219 research outputs found
Transition state theory for wave packet dynamics. II. Thermal decay of Bose-Einstein condensates with long-range interaction
We apply transition state theory to coupled Gaussian wave packets and
calculate thermal decay rates of Bose-Einstein condensates with additional
long-range interaction. The ground state of such a condensate is metastable if
the contact interaction is attractive and a sufficient thermal excitation may
lead to its collapse. The use of transition state theory is made possible by
describing the condensate within a variational framework and locally mapping
the variational parameters to classical phase space as has been demonstrated in
the preceding paper [A. Junginger, J. Main, and G. Wunner, submitted to J.
Phys. A]. We apply this procedure to Gaussian wave packets and present results
for condensates with monopolar 1/r-interaction comparing decay rates obtained
by using different numbers of coupled Gaussian trial wave functions as well as
different normal form orders.Comment: 14 pages, 4 figures, submitted to J. Phys.
Building capabilities in the construction sector:absorptive capacity of architectural and engineering medium-sized enterprises
Transition state theory for wave packet dynamics. I. Thermal decay in metastable Schr\"odinger systems
We demonstrate the application of transition state theory to wave packet
dynamics in metastable Schr\"odinger systems which are approached by means of a
variational ansatz for the wave function and whose dynamics is described within
the framework of a time-dependent variational principle. The application of
classical transition state theory, which requires knowledge of a classical
Hamilton function, is made possible by mapping the variational parameters to
classical phase space coordinates and constructing an appropriate Hamiltonian
in action variables. This mapping, which is performed by a normal form
expansion of the equations of motion and an additional adaptation to the energy
functional, as well as the requirements to the variational ansatz are discussed
in detail. The applicability of the procedure is demonstrated for a cubic model
potential for which we calculate thermal decay rates of a frozen Gaussian wave
function. The decay rate obtained with a narrow trial wave function agrees
perfectly with the results using the classical normal form of the corresponding
point particle. The results with a broader trial wave function go even beyond
the classical approach, i.e., they agree with those using the quantum normal
form. The method presented here will be applied to Bose-Einstein condensates in
the following paper [A. Junginger, M. Dorwarth, J. Main, and G. Wunner,
submitted to J. Phys. A].Comment: 21 pages, 3 figures, submitted to J. Phys.
Geometrical Models of the Phase Space Structures Governing Reaction Dynamics
Hamiltonian dynamical systems possessing equilibria of stability type display \emph{reaction-type
dynamics} for energies close to the energy of such equilibria; entrance and
exit from certain regions of the phase space is only possible via narrow
\emph{bottlenecks} created by the influence of the equilibrium points. In this
paper we provide a thorough pedagogical description of the phase space
structures that are responsible for controlling transport in these problems. Of
central importance is the existence of a \emph{Normally Hyperbolic Invariant
Manifold (NHIM)}, whose \emph{stable and unstable manifolds} have sufficient
dimensionality to act as separatrices, partitioning energy surfaces into
regions of qualitatively distinct behavior. This NHIM forms the natural
(dynamical) equator of a (spherical) \emph{dividing surface} which locally
divides an energy surface into two components (`reactants' and `products'), one
on either side of the bottleneck. This dividing surface has all the desired
properties sought for in \emph{transition state theory} where reaction rates
are computed from the flux through a dividing surface. In fact, the dividing
surface that we construct is crossed exactly once by reactive trajectories, and
not crossed by nonreactive trajectories, and related to these properties,
minimizes the flux upon variation of the dividing surface.
We discuss three presentations of the energy surface and the phase space
structures contained in it for 2-degree-of-freedom (DoF) systems in the
threedimensional space , and two schematic models which capture many of
the essential features of the dynamics for -DoF systems. In addition, we
elucidate the structure of the NHIM.Comment: 44 pages, 38 figures, PDFLaTe
The efficient computation of transition state resonances and reaction rates from a quantum normal form
A quantum version of a recent formulation of transition state theory in {\em
phase space} is presented. The theory developed provides an algorithm to
compute quantum reaction rates and the associated Gamov-Siegert resonances with
very high accuracy. The algorithm is especially efficient for
multi-degree-of-freedom systems where other approaches are no longer feasible.Comment: 4 pages, 3 figures, revtex
The Quantum-Classical Correspondence in Polygonal Billiards
We show that wave functions in planar rational polygonal billiards (all
angles rationally related to Pi) can be expanded in a basis of quasi-stationary
and spatially regular states. Unlike the energy eigenstates, these states are
directly related to the classical invariant surfaces in the semiclassical
limit. This is illustrated for the barrier billiard. We expect that these
states are also present in integrable billiards with point scatterers or
magnetic flux lines.Comment: 8 pages, 9 figures (in reduced quality), to appear in PR
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