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## 6.5.2 Particle-Mesh 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 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 :

 (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 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. 6.9).

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
 (6.16) (6.17)

Next we integrate the equation of motion 6.15:
 (6.18)

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 Table 6.1):

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 technique.

Next: 6.6 Stochastic Dynamics Up: 6.5 Particles and Fields Previous: 6.5.1 Ewald summation
Franz J. Vesely Oct 2005