1,306 research outputs found
A blob method for diffusion
As a counterpoint to classical stochastic particle methods for diffusion, we
develop a deterministic particle method for linear and nonlinear diffusion. At
first glance, deterministic particle methods are incompatible with diffusive
partial differential equations since initial data given by sums of Dirac masses
would be smoothed instantaneously: particles do not remain particles. Inspired
by classical vortex blob methods, we introduce a nonlocal regularization of our
velocity field that ensures particles do remain particles, and we apply this to
develop a numerical blob method for a range of diffusive partial differential
equations of Wasserstein gradient flow type, including the heat equation, the
porous medium equation, the Fokker-Planck equation, the Keller-Segel equation,
and its variants. Our choice of regularization is guided by the Wasserstein
gradient flow structure, and the corresponding energy has a novel form,
combining aspects of the well-known interaction and potential energies. In the
presence of a confining drift or interaction potential, we prove that
minimizers of the regularized energy exist and, as the regularization is
removed, converge to the minimizers of the unregularized energy. We then
restrict our attention to nonlinear diffusion of porous medium type with at
least quadratic exponent. Under sufficient regularity assumptions, we prove
that gradient flows of the regularized energies converge to solutions of the
porous medium equation. As a corollary, we obtain convergence of our numerical
blob method, again under sufficient regularity assumptions. We conclude by
considering a range of numerical examples to demonstrate our method's rate of
convergence to exact solutions and to illustrate key qualitative properties
preserved by the method, including asymptotic behavior of the Fokker-Planck
equation and critical mass of the two-dimensional Keller-Segel equation
How accurate is density functional theory at predicting dipole moments? An assessment using a new database of 200 benchmark values
Dipole moments are a simple, global measure of the accuracy of the electron
density of a polar molecule. Dipole moments also affect the interactions of a
molecule with other molecules as well as electric fields. To directly assess
the accuracy of modern density functionals for calculating dipole moments, we
have developed a database of 200 benchmark dipole moments, using coupled
cluster theory through triple excitations, extrapolated to the complete basis
set limit. This new database is used to assess the performance of 88 popular or
recently developed density functionals. The results suggest that double hybrid
functionals perform the best, yielding dipole moments within about 3.6-4.5%
regularized RMS error versus the reference values---which is not very different
from the 4% regularized RMS error produced by coupled cluster singles and
doubles. Many hybrid functionals also perform quite well, generating
regularized RMS errors in the 5-6% range. Some functionals however exhibit
large outliers and local functionals in general perform less well than hybrids
or double hybrids.Comment: Added several double hybrid functionals, most of which turned out to
be better than any functional from Rungs 1-4 of Jacob's ladder and are
actually competitive with CCS
Semilocal momentum-space regularized chiral two-nucleon potentials up to fifth order
We introduce new semilocal two-nucleon potentials up to fifth order in the
chiral expansion. We employ a simple regularization approach for the
pion-exchange contributions which (i) maintains the long-range part of the
interaction, (ii) is implemented in momentum space and (iii) can be
straightforwardly applied to regularize many-body forces and current operators.
We discuss in detail the two-nucleon contact interactions at fourth order and
demonstrate that three terms out of fifteen used in previous calculations can
be eliminated via suitably chosen unitary transformations. The removal of the
redundant contact terms results in a drastic simplification of the fits to
scattering data and leads to interactions which are much softer (i.e. more
perturbative) than our recent semilocal coordinate-space regularized
potentials. Using the pion-nucleon low-energy constants from matching
pion-nucleon Roy-Steiner equations to chiral perturbation theory, we perform a
comprehensive analysis of nucleon-nucleon scattering and the deuteron
properties up to fifth chiral order and study the impact of the leading F-wave
two-nucleon contact interactions which appear at sixth order. The resulting
chiral potentials lead to an outstanding description of the proton-proton and
neutron-proton scattering data from the self-consistent Granada-2013 database
below the pion production threshold, which is significantly better than for any
other chiral potential. For the first time, the chiral potentials match in
precision and even outperform the available high-precision phenomenological
potentials, while the number of adjustable parameters is, at the same time,
reduced by about ~40%. Last but not least, we perform a detailed error analysis
and, in particular, quantify for the first time the statistical uncertainties
of the fourth- and the considered sixth-order contact interactions.Comment: 57 pages, 17 figures, 19 table
S-, P- and D-wave resonances in positronium-sodium and positronium-potassium scattering
Scattering of positronium (Ps) by sodium and potassium atoms has been
investigated employing a three-Ps-state coupled-channel model with Ps(1s,2s,2p)
states using a time-reversal-symmetric regularized electron-exchange model
potential fitted to reproduce accurate theoretical results for PsNa and PsK
binding energies. We find a narrow S-wave singlet resonance at 4.58 eV of width
0.002 eV in the Ps-Na system and at 4.77 eV of width 0.003 eV in the Ps-K
system. Singlet P-wave resonances in both systems are found at 5.07 eV of width
0.3 eV. Singlet D-wave structures are found at 5.3 eV in both systems. We also
report results for elastic and Ps-excitation cross sections for Ps scattering
by Na and K.Comment: 9 pages, 5 figures, Accepted in Journal of Physics
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