Next: The white noise approximation Up: The cavity system Previous: Form of correlation spectrum

## Conversion of time averages to averages over collision parameters

This section shows how ergodicity can be used to go from a time average to a collision parameter average, for a quantity which takes values at each of the collisions. This will prove useful for approximating in the next section, but it can be skipped on first reading. (It is presented in a more terse form in [46]).

I treat time-averages of quantities which are non-zero only during the collisions, and whose value during a collision is a function of the parameters of the collision (angle from the normal, and surface location) only. They can be converted to collision-parameter averages, eliminating the time variable, as follows. For a hard-walled billiard, imagine a function of phase-space

 (3.6)

which exists only in the thin shell of thickness near the wall. The position vector is represented by where is a surface coordinate and an outward radial' coordinate normal to the surface. is the angle of the vector from the local normal. This leaves independent of the other momentum degrees of freedom. The function is assumed constant upon reflection, i.e. it is even about . The time-dependence of along a given trajectory will be a series of impulses due to passing through the shell; the duration of impulse will be , and during this time its value is . Therefore the cos factors cancel, and the time-average is
 (3.7)

By ergodicity one can equate this to the phase-space average of (3.6) at the energy ,
 (3.8)

Here I used the fact that in the limit the fraction of position space occupied by the shell is . This collapses the average to be over and over the dimensional solid angle . Choosing the function gives a useful expression for the mean collision rate (the inverse of the ballistic time),
 (3.9)

which is a form of Sabine's formula3.1. The geometrical factor is the solid angle average for dimension (see Appendix D of [46]). Or, choosing gives
 (3.10)

where we used the fact that for this choice the averages over surface location and solid angle separate. Again, formulae exist for the dimension-dependent geometric factor for [46].

Table 3.1: Key to deformation types used for numerical 2D billiard experiments in this and the following chapter. is the billiard perimeter. The deformation is described by a function , where is measured counter-clockwise along the perimeter with at the upper left corner. In the fracture' and `shift-x' cases we use the horizontal Cartesian coordinate .
 key description surface deformation function CO constant 1 W periods DF diffuse random[-1,1] (equivalent to W) FR fracture sgn if on top or bottom, else 0 SX shift-x sgn if on left or right, else 0 P1 piston 1 , P2 piston 2 , WG wiggle ,

Next: The white noise approximation Up: The cavity system Previous: Form of correlation spectrum
Alex Barnett 2001-10-03