and time t. This is a macroscopic, or hydrodynamic, picture of
the fluid, where microscopic details such as the atomic nature of
matter, and the fluctuations of molecular velocities are washed
out of the picture. At the other extreme is the microscopic
picture, in which one follows the trajectories of each individual
molecule. This is described by means of the Liouville equation,
which in the case of an isolated system, generates Newton's laws
of motion for each molecule. More will be said on this in section
. The third picture, called the mesoscopic
picture, [Serra et al.
[1986]]
treats the velocity fluctuations by a phase-space
distribution function 
that depends on
position 
and velocity 
at time t, where 
is the number of
particles contained in the phase space volume element 
. The rate of change of the distribution function is
made up of a streaming term due to particles entering and
leaving the volume 
, another term due to collisions between
particles, and in the presence of an external field,
an acceleration term describing the particle flux in the
volume 
. Symbolically, this
reads
When Boltzmann first derived this equation, he considered a
dilute gas in which only two gas molecules at a time are
involved in the collisions. This implies that 
is a
quadratic operator, and this equation is known as the Boltzmann
equation. For a more detailed discussion on the form of 
, and a derivation, see Dorfman and van Beijeren
[1977]. Attempts have been made to
generalize
the Boltzmann
equation to handle denser gases by including collisions involving
three or more bodies.
The simplest and clearest formulation was given by Bogolubov in
1945 [see Cohen
[1962]]
. However, in the middle of
the
1960s, it was discovered that Bogolubov's generalization of the
Boltzmann equation could not be correct, since the fourth and
all higher order collision terms diverged. In two dimensions,
even the third order term diverges. It is believed that the higher order
collision operators may be resummed to obtain convergent collision
integrals.
See Cohen [1983] and references therein for a discussion.
Since the mesoscopic approach fails for fluids of high density, a
different
approach is required for computing the transport coefficients from details
of
the microscopic interactions. A method was pioneered by Green and Kubo
[see Zwanzig
[1965]]
, which relates the transport
coefficients
to certain time correlation functions that can be computed by simulating
the
motion of the molecules due to their interactions. This expression is
exact
for arbitrary density, but is limited to the linear coefficients such as
the
in (). Recently, Evans and
Lynden-Bell
[1988] have produced expressions for the
non-linear
Burnett coefficients. The work in this thesis implements the calculation
for
the simple model of electrical conductivity outlined in that paper.
In swarm physics, the swarm is dilute in comparison with the
background neutral gas. The collisions between the charged
particles and the neutral gas dominate over the collisions of
the charged particles amongst themselves. This leads to a
modelling of swarm physics by the linear Boltzmann equation,
where 
is taken to be a linear operator acting on
f. This allows us to make use of the vast array of
techniques available for linear operators.