Boltzmann Equation

As a generalized form of the Continuity Equation, the Boltzmann equation gives an exact description for the density of a plasma constituent both in real and velocity space. It is obtained by equating the total time differential for the density distribution function n(r,v,t) with the local production and loss rates, i.e.

∂/∂t(n(r,v,t)) +v^.grad_r(n(r,v,t)) +F/M^.grad_v(n(r,v,t)) = q^Ion(r,v,t) -l^Rec(r,v,t) +C(r,v,t) .

The steady-state (time independent) equation is obtained by setting ∂/∂t(n(r,v,t))=0. In this case, the production and loss rates due to convection (transport) in geometrical and velocity space (where F contains all external forces on the particle with mass M, i.e. electric, magnetic and gravitational forces), are exactly balanced by the local production and loss rates due to ionization (q^Ion), recombination (l^Rec) and velocity changing collisions (C) (in general, these last three terms do also depend on n(r,v,t) which has not been written here).
The ionization and recombination terms are usually neglected in standard treatments. However, they are vitally important as they are responsible for the inhomogeneities of the plasma density and affect therefore the velocity distribution function through the convection terms in the equation (see link below).
Also, one should note that the usual formulation in terms of the normalized distribution function f(r,v,t)= n(r,v,t)/N(r,t) (with N(r,t) = ∫d³v n(r,v,t) ) is in general not sufficient because of the dependence of N(r,t) on r.
For the one-dimensional case, the Boltzmann equation can be written as a first order linear differential equation in either the spatial or velocity variable. Formal solution yields a non-linear integral equation which can then be solved numerically (see /research/#A6 for an application to ion diffusion in the earth's ionosphere).