next up previous
Next: 6.6 Stochastic Dynamics Up: 6.5 Particles and Fields Previous: 6.5.1 Ewald summation

6.5.2 Particle-Mesh Methods (PM and P3M):

Huge model systems with $1/r$ potentials (celestial masses or microscopic charged particles) may be described by introducing ``superparticles''consisting of some $10^{4}-10^{8}$ 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 $1/r$ potential.

Consider a square cell of $M \times M$ subcells of side length $\Delta x = \Delta y = \Delta l$. Each subcell should contain an average of $10-100$ superparticles.
Equation of motion for superparticle $k$:

\ddot{\mbox{$\bf r$}}_{k} = -\frac{q_{k}}{m_{k}} \mbox{$\bf\...
...\Phi =
\frac{q_{k}}{m_{k}}\mbox{$\bf E$}(\mbox{$\bf r$}_{k})
\end{displaymath} (6.15)

where $\Phi(\mbox{$\bf r$})$ is the electrostatic or gravitational potential. It is determined by charge (or mass) density $\rho(\mbox{$\bf r$},t)$ defined by the positions of all superparticles.

To compute $\Phi(\mbox{$\bf r$})$ at time $t_{n}$ at the centers of the subcells, the given configuration of superions is first replaced by lattice-like charge distribution $\rho_{i,j}$. The easiest discretization method is the nearest grid point (NGP) rule:

\rho_{i,j} =\frac{1}{(\Delta l)^{2}} \sum_{k=1}^{N} q_{k}
...l}-i \right)
\,\delta \left( \frac{y_{k}}{\Delta l}-j \right)

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 $\Delta l$, centered around the particle under consideration, and the respective neighbor cells (see Fig. 6.9).

Figure 6.9: 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 $\Phi_{i,j}$ at the cell centers. The field strength at the position $\mbox{$\bf r$}_{k}$ of some superparticle $k$ in cell $(i,j)$ is then
$\displaystyle E_{x}$ $\textstyle =$ $\displaystyle -\left[ \Phi_{i+1,j} - \Phi_{i-1,j} \right]\left/ 2\Delta l \right.$ (6.16)
$\displaystyle E_{y}$ $\textstyle =$ $\displaystyle -\left[ \Phi_{i,j+1} - \Phi_{i,j-1} \right]\left/ 2\Delta l \right.$ (6.17)

Next we integrate the equation of motion 6.15:
$\displaystyle \mbox{$\bf r$}_{k}^{n+1}$ $\textstyle =$ $\displaystyle 2 \mbox{$\bf r$}_{k}^{n}-\mbox{$\bf r$}_{k}^{n-1}
+\frac{q_{k}}{m_{k}} (\Delta t)^{2} \, \mbox{$\bf E$}_{i,j}^{n}$ (6.18)
$\displaystyle \mbox{$\bf v$}_{k}^{n}$ $\textstyle =$ $\displaystyle \left[ \mbox{$\bf r$}_{k}^{n+1}-\mbox{$\bf r$}_{k}^{n-1}\right] \left/
2\Delta t \right.$  

which completes the time step. Here is the prescription once more:
Figure 6.10: Particle-mesh method
% latex2html id marker 21429
\begin{minipage}{510 pt}
{\bf Particle-mesh...
...^{2} \, \mbox{$\bf E$}_{i,j}^{n}

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):

U(r) = \frac{q_{i}q_{j}}{4\pi \epsilon_{0}r}+B_{ij}
...e -\alpha_{ij}r} - \frac{C_{ij}}{r^{6}} - \frac{D_{ij}}{r^{8}}

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 $\rm P^{3}M$ technique.
next up previous
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," Kluwer-Plenum 2001