Fine structure of distributions and central limit theorem in diffusive billiards

We investigate deterministic diffusion in periodic billiard models, in terms of the convergence of rescaled distributions to the limiting normal distribution required by the central limit theorem; this is stronger than the usual requirement that the mean-square displacement grow asymptotically linearly in time. The main model studied is a chaotic Lorentz gas where the central limit theorem has been rigorously proved. We study one-dimensional position and displacement densities describing the time evolution of statistical ensembles in a channel geometry, using a more refined method than histograms. We find a pronounced oscillatory fine structure, and show that this has its origin in the geometry of the billiard domain. This fine structure prevents the rescaled densities from converging pointwise to Gaussian densities; however, demodulating them by the fine structure gives new densities which seem to converge uniformly. We give an analytical estimate of the rate of convergence of the original distributions to the limiting normal distribution, based on the analysis of the fine structure, which agrees well with simulation results. We show that using a Maxwellian (Gaussian) distribution of velocities in place of unit speed velocities does not affect the growth of the mean-square displacement, but changes the limiting shape of the distributions to a non-Gaussian one. Using the same methods, we give numerical evidence that a nonchaotic polygonal channel model also obeys the central limit theorem, but with a slower convergence rate.

Figure 1

(a) Part of the infinite system, constructed from two square lattices of disks shown in different shades of gray; dashed lines indicate several unit cells and an elastic collision is shown. (b) A single unit cell, defining the geometrical parameters. The billiard domain is the area $Q$ exterior to the disks.

Figure 2

1D channel obtained by unfolding a torus in the $x$ direction.

Figure 3

Lorentz channel studied in 19, 20 with hard upper and lower boundaries; dotted lines indicate unit cells. (b) Fully unfolded triangular Lorentz gas. Dotted lines indicate unit cells forming a channel with periodic upper and lower boundaries.

Figure 4

(a) 2D position distribution; (b) 2D displacement distribution. $r=2.5$; $b=0.4$; $t=50$; $N=5\times {10}^4$ initial conditions.

Figure 5

(Color online) (a) Time evolution of displacement distribution functions. (b) Time evolution of displacement densities, calculated by numerically differentiating a cubic spline approximations to the distribution functions. $r=2.1$; $b=0.2$. The inset in (a) shows the definition of the set $H\left(x\right)$ required later.

Figure 6

(Color online) Position density $f_t$ exhibiting a pronounced fine structure, together with the demodulated slowly varying function ${\overline{\rho }}_t$ and a Gaussian with variance $2Dt$. The inset shows one period of the demodulating fine-structure function $h$. $r=2.3$; $b=0.5$; $t=50$.

Figure 7

(Color online) Partial sums ${\varphi }_m$ up to $m$ terms of the Fourier series for $\varphi$, with $r=2.3$ and $b=0.5$.

Figure 8

(Color online) Displacement density $g_t$, with demodulated ${\overline{\eta }}_t$ compared to a Gaussian of variance $2D$. The inset in (a) shows the fine-structure function $\varphi$ for these geometrical parameters. $r=2.1$; $b=0.2$; $t=50$.

Figure 9

(Color online) Displacement density $g_t$ and demodulated ${\overline{\eta }}_t$, both rescaled by $\sqrt{t}$, at $t=200$ and $t=1000$, compared to a Gaussian of variance $2D$. The inset again shows the fine-structure function $\varphi$. $r=2.01$; $b=0.1$.

Figure 10

(Color online) Displacement densities as in Fig. 5b after rescaling by $\sqrt{t}$, compared to a Gaussian density with mean 0 and variance $2D$. $r=2.1$; $b=0.2$.

Figure 11

(Color online) Difference between rescaled distribution functions and limiting normal distribution with variance $2D$. $r=2.1$; $b=0.2$.

Figure 12

(Color online) Distance of rescaled distribution functions ${\tilde{G}}_t$ from limiting normal distribution $N_{2D}$ in log-log plot. The straight line is a fit to the long-time decay of the data for $r=2.05$.

Figure 13

(Color online) The radial density function ${\tilde{f}}_t^{\mathrm{rad}}$ compared to the numerically calculated radial fine-structure function ${\varphi }^{\mathrm{rad}}$, rescaled to converge to 1 and then vertically shifted for clarity. The demodulated radial density ${\tilde{\rho }}_t^{\mathrm{rad}}$ is also shown. $r=2.3$; $b=0.5$; $t=100$.

Figure 14

(Color online) Comparison of the demodulated radial density ${\tilde{\rho }}_t^{\mathrm{rad}}$ with the exact Meijer-$G$ representation, the large-$r$ asymptotic approximation, and the radial Gaussian with variance $2D$.

Figure 15

(Color online) Rescaled 1D marginal of the displacement density ${\tilde{g}}_t$ and the demodulated version ${\tilde{\overline{\eta }}}_t$ compared to the Gaussian with variance $2D$ and to the asymptotic expression. The latter is not shown close to $x=0$, where it drops to 0. $r=2.3$; $b=0.5$.

Figure 16

(a) The geometry of the polygonal billiard unit cell, shown to scale with ${\varphi }_2=\pi ∕\left(2e\right)$. (b) Part of the polygonal channel with the same parameters.

Figure 17

(Color online) Position density at $t=50$ in the polygonal model with ${\varphi }_2=\pi ∕\left(2e\right)$. The inset shows $h\left(x\right)$ over two periods.

Figure 18

(Color online) Rescaled displacement densities compared to the Gaussian with variance $2D$. The inset shows the function $\varphi$ for this geometry.

Figure 19

(Color online) Demodulated densities ${\tilde{\overline{\eta }}}_t$ for $t=100$, $t=1000$, and $t=10000$, compared to a Gaussian with variance $2D$. The inset shows a detailed view of the peak near $x=0$.

Figure 20

(Color online) Distance of the rescaled distribution functions from the limiting normal distribution for the polygonal model with different values of ${\varphi }_2$. The straight line is a fit to the long-time decay of the irrational case ${\varphi }_2=\pi ∕\left(2e\right)$.