Huge model systems with potentials (celestial masses or microscopic
charged particles) may be described by introducing
``superparticles''consisting of some ions, electrons,
or stars.[HOCKNEY 81]
Neglecting the short-range interactions, Hockney suggested a method
to speed up the dynamics due to the far-reaching potential.
Consider a square cell of subcells of side length
. Each subcell should contain
an average of superparticles.
Equation of motion for superparticle :
is the electrostatic or gravitational potential.
It is determined by charge (or mass) density
defined by the positions of all superparticles.
at time at the centers of the
subcells, the given configuration of superions is first replaced by
lattice-like charge distribution . The easiest
discretization method is the
nearest grid point (NGP) rule:
A more refined method of charge assignment than the NGP rule is
the cloud in cell (CIC) prescription:
Appropriate fractions of each charge are distributed to the four (2D) or
eight (3D) nearest cell centers.
These fractions, or weights, are assigned in proportion to the overlap
areas of a square of side length , centered around the particle
under consideration, and the respective neighbor cells (see Fig. 9.2).
Area weighting according to the CIC (cloud-in-cell) rule
The next step is the calculation of the potential produced by the
charge lattice. Any of the relaxation method of Chapter 5 may be
applied, but the FACR technique as developed by Hockney
is preferred.[HOCKNEY 81]
Result: values of the potential at the cell
centers. The field strength at the position
superparticle in cell is then
which completes the time step. Here is the prescription once more:
The PM technique considers only the action of the total field
by the distant superparticles. If the short-ranged forces may
not be neglected, as in the simulation of molten salts, the
Born, Huggins and Mayer potential is included (see
Combining the PM method and the molecular dynamics technique [HOCKNEY 81],
we may take into account the short-ranged forces up to a certain interparticle
distance, while the long-ranged contributions are included
by the particle-mesh procedure. This combination of particle-particle and
particle-mesh methods has come to be called PPPM- or
F. J. Vesely / University of Vienna