Test Particle Method in Kinetic Theory of a Plasma

A test particle of coordinates X = (x, v) is surrounded by a shield cloud of field particles of coordinates X′ characterized by a conditional probability function P(X|X′t). A relationship has been found between this function, the one‐particle function f(X, t) and the two‐particle correlation function G(X, X′; t). It is $$\mathrm{G(X,\hspace{0.17em}X\prime ;t)=f(Xt)P(X\hspace{0.17em}|\hspace{0.17em}X\prime t)+f(X\prime t)P(X\prime \hspace{0.17em}|\hspace{0.17em}Xt)+n}{\displaystyle \int }\mathrm{dX″\hspace{0.17em}f(X″,t)P(X″\hspace{0.17em}|\hspace{0.17em}Xt)P(X″\hspace{0.17em}|\hspace{0.17em}X\prime t)}$$ . The first two terms indicate that each of the two particles involved is a test particle as well as part of the shield cloud of the other particle. The last term corresponds to the two particles shielding a third particle. This relation has been established without solving explicitly for anything and has none of the usual restrictions such as spatial homogeneity, adiabatic time behavior, etc., usually necessary for obtaining explicit solutions. It is useful because the problem of kinetic theory is reduced to determining P which involves only the Vlasov equation. In addition, superposition principles for fluctuations, etc., are apparent at the outset.