Particle Mesh
Encyclopedia
Particle Mesh is a computational method for determining the forces in a system of particles. These particles could be atoms, stars, or fluid components and so the method is applicable to many fields, including molecular dynamics and astrophysics. The basic principle is that a system of particles is converted into a grid (or "mesh") of density values. The potential is then solved for this density grid, and forces are applied to each particle based on what cell it is in, and where in the cell it lies.
Various methods for converting a system of particles into a grid of densities exist. One method is that each particle simply gives its mass to the closest point in the mesh. Another method is the Cloud-in-Cell (CIC) method, where the particles are modelled as constant density cubes, and one particle can contribute mass to several cells.
Once the density distribution is found, the potential energy of each point in the mesh can be determined using Fourier transform
techniques. Thus it is faster to do a PM calculation than to simply add up all the interactions on a particle due to all other particles for two reasons: firstly, there are usually fewer grid points than particles, so the number of interactions to calculate is smaller, and secondly the grid technique permits the use of Fourier transform techniques to evaluate the potential, and these can be very fast.
PM is considered an obsolete method as it does not model close interaction between particles well. It has been supplanted by the Particle-Particle Particle-Mesh
method, which uses a straight particle-particle sum between nearby particles in addition to the PM calculation.
Various methods for converting a system of particles into a grid of densities exist. One method is that each particle simply gives its mass to the closest point in the mesh. Another method is the Cloud-in-Cell (CIC) method, where the particles are modelled as constant density cubes, and one particle can contribute mass to several cells.
Once the density distribution is found, the potential energy of each point in the mesh can be determined using Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
techniques. Thus it is faster to do a PM calculation than to simply add up all the interactions on a particle due to all other particles for two reasons: firstly, there are usually fewer grid points than particles, so the number of interactions to calculate is smaller, and secondly the grid technique permits the use of Fourier transform techniques to evaluate the potential, and these can be very fast.
PM is considered an obsolete method as it does not model close interaction between particles well. It has been supplanted by the Particle-Particle Particle-Mesh
P3M
Particle-Particle-Particle Mesh is a Fourier-based Ewald summation method to calculate potentials in N-body simulations.The potential could be the electrostatic potential among N point charges i.e. molecular dynamics, the gravitational potential among N gas particles in e.g. smoothed particle...
method, which uses a straight particle-particle sum between nearby particles in addition to the PM calculation.
See also
- P3MP3MParticle-Particle-Particle Mesh is a Fourier-based Ewald summation method to calculate potentials in N-body simulations.The potential could be the electrostatic potential among N point charges i.e. molecular dynamics, the gravitational potential among N gas particles in e.g. smoothed particle...
- Particle mesh method
- Particle mesh Ewald method
- Madelung constantMadelung constantThe Madelung constant is used in determining the electrostatic potential of a single ion in a crystal by approximating the ions by point charges. It is named after Erwin Madelung, a German physicist....
- Poisson summation formulaPoisson summation formulaIn mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples...
- Paul Peter EwaldPaul Peter EwaldPaul Peter Ewald was a German-born U.S. crystallographer and physicist, a pioneer of X-ray diffraction methods.-Education:...