diabatic
couplings, spin-orbit couplings, or couplings to external fields, and to
evaluate the accuracy of quantum dynamics with an approximate nonadiabatic
Hamiltonian. The method is tested on three model problems introduced by Tully,
on a two-surface model of dissociation of NaI, and a three-surface model
including spin-orbit interactions. An example is presented that demonstrates
the importance of often-neglected second-order nonadiabatic couplings.Comment: 14 pages, 4 figures, submitted to J. Chem. Phy
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