321 research outputs found
An Efficient Runge-Kutta (4,5) pair
A pair of explicit Runge-Kutta formulas of orders 4 and 5 is derived. It is significantly more efficient than the Fehlberg and Dormand-Prince pairs, and by standard measures it is of at least as high quality. There are two independent estimates of the local error. The local error of the interpolant is, to leading order, a problem-independent function of the local error at the end of the step
Path integrals and symmetry breaking for optimal control theory
This paper considers linear-quadratic control of a non-linear dynamical
system subject to arbitrary cost. I show that for this class of stochastic
control problems the non-linear Hamilton-Jacobi-Bellman equation can be
transformed into a linear equation. The transformation is similar to the
transformation used to relate the classical Hamilton-Jacobi equation to the
Schr\"odinger equation. As a result of the linearity, the usual backward
computation can be replaced by a forward diffusion process, that can be
computed by stochastic integration or by the evaluation of a path integral. It
is shown, how in the deterministic limit the PMP formalism is recovered. The
significance of the path integral approach is that it forms the basis for a
number of efficient computational methods, such as MC sampling, the Laplace
approximation and the variational approximation. We show the effectiveness of
the first two methods in number of examples. Examples are given that show the
qualitative difference between stochastic and deterministic control and the
occurrence of symmetry breaking as a function of the noise.Comment: 21 pages, 6 figures, submitted to JSTA
Singularity subtraction for nonlinear weakly singular integral equations of the second kind
The singularity subtraction technique for computing an approximate solution of a linear weakly singular Fredholm integral equation of the second kind is generalized to the case of a nonlinear integral equation of the same kind. Convergence of the sequence of approximate solutions to the exact one is proved under mild standard hypotheses on the nonlinear factor, and on the sequence of quadrature rules used to build an approximate equation. A numerical example is provided with a Hammerstein operator to illustrate some practical aspects of effective computations.The third author was partially supported by CMat (UID/MAT/00013/2013), and the second and fourth authors were partially supported by CMUP (UID/ MAT/ 00144/2013), which are funded by FCT (Portugal) with national funds (MCTES) and European structural funds (FEDER) under the partnership agreement PT2020
Theoretical analysis of the implementation of a quantum phase gate with neutral atoms on atom chips
We present a detailed, realistic analysis of the implementation of a proposal
for a quantum phase gate based on atomic vibrational states, specializing it to
neutral rubidium atoms on atom chips. We show how to create a double--well
potential with static currents on the atom chips, using for all relevant
parameters values that are achieved with present technology. The potential
barrier between the two wells can be modified by varying the currents in order
to realize a quantum phase gate for qubit states encoded in the atomic external
degree of freedom. The gate performance is analyzed through numerical
simulations; the operation time is ~10 ms with a performance fidelity above
99.9%. For storage of the state between the operations the qubit state can be
transferred efficiently via Raman transitions to two hyperfine states, where
its decoherence is strongly inhibited. In addition we discuss the limits
imposed by the proximity of the surface to the gate fidelity.Comment: 9 pages, 5 color figure
Quantum control theory for coupled 2-electron dynamics in quantum dots
We investigate optimal control strategies for state to state transitions in a
model of a quantum dot molecule containing two active strongly interacting
electrons. The Schrodinger equation is solved nonperturbatively in conjunction
with several quantum control strategies. This results in optimized electric
pulses in the THz regime which can populate combinations of states with very
short transition times. The speedup compared to intuitively constructed pulses
is an order of magnitude. We furthermore make use of optimized pulse control in
the simulation of an experimental preparation of the molecular quantum dot
system. It is shown that exclusive population of certain excited states leads
to a complete suppression of spin dephasing, as was indicated in Nepstad et al.
[Phys. Rev. B 77, 125315 (2008)].Comment: 24 pages, 9 figure
A ferromagnet with a glass transition
We introduce a finite-connectivity ferromagnetic model with a three-spin
interaction which has a crystalline (ferromagnetic) phase as well as a glass
phase. The model is not frustrated, it has a ferromagnetic equilibrium phase at
low temperature which is not reached dynamically in a quench from the
high-temperature phase. Instead it shows a glass transition which can be
studied in detail by a one step replica-symmetry broken calculation. This spin
model exhibits the main properties of the structural glass transition at a
solvable mean-field level.Comment: 7 pages, 2 figures, uses epl.cls (included
Vesicle shape, molecular tilt, and the suppression of necks
Can the presence of molecular-tilt order significantly affect the shapes of
lipid bilayer membranes, particularly membrane shapes with narrow necks?
