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6.5.2 ParticleMesh Methods (PM and P3M):
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 shortrange interactions, Hockney suggested a method
to speed up the dynamics due to the farreaching potential.
Consider a square cell of subcells of side length
. Each subcell should contain
an average of superparticles.
Equation of motion for superparticle :

(6.15) 
where
is the electrostatic or gravitational potential.
It is determined by charge (or mass) density
defined by the positions of all superparticles.
To compute
at time at the centers of the
subcells, the given configuration of superions is first replaced by
latticelike 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. 6.9).
Figure 6.9:
Area weighting according to the CIC (cloudincell) 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
of some
superparticle in cell is then
Next we integrate the equation of motion 6.15:
which completes the time step. Here is the prescription once more:
Figure 6.10:
Particlemesh method

The PM technique considers only the action of the total field
by the distant superparticles. If the shortranged forces may
not be neglected, as in the simulation of molten salts, the
Born, Huggins and Mayer potential is included (see
Table 6.1):
Combining the PM method and the molecular dynamics technique [HOCKNEY 81],
we may take into account the shortranged forces up to a certain interparticle
distance, while the longranged contributions are included
by the particlemesh procedure. This combination of particleparticle and
particlemesh methods has come to be called PPPM or
technique.
Next: 6.6 Stochastic Dynamics
Up: 6.5 Particles and Fields
Previous: 6.5.1 Ewald summation
Franz J. Vesely Oct 2005
See also: "Computational Physics  An Introduction," KluwerPlenum 2001