Drift of particles in self-similar systems and its Liouvillian interpretation

We study the dynamics of classical particles in different classes of spatially extended self-similar systems, consisting of (i) a self-similar Lorentz billiard channel, (ii) a self-similar graph, and (iii) a master equation. In all three systems, the particles typically drift at constant velocity and spread ballistically. These transport properties are analyzed in terms of the spectral properties of the operator evolving the probability densities. For systems (i) and (ii), we explain the drift from the properties of the Pollicott-Ruelle resonance spectrum and corresponding eigenvectors.

Figure 1

The self-similar billiard and a trajectory for $\mu >1$, $\mu =1$, and $\mu <1$, respectively. Notice the symmetry of the billiard under $\mu \to 1∕\mu$ and $x\to -x$.

Figure 2

Upper panel: The reference cell geometry. A detailed description is presented in Appendix . Here it is sufficient to note that $\Delta =\sqrt{3}D-2R$, and that with this choice the usual Lorentz channel is retrieved for $\mu =1$. Lower panel: The arrows represent schematically the matching condition. A trajectory leaving a cell from the right is reinjected to the left with a change of vertical coordinate $y=h\to y=h∕\mu$ and the velocity changed according to $v\to v∕\mu$.

Figure 3

(a) Average horizontal position ${⟨X⟩}_t$ in the billiard, as a function of time for different values of $\mu$. We see that for each value of $\mu$ there is a well defined slope (average velocity) and as $\mu$ increases so does the slope. (b) Plot of the mean square displacement ${⟨\Delta X^2⟩}_t$, for the same values of $\mu$. We observe a clear quadratic behavior $\sim t^2$ (upper solid line) for all values of $\mu$, except for $\mu =1$ (lower curve) where the mean square displacement behaves linearly with time. For panel (a) the data corresponds to $\mu =1∕1.53$ (▿), 1/1.5 (−), $1∕1.4$ (×), $1∕1.3$ (◇), $1∕1.2$ (◻), $1∕1.1$ (▵), 1 (엯), 1.1 (▵), 1.2 (◻), 1.3 (◇), 1.4 (×), 1.5 (−), 1.53 (▿). Note that these values are chosen such that we have pairs $\mu$ and $1∕\mu$. The symmetry $v\to -v$ when $\mu \to 1∕\mu$ is seen in (a). In (b) the data for $\mu$ and $1∕\mu$ almost superpose; thus, we have plotted values only for $\mu ⩾1$. We recall the parameter values are $R∕D=0.48$ and $r∕D=0.395$.

Figure 4

Dots: Plot of the velocity $v$ as a function of $\mu$. The solid line is the function $c(\mu -1∕\mu )(\sqrt{\mu }+1∕\sqrt{\mu })$, with $c=0.05$ a constant we use to fit the data close to $\mu =1$ (the reason to plot this function will become clear in Sec. 4). The dashed line is a straight line with slope 0.21. Note that for values of $\mu$ away from $\mu =1$ the dots are not well approximated by the line.

Figure 5

Schematic representation of a self-similar graph with $\mu =2$.

Figure 6

(a) Average horizontal position ${⟨X⟩}_t$ in the graph vs time, for different values of $\mu$. We see that for each value of $\mu$ there is a well-defined slope (velocity) and as $\mu$ increases so does the slope. (b) Mean square displacement ${⟨\Delta X^2⟩}_t$, for the same values of $\mu$. We observe a clear $\sim t^2$ (upper line) for all values of $\mu$ except for $\mu =1$ (lower curve) where the behavior is $\sim t$. The same values of $\mu$ as in Fig. 3 were used.

Figure 7

Plot of the velocity $v$ as a function of $\mu$. The line is the function $c(\mu -1∕\mu )(\sqrt{\mu }+1∕\sqrt{\mu })$, with $c$ a constant we use to fit the data. The reason to plot this function will become clear in Sec. 4.

Figure 8

From Eqs. (A14, A15). The gray region is forbidden when $1∕2<\rho <\sqrt{3}∕2$.

Figure 9

Geometry and definition of $\alpha$, $\beta$, $\gamma$, and $x$.

Figure 10

Values of $\mu$ vs $\lambda$, for fixed $\rho =0.48$ (a), 0.52 (b), which are consistent with Eqs. (A9, A10, A11, A23).

Figure 11

Values of $\mu$ vs $\rho$, for fixed $\lambda =0.395$ which are consistent with Eqs. (A9, A10, A11, A23).