(4.4) |

with factors of . Clearly, is a tensor of rank (Riley 1974).

The set , for , can be viewed as an alternative description of the distribution function that uniquely specifies when the latter is sufficiently smooth. For example, a (displaced) Gaussian distribution function is uniquely specified by three moments: , the vector , and the scalar formed by contracting .

The low-order moments all have simple physical interpretations.
First, we have the particle *number density*,

(4.5) |

and the particle

(4.6) |

The quantity is, of course, the

(4.7) | ||

(4.8) |

The second-order moment, describing the flow of momentum in the
laboratory frame, is called the *stress tensor*, and takes the form

(4.9) |

Finally, there is an important third-order moment measuring the

(4.10) |

It is often convenient to measure the second- and third-order moments in the rest-frame of the species under consideration. In this case, the
moments have different names. The stress tensor measured in the rest-frame
is called the *pressure tensor*,
, whereas the energy flux
density becomes the *heat flux density*,
. We introduce the
relative velocity,

(4.11) |

in order to write

(4.12) |

and

(4.13) |

The trace of the pressure tensor measures the ordinary (or scalar) pressure,

(4.14) |

In fact, is the kinetic energy density of species : that is,

In thermodynamic equilibrium, the distribution function becomes a Maxwellian characterized by some temperature , and Equation (4.15) yields . It is, therefore, natural to define the (kinetic) temperature as

(4.16) |

Of course, the moments measured in the two different frames are related. By direct substitution, it is easily verified that