Form of correlation spectrum and timescales

As presented in Section 2.1, the generalized time-dependent `force' associated with the parameter $x$ is ${\mathcal{F}}(t) = - \partial {\mathcal{H}}({\mathbf r},{\mathbf p};x)/\partial x - F(x)$, where $F(x)$ is a constant such that the time-average is zero. It has the $t$-independent auto-correlation function

$$C_{{\mbox{\tiny E}}}(\tau) \; \equiv \; \langle {\mathcal{F}}(t) {\mathcal{F}}(t+\tau) \rangle_{{\mbox{\tiny E}}}$$ (3.2)

The subscript ${\mbox{\tiny E}}$, whenever used, suggests that the average over initial conditions is of microcanonical type, with energy $E$. For the Hamiltonian (3.1) we can write

$$-\frac{\partial \mathcal{H}}{\partial x}(t) \; = \; {\mathbf D}{(... ...nabla}} U{({\mathbf r})}= - {\mathbf D}{({\mathbf r})}\cdot \dot{{\mathbf p}} .$$ (3.3)

Recognizing $\dot{{\mathbf p}}$ as the force on the gas particle, we see that ${\mathcal{F}}(t)$ is a train of spikes superimposed on a constant average value (see Fig. 3.3a):

$${\mathcal{F}}(t) \ = \ \sum_{i} 2m v \ \cos(\theta_i) \ D_i \ \delta(t-t_i) \; - \; F(x)$$ (3.4)

where $i$ labels collisions: $t_i$ is the time of a collision, $D_i$ stands for $D({\mathbf s}_i)$ at the location ${\mathbf s}_i$ of a collision, and $v \cos(\theta_i)$ is the normal component of the particle's collision velocity. If the deformation is volume-preserving then $\langle {\mathcal{F}}(t)\rangle =0$, otherwise it is convenient to subtract the (constant) average value $F(x)$.

The auto-correlation function $C_{{\mbox{\tiny E}}}(\tau)$ 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 $C_{{\mbox{\tiny E}}}(\tau)$ and its Fourier transform $\tilde{C}_{{\mbox{\tiny E}}}(\omega)\equiv \int C_{{\mbox{\tiny E}}}(\tau) \exp(i \omega \tau) d\tau$ are illustrated schematically in Figs. 3.3c and 3.3d. The removal of the average value of ${\mathcal{F}}(t)$ ensures that there is no $\delta$-function spike at $\omega=0$ in $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$. This is required so that $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ has a well-defined $\omega\rightarrow0$ limit, namely the fluctuations intensity $\nu_{{\mbox{\tiny E}}}$.

The auto-correlation function $C_{{\mbox{\tiny E}}}(\tau)$ consists of a $\tau=0$ (`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 $F(x)$ contributes nothing to the `self' $\delta(\tau)$ peak, so its only effect is on the `non-self' component). Thus two time scales are involved: the short time scale is $\tau_0 = 2m v /f \rightarrow 0$ in the hard wall limit, and the other time scale is $\tau_{{\mbox{\tiny col}}}$, which involves the time for succesive collisions with the deforming part of the boundary. Note that $\tau_{{\mbox{\tiny col}}}$ can be much larger than the ballistic time $\tau_{{\mbox{\tiny bl}}}$ (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 $1/\tau_{{\mbox{\tiny col}}}$ is postponed to Section 3.2.

I shall be most interested in the noise intensity $\nu_{{\mbox{\tiny E}}}$ defined by (2.14). Observing that ${\mathcal{F}}(t)$ is linear in $D{({\mathbf s})}$, it follows that the noise intensity must (exactly) be a general quadratic functional

$$\nu_{{\mbox{\tiny E}}} \; = \; \oint \! \! \oint d{\mathbf ... ...box{\tiny E}}}({\mathbf s}_1,{\mathbf s}_2) D({\mathbf s}_2) ,$$ (3.5)

where the kernel $\gamma_{{\mbox{\tiny E}}}$ depends on both the cavity shape and the particle energy $E$ [120]. Furthermore, billiards are scaling systems in the sense that a change in $E$ leaves the trajectories unchanged. From this and (3.4) we have the scaling relation $\gamma_{{\mbox{\tiny E}}}({\mathbf s}_1,{\mathbf s}_2) = m^2 v_{{\mbox{\tiny E}}}^3 \cdot \hat{\gamma}({\mathbf s}_1,{\mathbf s}_2)$, where the scaled kernel depends entirely on the geometrical shape of the cavity. However, the reason for being interested in approximations for $\nu_{{\mbox{\tiny E}}}$ is that the exact result for the kernel $\hat{\gamma}$ is not analytic: it is very complicated to evaluate, and involves a sum over all classical paths from ${\mathbf s}_1$ to ${\mathbf s}_2$ (see [120]).

Figure: a) The fluctuating force ${\mathcal{F}}(t)$ is a series of impulses of maximum duration $\tau_0$. In the hard-walled limit $\tau_0 \rightarrow 0$. The average value $F(x)$ has been subtracted. b) The autocorrelation function $C(\tau)$ can be viewed as the average projection of the function ${\mathcal{F}}(t_1){\mathcal{F}}(t_2)$ onto the $\tau \equiv t_2-t_1$ axis (the `self' components are shaded). c) The autocorrelation function has a `self' peak of half-width $\tau_0$, and becomes negligible beyond the ergodic time $t_{{\mbox{\tiny erg}}}$. d) The noise power spectrum $\tilde{C}(\omega)$. Its $\omega\rightarrow0$ limit is equal to the area under $C(\tau)$. The power spectrum typically shows billiard-specific features around the frequency $\omega \sim 1/\tau_{{\mbox{\tiny bl}}}$, where $\tau_{{\mbox{\tiny bl}}}$ is the ballistic time. Note that in some examples $\tau_{{\mbox{\tiny col}}}$ is much longer than $\tau_{{\mbox{\tiny bl}}}$.