798 research outputs found
Explicit large nuclear charge limit of electronic ground states for Li, Be, B, C, N, O, F, Ne and basic aspects of the periodic table
This paper is concerned with the Schrödinger equation for atoms and ions with to 10 electrons. In the asymptotic limit of large nuclear charge , we determine explicitly the low-lying energy levels and eigenstates. The asymptotic energies and wavefunctions are in good quantitative agreement with experimental data for positive ions, and in excellent qualitative agreement even for neutral atoms (). In particular, the predicted ground state spin and angular momentum quantum numbers ( for He, Be, Ne, for H and Li, for N, for B and F, and for C and O) agree with experiment in every case. The asymptotic Schrödinger ground states agree, up to small corrections, with the semiempirical hydrogen orbital configurations developed by Bohr, Hund, and Slater to explain the periodic table. In rare cases where our results deviate from this picture, such as the ordering of the lowest and states of the carbon isoelectronic sequence, experiment confirms our predictions and not Hund's
A derivation of the Liouville equation for hard particle dynamics with non-conservative interactions
The Liouville equation is of fundamental importance in the derivation of
continuum models for physical systems which are approximated by interacting
particles. However, when particles undergo instantaneous interactions such as
collisions, the derivation of the Liouville equation must be adapted to exclude
non-physical particle positions, and include the effect of instantaneous
interactions. We present the weak formulation of the Liouville equation for
interacting particles with general particle dynamics and interactions, and
discuss the results using an example
Modelling inelastic granular media using Dynamical Density Functional Theory
We construct a new mesoscopic model for granular media using Dynamical
Density Functional Theory (DDFT). The model includes both a collision operator
to incorporate inelasticity and the Helmholtz free energy functional to account
for external potentials, interparticle interactions and volume exclusion. We
use statistical data from event-driven microscopic simulations to determine the
parameters not given analytically by the closure relations used to derive the
DDFT. We numerically demonstrate the crucial effects of each term in the DDFT,
and the importance of including an accurately parametrised pair correlation
function
The Singular Hydrodynamic Interactions Between Two Spheres In Stokes Flow
We study exact solutions for the slow viscous flow of an infinite liquid
caused by two rigid spheres approaching each either along or parallel to their
line of centres, valid at all separations. This goes beyond the applicable
range of existing solutions for singular hydrodynamic interactions (HIs) which,
for practical applications, are limited to the near-contact or far field region
of the flow. For the normal component of the HI, by use of a bipolar coordinate
system, we derive the stream function for the flow as and a formula
for the singular (squeeze) force between the spheres as an infinite series. We
also obtain the asymptotic behaviour of the forces as the nondimensional
separation between the spheres goes to zero and infinity, rigorously confirming
and improving upon known results relevant to a widely accepted lubrication
theory. Additionally, we recover the force on a sphere moving perpendicularly
to a plane as a special case. For the tangential component, again by using a
bipolar coordinate system, we obtain the corresponding infinite series
expression of the (shear) singular force between the spheres. All results hold
for retreating spheres, consistent with the reversibility of Stokes flow. We
demonstrate substantial differences in numerical simulations of colloidal
fluids when using the present theory compared with existing multipole methods.
Furthermore, we show that the present theory preserves positive definiteness of
the resistance matrix in a number of situations in which
positivity is destroyed for multipole/perturbative methods.Comment: 28 pages, 12 Figure
MultiShape: A Spectral Element Method, with Applications to Dynamic Density Functional Theory and PDE-Constrained Optimization
A numerical framework is developed to solve various types of PDEs on
complicated domains, including steady and time-dependent, non-linear and
non-local PDEs, with different boundary conditions that can also include
non-linear and non-local terms. This numerical framework, called MultiShape, is
a class in Matlab, and the software is open source. We demonstrate that
MultiShape is compatible with other numerical methods, such as
differential--algebraic equation solvers and optimization algorithms. The
numerical implementation is designed to be user-friendly, with most of the
set-up and computations done automatically by MultiShape and with intuitive
operator definition, notation, and user-interface. Validation tests are
presented, before we introduce three examples motivated by applications in
Dynamic Density Functional Theory and PDE-constrained optimization,
illustrating the versatility of the method
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