Introduction

Plasma physics can be viewed formally as a closure of Maxwell's equations
by means of *constitutive relations*: *i.e.*, expressions
for the charge density, , and the current density, , in terms of
the electric and magnetic fields, and . Such relations are
easily expressed in terms of the microscopic distribution functions,
, for each plasma species. In fact,

Here, is the exact, ``microscopic'' phase-space density of plasma species (charge , mass ) near point at time . The distribution function is normalized such that its velocity integral is equal to the particle density in coordinate space. Thus,

(168) |

If we could determine each
in terms of the
electromagnetic fields, then Eqs. (166)-(167) would immediately give us the
desired constitutive relations. Furthermore, it is easy to see, in principle,
how each distribution function evolves. Phase-space conservation
requires that

(170) |

It would appear that the distribution functions for the various plasma species, from which the constitutive relations are trivially obtained, are determined by a set of rather harmless looking first-order partial differential equations. At this stage, we might wonder why, if plasma dynamics is apparently so simple when written in terms of distribution functions, we need a fluid description of plasma dynamics at all. It is not at all obvious that fluid theory represents an advance.

The above argument is misleading for several reasons. However, by far the
most serious flaw is the view of Eq. (169) as a tractable equation.
Note that this equation is easy to derive, because it is exact, taking into
account all scales from the microscopic to the macroscopic. Note, in
particular, that there is *no* statistical averaging involved in Eq. (169).
It follows that the microscopic distribution function is
essentially a sum of Dirac delta-functions, each following the
detailed trajectory of a single particle. Furthermore, the electromagnetic
fields in Eq. (169) are horribly spiky and chaotic on microscopic scales. In other words, solving Eq. (169) amounts to nothing less than solving the classical
electromagnetic many-body problem--a completely hopeless task.

A much more useful and tractable equation can be extracted from Eq. (169)
by ensemble averaging. The average distribution function,

(171) |

(172) |

(173) |

The traditional goal of kinetic theory is to analyze the correlations,
using approximations tailored to the parameter regime of interest, and
thereby express the average acceleration term in terms of
and
the average electromagnetic fields alone. Let us assume that this ambitious
task has already been completed, giving an expression of the form

(174) |

In general, the above equation is very difficult to solve, because of the
complexity of the collision operator. However, there are some situations
where collisions can be completely neglected. In this case, the
apparent simplicity of Eq. (169) is not deceptive. A useful kinetic
description is obtained by just ensemble averaging this equation to
give

(176) |

Firstly, fluid equations possess the key simplicity of involving fewer dimensions: three spatial dimensions instead of six phase-space dimensions. This advantage is especially important in computer simulations.

Secondly, the fluid description is intuitively appealing. We immediately understand the significance of fluid quantities such as density and temperature, whereas the significance of distribution functions is far less obvious. Moreover, fluid variables are relatively easy to measure in experiments, whereas, in most cases, it is extraordinarily difficult to measure a distribution function accurately. There seems remarkably little point in centering our theoretical description of plasmas on something that we cannot generally measure.

Finally, the kinetic approach to plasma physics is spectacularly inefficient. The species distribution functions provide vastly more information than is needed to obtain the constitutive relations. After all, these relations only depend on the two lowest moments of the species distribution functions. Admittedly, fluid theory cannot generally compute and without reference to other higher moments of the distribution functions, but it can be regarded as an attempt to impose some efficiency on the task of dynamical closure.