Collision Operator

Plasma physics can be regarded formally as a closure of Maxwell's equations by means of constitutive relations: that is, expressions specifying the charge density, $\rho_c$, and the current density, ${\bf j}$, in terms of the electric and magnetic fields, ${\bf E}$ and ${\bf B}$ (Hazeltine and Waelbroeck 2004). Such relations can be expressed in terms of the microscopic distribution functions, ${\cal F}_s$, for each plasma species:

$$\rho_c = \sum_s e_s\! \int\! {\cal F}_s({\bf r}, {\bf v}, t)\,d^3{\bf v},$$ (3.1)

$${\bf j} = \sum_s e_s \!\int {\bf v}\,{\cal F}_s({\bf r}, {\bf v}, t)\,d^3 {\bf v}.$$ (3.2)

Here, ${\cal F}_s({\bf r}, {\bf v}, t)$ is the exact microscopic phase-space density of plasma species $s$ (with charge $e_s$ and mass $m_s$) near point $({\bf r},\, {\bf v})$ at time $t$ (Reif 1965). The distribution function ${\cal F}_s$ is normalized such that its velocity integral is equal to the particle number density in coordinate space. In other words,

$$\int \!{\cal F}_s({\bf r}, {\bf v}, t)\,d^3 {\bf v} = n_s({\bf r}, t),$$ (3.3)

where $n_s({\bf r},t)$ is the number (per unit volume) of species-$s$ particles near point ${\bf r}$ at time $t$.

If we could determine each ${\cal F}_s({\bf r}, {\bf v}, t)$ in terms of the electromagnetic fields then Equations (3.1) and (3.2) would give us the desired constitutive relations. In fact, the time evolution of the various distribution functions is determined by particle conservation in phase-space, which requires that (Reif 1965)

$$\frac{\partial{\cal F}_s}{\partial t} + {\bf v}\cdot \frac{\parti... ...tial {\bf r}} + {\bf a}_s\cdot\frac{\partial {\cal F}_s}{\partial {\bf v}} = 0,$$ (3.4)

where

$${\bf a}_s = \frac{e_s}{m_s}\,({\bf E} + {\bf v}\times{\bf B})$$ (3.5)

is the species-$s$ particle acceleration under the influence of the ${\bf E}$ and ${\bf B}$ fields.

Equation (3.4) is easy to derive because it is exact, taking into account all lengthscales from the microscopic to the macroscopic. Note, in particular, that there is no statistical averaging involved in Equation (3.4). It follows that the microscopic distribution function, ${\cal F}_s$, is essentially a sum of Dirac delta-functions, each following the detailed trajectory of a single particle. Consequently, the electromagnetic fields appearing in Equation (3.4) are extremely spiky on microscopic scales. In fact, solving Equation (3.4) is equivalent to solving the classical electromagnetic many-body problem, which is a completely hopeless task.

A much more useful equation can be extracted from Equation (3.4) by ensemble averaging (Reif 1965). The average distribution function,

$$\langle {\cal F}_s\rangle\equiv f_s,$$ (3.6)

is smooth on microscopic lengthscales, and is closely related to actual experimental measurements. Here, angle brackets denote an ensemble average. Similarly, the ensemble-averaged electromagnetic fields are also smooth. Unfortunately, the extraction of an ensemble-averaged equation from Equation (3.4) is mathematically challenging, and invariably involves some level of approximation (Balescu 1960; Boboliubov 1946; Lenard 1960). The problem is that, because the exact electromagnetic fields depend on particle trajectories, ${\bf E}$ and ${\bf B}$ are not statistically independent of ${\cal F}_s$. In other words, as a consequence of correlations between the distribution function and the electromagnetic fields on microscopic lengthscales, the ensemble average of the nonlinear acceleration term in Equation (3.4) is such that

$$\left\langle{\bf a}_s\cdot\frac{\partial {\cal F}_s}{\partial {\b... ...t\rangle \neq \langle{\bf a}_s\rangle\cdot\frac{\partial f_s}{\partial{\bf v}}.$$ (3.7)

It is convenient to write

$$\left\langle{\bf a}_s\cdot\frac{\partial {\cal F}_s}{\partial{\bf... ...le = \langle{\bf a}_s\rangle \cdot\frac{\partial f_s}{\partial {\bf v}}-C_s(f),$$ (3.8)

where $C_s$ is an operator that accounts for the correlations. Because the most important correlations result from close encounters between particles, $C_s$ is known as the collision operator (for species $s$). It is not generally a linear operator, and usually involves the distribution functions of both colliding species (the subscript in the argument of $C_s$ is omitted for this reason). Hence, the ensemble-averaged version of Equation (3.4) is written

$$\frac{\partial{f_s}}{\partial t} + {\bf v}\cdot\frac{\partial f_s... ...f E} + {\bf v}\times{\bf B})\cdot\frac{\partial f_s}{\partial{\bf v}} = C_s(f),$$ (3.9)

where ${\bf E}$ and ${\bf B}$ are now understood to be the smooth, ensemble-averaged electromagnetic fields. Of course, in a weakly coupled plasma, the dominant collisions are two-particle Coulomb collisions. Equation (3.9) is generally known as the kinetic equation.