2.1 Boltzmann's Transport Equation

The evolution of the distribution density in space,
, is described by
**Boltzmann's transport equation**. A thorough treatment of this
beautiful achievement is beyond the scope of our discussion.
But we may sketch the basic ideas used in its derivation.

- If there were no collisions at all, the swarm of particles
in space would flow according to

where denotes an eventual external force acting on particles at point . The time derivative of is therefore, in the collisionless case,

where

(2.3)

(2.4) A very simple mu-space

At time a differential ``volume element'' at contains, on the average, particles. The temporal change of is then given by

(2.5)

(2.6) (2.7) (2.8) (2.9) (2.10) (2.11)

The relation is then easily generalized to the case of a non-vanishing force,

(2.12)

All this is for collisionless flow only. - In order to account for collisions a term
is added on the right hand side:

(2.13)

- only binary collisions need be considered (dilute gas);

- the influence of container walls may be neglected;

- the influence of the external force (if any) on the rate of collisions is negligible;

- velocity and position of a molecule are uncorrelated (assumption of**molecular chaos**).The effect of the binary collisions is expressed in terms of a ``differential scattering cross section'' which describes the probability density for a certain change of velocities,

(2.14)

Under all these assumptions, and by a linear expansion of the left hand side of equ. 2.1 with respect to time, the Boltzmann equation takes on the following form:

(2.15) |

Chapman and Enskog developed a general procedure for the approximate solution of Boltzmann's equation. For certain simple model systems such as hard spheres their method produces predictions for (or its moments) which may be tested in computer simulations. Another more modern approach to the numerical solution of the transport equation is the ``Lattice Boltzmann'' method in which the continuous variables and are restricted to a set of discrete values; the time change of these values is then described by a modified transport equation which lends itself to fast computation.

The initial distribution density
may be
of arbitrary shape. To consider a simple example, we may have
all molecules assembled in the left half of a container - think
of a removable shutter - and at time make the rest of the
volume accessible to the gas particles:

(2.16) |

Applet BM: Start |

The

(2.17) |

It may be shown that this equilibrium distribution is given by

(2.18) |

If there are no external forces such as gravity or electrostatic interactions
we have
. In case the temperature is also
independent of position, and if the gas as a whole is not moving
(), then
, with

2005-01-25