Motivated by the propensity for tilt order and the common occurrence of narrow
necks in the intermediate stages of biological processes such as endocytosis
and vesicle trafficking, we examine how tilt order inhibits the formation of
necks in the equilibrium shapes of vesicles. For vesicles with a spherical
topology, point defects in the molecular order with a total strength of
are required. We study axisymmetric shapes and suppose that there is a
unit-strength defect at each pole of the vesicle. The model is further
simplified by the assumption of tilt isotropy: invariance of the energy with
respect to rotations of the molecules about the local membrane normal. This
isotropy condition leads to a minimal coupling of tilt order and curvature,
giving a high energetic cost to regions with Gaussian curvature and tilt order.
Minimizing the elastic free energy with constraints of fixed area and fixed
enclosed volume determines the allowed shapes. Using numerical calculations, we
find several branches of solutions and identify them with the branches
previously known for fluid membranes. We find that tilt order changes the
relative energy of the branches, suppressing thin necks by making them costly,
leading to elongated prolate vesicles as a generic family of tilt-ordered
membrane shapes.Comment: 10 pages, 7 figures, submitted to Phy. Rew.
Infrared emission spectrum and potentials of and states of Xe excimers produced by electron impact
We present an investigation of the Xe excimer emission spectrum
observed in the near infrared range about 7800 cm in pure Xe gas and in
an Ar (90%) --Xe (10%) mixture and obtained by exciting the gas with energetic
electrons. The Franck--Condon simulation of the spectrum shape suggests that
emission stems from a bound--free molecular transition never studied before.
The states involved are assigned as the bound state with atomic limit and the dissociative state with limit. Comparison with the spectrum simulated by using theoretical
potentials shows that the dissociative one does not reproduce correctly the
spectrum features.Comment: 4 pages, 3 figures, submitted to Phys. Rev. Let
Josephson Junctions as Threshold Detectors of the Full Counting Statistics: Open issues
I study the dynamics of a Josephson junction serving as a threshold detector
of fluctuations which is subjected to a general non-equilibrium electronic
noise source whose characteristics is to be determined by the junction. This
experimental setup has been proposed several years ago as a prospective scheme
for determining the Full Counting Statistics of the electronic noise source.
Despite of intensive theoretical as well as experimental research in this
direction the promise has not been quite fulfilled yet and I will discuss what
are the unsolved issues. First, I review a general theory for the calculation
of the exponential part of the non-equilibrium switching rates of the junction
and compare its predictions with previous results found in different limiting
cases by several authors. I identify several possible weak points in the
previous studies and I report a new analytical result for the linear correction
to the rate due to the third cumulant of a non-Gaussian noise source in the
limit of a very weak junction damping. The various analytical predictions are
then compared with the results of the developed numerical method. Finally, I
analyze the status of the so-far publicly available experimental data with
respect to the theoretical predictions and discuss briefly the suitability of
the present experimental schemes in view of their potential to measure the
whole FCS of non-Gaussian noise sources as well as their relation to the
available theories.Comment: 15 pages, 2 figures; Proceedings of UPoN 2008, Lyon, June 2008; v2:
minor text changes, as close to the published version as possibl
Stationary solutions of driven fourth- and sixth-order Cahn-Hilliard type equations
New types of stationary solutions of a one-dimensional driven sixth-order
Cahn-Hilliard type equation that arises as a model for epitaxially growing
nano-structures such as quantum dots, are derived by an extension of the method
of matched asymptotic expansions that retains exponentially small terms. This
method yields analytical expressions for far-field behavior as well as the
widths of the humps of these spatially non-monotone solutions in the limit of
small driving force strength which is the deposition rate in case of epitaxial
growth. These solutions extend the family of the monotone kink and antikink
solutions. The hump spacing is related to solutions of the Lambert
function. Using phase space analysis for the corresponding fifth-order
dynamical system, we use a numerical technique that enables the efficient and
accurate tracking of the solution branches, where the asymptotic solutions are
used as initial input. Additionally, our approach is first demonstrated for the
related but simpler driven fourth-order Cahn-Hilliard equation, also known as
the convective Cahn-Hilliard equation
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