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 $\nu_{{\mbox{\tiny E}}}$ 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

$$B({\mathbf r},{\mathbf p}) \; \equiv \; \left\{ \begin{arr... ...z>-\epsilon$} \\ 0 & \mbox{otherwise} , \end{array} \right.$$ (3.6)

which exists only in the thin shell of thickness $\epsilon$ near the wall. The position vector ${\mathbf r}$ is represented by $({\mathbf s},z)$ where ${\mathbf s}$ is a surface coordinate and $z$ an outward `radial' coordinate normal to the surface. $\theta$ is the angle of the ${\mathbf p}$ vector from the local normal. This leaves $B$ independent of the other $d{-}1$ momentum degrees of freedom. The function $b({\mathbf s},\theta)$ is assumed constant upon reflection, i.e. it is even about $\theta = -\pi/2$. The time-dependence of $B$ along a given trajectory will be a series of impulses due to passing through the shell; the duration of impulse $i$ will be $2\epsilon / (v_{{\mbox{\tiny E}}} \cos\theta_i)$, and during this time its value is $\vert\cos\theta_i\vert\, b({\mathbf s}_i,\theta_i)$. Therefore the cos factors cancel, and the time-average is

$$\langle B \rangle_t \; = \; \frac{2\epsilon}{v_{{\mbox{\tin... ...m_i b({\mathbf s}_i,\theta_i) \ \delta(t-t_i) \right\rangle_t.$$ (3.7)

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

$$\langle B \rangle_{{\mbox{\tiny E}}} \; = \; \frac{\epsilon... ... \ b({\mathbf s},\theta) \right\rangle_{{\mathbf s},\Omega} .$$ (3.8)

Here I used the fact that in the $\epsilon\rightarrow0$ limit the fraction of position space occupied by the shell is $\epsilon{\mathsf{A}}/{\mathsf{V}}$. This collapses the average to be over ${\mathbf s}$ and over the $d{-}1$ dimensional solid angle $\Omega$. Choosing the function $b = 1$ gives a useful expression for the mean collision rate (the inverse of the ballistic time),

$$\tau_{{\mbox{\tiny bl}}}^{-1} \;\equiv \; \left\langle \sum... ...f{A}}}{2 {\mathsf{V}}} \ \langle \vert\cos\theta\vert \rangle,$$ (3.9)

which is a form of Sabine's formula^3.1. The geometrical factor is the solid angle average $\langle\vert\cos\theta\vert\rangle = 2/\pi, 1/2, \cdots$ for dimension $d=2,3, \cdots$ (see Appendix D of [46]). Or, choosing $b = \cos^2\theta \ D{({\mathbf s})}^2$ gives

$$\left\langle \sum_i \cos^2\theta_i \ D({\mathbf s}_i) ^2 \ ... ...3 \rangle \, \langle D{({\mathbf s})}^2 \rangle_{{\mathbf s}}$$ (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 $\langle\vert\cos\theta\vert^3\rangle = 4/(3\pi), 1/4, \cdots$ for $d=2,3, \cdots$ [46].

Figure: The white-noise approximation estimate (WNA is the horizontal dotted line) compared to actual $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ power spectra for some example deformations of the 2D generalized Sinai billiard, with $m=v=1$. (The RMS estimation error of 3% can be seen as multiplicative noise with short correlation length in $\omega$). The deformations (sketched on the right), from worst to best agreement of the WNA are: FR (for which sensitivity to the least-unstable vertical periodic orbit causes large correlation effects and large deviations from WNA), W8 (oscillatory deformation changes sign enough to be close to WNA), P1 (localized `piston' type deformation for which WNA is good), and DF (random function of zero correlation-length along the perimeter, showing complete WNA agreement). Deformation functions are defined in Table 3.1.

Table 3.1: Key to deformation types used for numerical 2D billiard experiments in this and the following chapter. $L$ is the billiard perimeter. The deformation is described by a function $D(s)$, where $s$ is measured counter-clockwise along the perimeter with $s=0$ at the upper left corner. In the `fracture' and `shift-x' cases we use the horizontal Cartesian coordinate $x(s)$.

key	description	surface deformation function $D(s)$
CO	constant	1
W$n$	$n$ periods	$\cos(2 \pi n s/L)$
DF	diffuse	random[-1,1] (equivalent to W$\infty$)
FR	fracture	sgn$(x(s))$ if on top or bottom, else 0
SX	shift-x	sgn$(x(s))$ if on left or right, else 0
P1	piston 1	$10 \exp(-\mbox{\small $\frac{1}{2}$}\alpha^2)$, $\ \alpha{=}(s/L - 0.3)/0.01$
P2	piston 2	$10 \exp(-\mbox{\small $\frac{1}{2}$}\alpha^2)$, $\ \alpha{=}(s/L - 0.6)/0.005$
WG	wiggle	$5\alpha \exp(-\mbox{\small $\frac{1}{2}$}\alpha^2)$, $\ \alpha{=}(s/L - 0.25)/0.02$