Novel simulation methods for Coulomb and hydrodynamic interactions
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Abstract
This thesis presents new methods to simulate systems with
nhydrodynamic and electrostatic interactions.
Part 1 is devoted to computer simulations of Brownian particles with
hydrodynamic interactions. The main influence of the solvent on the
dynamics of Brownian particles is that it mediates hydrodynamic interactions. In
the method, this is simulated by numerical solution of the
Navier--Stokes equation on a lattice. To this end, the Lattice--Boltzmann method is used, namely its
D3Q19 version. This model is capable to simulate compressible flow. It
gives us the advantage to treat dense systems, in particular away from
thermal equilibrium. The Lattice--Boltzmann equation is coupled to the
particles via a friction force. In addition to this force, acting on {\it point} particles, we construct another coupling force, which
comes from the pressure tensor. The coupling is purely local,
i.~e. the algorithm scales linearly with the total number of
particles. In order to be able to map the physical properties of the
Lattice--
Boltzmann fluid onto a Molecular Dynamics (MD) fluid, the
case of an almost incompressible flow is considered. The
Fluctuation--Dissipation theorem for the hybrid coupling is analyzed,
and a geometric interpretation of the friction coefficient in terms of
a Stokes radius is given.
Part 2 is devoted to the simulation of charged particles. We present
a novel method for obtaining Coulomb interactions as the potential of
mean force between charges which are dynamically coupled to a local
electromagnetic field. This algorithm scales linearly, too. We focus on the Molecular Dynamics version
of the method and show that it is intimately related to the Car--Parrinello approach, while being equivalent to solving Maxwell's
equations with freely adjustable speed of light. The Lagrangian
formulation of the coupled particles--fields system is derived. The
quasi--Hamiltonian dynamics of the system is studied in great
detail. For implementation on the computer, the equations of
motion are
discretized with respect to both space and time. The discretization of
the electromagnetic fields on a lattice, as well as the
interpolation of the particle charges on the lattice is given. The algorithm is as
local as possible: Only nearest neighbors sites of the lattice are
interacting with a charged particle. Unphysical self--energies arise
as a result of the lattice interpolation of charges, and are corrected
by a subtraction scheme based on the exact lattice Green's
function. The method allows easy parallelization using standard domain
decomposition. Some benchmarking results of the algorithm are presented
and discussed.