156 research outputs found
Screened exchange corrections to the random phase approximation from many-body perturbation theory
The random phase approximation (RPA) systematically overestimates the
magnitude of the correlation energy and generally underestimates cohesive
energies. This originates in part from the complete lack of exchange terms,
which would otherwise cancel Pauli exclusion principle violating (EPV)
contributions. The uncanceled EPV contributions also manifest themselves in
form of an unphysical negative pair density of spin-parallel electrons close to
electron-electron coalescence.
We follow considerations of many-body perturbation theory to propose an
exchange correction that corrects the largest set of EPV contributions while
having the lowest possible computational complexity. The proposed method
exchanges adjacent particle/hole pairs in the RPA diagrams, considerably
improving the pair density of spin-parallel electrons close to coalescence in
the uniform electron gas (UEG). The accuracy of the correlation energy is
comparable to other variants of Second Order Screened Exchange (SOSEX)
corrections although it is slightly more accurate for the spin-polarized UEG.
Its computational complexity scales as or
in orbital space or real space, respectively. Its memory requirement scales as
Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces
In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted Lq-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt A-class. In the Besov space case we have the restriction that the microscopic parameter equals to q. Going beyond the A-range, where p is the integrability parameter of the Besov or Triebel-Lizorkin space under consideration, yields extra flexibility in the sharp regularity of the boundary inhomogeneities. This extra flexibility allows us to treat rougher boundary data and provides a quantitative smoothing effect on the interior of the domain. The main ingredient is an analysis of anisotropic Poisson operators
Reduction Methods in Climate Dynamics -- A Brief Review
We review a range of reduction methods that have been, or may be useful for
connecting models of the Earth's climate system of differing complexity. We
particularly focus on methods where rigorous reduction is possible. We aim to
highlight the main mathematical ideas of each reduction method and also provide
several benchmark examples from climate modelling
Geometric blow-up of a dynamic Turing instability in the Swift-Hohenberg equation
We present a rigorous analysis of the slow passage through a Turing
bifurcation in the Swift-Hohenberg equation using a novel approach based on
geometric blow-up. We show that the formally derived multiple scales ansatz
which is known from classical modulation theory can be adapted for use in the
fast-slow setting, by reformulating it as a blow-up transformation. This leads
to dynamically simpler modulation equations posed in the blown-up space, via a
formal procedure which directly extends the established approach to the
time-dependent setting. The modulation equations take the form of
non-autonomous Ginzburg-Landau equations, which can be analysed within the
blow-up. The asymptotics of solutions in weighted Sobelev spaces are given in
two different cases: (i) A symmetric case featuring a delayed loss of
stability, and (ii) A second case in which the symmetry is broken by a source
term. In order to characterise the dynamics of the Swift-Hohenberg equation
itself we derive rigorous estimates on the error of the dynamic modulation
approximation. These estimates are obtained by bounding weak solutions to an
evolution equation for the error which is also posed in the blown-up space.
Using the error estimates obtained, we are able to infer the asymptotics of a
large class of solutions to the dynamic Swift-Hohenberg equation. We provide
rigorous asymptotics for solutions in both cases (i) and (ii). We also prove
the existence of the delayed loss of stability in the symmetric case (i), and
provide a lower bound for the delay time.Comment: 69 pages. A notational misprint in equation (17) has been correcte
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