Particle Mesh (PM) 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 from the differential form of

Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...

, which—after identifying the

electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...

as the negative gradient of the

electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...

—gives rise to a

Poisson equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...

that is easily solved after applying the Fourier transform. 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.

See also