Plasma Simulation Theory

The fundamental physics governing the dynamics of a plasma have been well understood for over a century, and yet plasma physics remains an active area of research. This is because the dynamics of a plasma are highly nonlinear, and it is therefore difficult to make analytic statements about how a given plasma will behave over long time-spans. Instead, theoretical plasma physicists often rely on simulation to understand plasma dynamics, and to make general statements about the behaviors of particular classes of plasmas.

Unfortunately, the simulation of plasmas is itself a quite challenging task because plasmas are composed of many, many, charged particles. As a result, there are far too many degrees of freedom to exactly solve the full equations of motion for a given plasma. Instead, physicists rely on approximations to derive physically relevant models for the systems in question.

The most "realistic" class of models are called kinetic models. In these models, the individual charged particles of each species are averaged to create distribution functions that depends on both position and velocity. The distribution functions give the likelihood of finding a charged particle of a particular species at any point in phase space. These distribution functions, along with a method for calculating the self-consistent interaction between the particles, yields a set of approximate equations of motion describing the plasma dynamics.

If the particle species are nonrelativistic, then the particles are not influenced by self-consistent magnetic fields, and so the interactions can be modeled using electrostatics. However, once the particles (typically the lightweight electrons) become relativistic, the sourced magnetic field must be taken into account, and so full electromagnetic interactions are required.

Over long periods of time, the plasma will begin to thermalize–-the distribution of particle velocities will become closer and closer to Maxwellian. This fact can be used to drastically improve the size and speed of simulations by using the two-fluid and magnetohydrodynamic (MHD) approximations of plasma dynamics. As this package does not implement algorithms for these simulation methods, we will not discuss the details of these methods further.

In the following sections, we describe the details of particle-in-cell algorithms, a specific type of kinetic simulation algorithm. For a more in depth introduction of PIC simulation theory, see the books by Birdsall and Langdon (2004) and Hockney and Eastwood (1989).