   Next: Conversion of time averages Up: The cavity system Previous: The cavity system

## Form of correlation spectrum and timescales

As presented in Section 2.1, the generalized time-dependent force' associated with the parameter is , where is a constant such that the time-average is zero. It has the -independent auto-correlation function (3.2)

The subscript , whenever used, suggests that the average over initial conditions is of microcanonical type, with energy . For the Hamiltonian (3.1) we can write (3.3)

Recognizing as the force on the gas particle, we see that is a train of spikes superimposed on a constant average value (see Fig. 3.3a): (3.4)

where labels collisions: is the time of a collision, stands for at the location of a collision, and is the normal component of the particle's collision velocity. If the deformation is volume-preserving then , otherwise it is convenient to subtract the (constant) average value .

The auto-correlation function can be handled as a time-average of a single trajectory rather than an ensemble-average over trajectories (by ergodicity, as discussed in Section 2.1.6); the resulting construction is illustrated in Fig. 3.3b. The forms of and its Fourier transform are illustrated schematically in Figs. 3.3c and 3.3d. The removal of the average value of ensures that there is no -function spike at in . This is required so that has a well-defined limit, namely the fluctuations intensity .

The auto-correlation function consists of a (self') peak due to the self-correlation of the spikes, and of an additional smooth (non-self') component due to correlations between successive bounces. (Note that in the hard walled limit the constant value contributes nothing to the self' peak, so its only effect is on the `non-self' component). Thus two time scales are involved: the short time scale is in the hard wall limit, and the other time scale is , which involves the time for succesive collisions with the deforming part of the boundary. Note that can be much larger than the ballistic time (the average time to cross the billiard) in the case that only a small piece of the boundary is being deformed. The quantitative definition of the collision rate is postponed to Section 3.2.

I shall be most interested in the noise intensity defined by (2.14). Observing that is linear in , it follows that the noise intensity must (exactly) be a general quadratic functional (3.5)

where the kernel depends on both the cavity shape and the particle energy . Furthermore, billiards are scaling systems in the sense that a change in leaves the trajectories unchanged. From this and (3.4) we have the scaling relation , where the scaled kernel depends entirely on the geometrical shape of the cavity. However, the reason for being interested in approximations for is that the exact result for the kernel is not analytic: it is very complicated to evaluate, and involves a sum over all classical paths from to (see ).    Next: Conversion of time averages Up: The cavity system Previous: The cavity system
Alex Barnett 2001-10-03