Ruelle's thermodynamic formalism assigns a dynamical partition
function to a chaotic system by raising its expansion factor along
unstable manifolds to a power 1- β, and averaging over all initial points on the
relevant shell in phase space. For a closed system its logarithm over
t, usually called the topological
pressure, yields the Kolmogorov-Sinai entropy as a derivative at
β=1, and is called the topological
entropy for β=0. For open systems the thermodynamic pressure at
β=1 also gives the average escape rate from the system.
For a dilute disordered Lorentz gas at equilibrium (that is, a system
of fixed hard spherical scatterers with one light particle moving
elastically among them) the thermodynamic pressure may be calculated
explicitly, yielding results in agreement with previous calculations.
For β-values different from unity the topological pressure for large
enough systems always becomes dominated by orbits confined either to
the direct neighborhood of a periodic orbit or to a small subspace with
a higher than average collision rate. For example the topological
entropy with increasing system size soon is determined exclusively by
orbits confined to a very small subsystem of the total system.
In the presence of a driving field combined with a gaussian thermostat
the calculation of the dynamic partition function involves a simple
transfer matrix formalism. The same holds for a system with open
boundaries. In the latter case it is helpful mapping the problem to a
random flight model with escape through the boundaries.