6.5 Particles and Fields

In many cases the interparticle potential may be safely neglected beyond a few particle diameters. Defining a cutoff distance $r_{co}$, we may estimate the importance of the neglected ``tail'' of a potential $u(r)$ by

$$\int_{r_{co}}^{\infty} u(r) \, 4 \pi r^{2}\, dr$$

If the decay of $u(r)$ is slow, this integral may be non-negligible. For example, the interaction between charged particles decays only as $r^{-1}$, and any cutoff will introduce a considerable error. Another instance is astrophysics, where the gravitational potential reaches too far to permit a simple cutoff.

For ionic systems that are globally neutral the Ewald summation method known from solid state physics may be invoked. In the case of hot plasmas or gravitating systems the Particle-Mesh methods are appropriate.