Fractality of the nonequilibrium stationary states of open volume-preserving systems. II. Galton boards

Galton boards are models of deterministic diffusion in a uniform external field, akin to driven periodic Lorentz gases, here considered in the absence of dissipation mechanism. Assuming a cylindrical geometry with axis along the direction of the external field, the two-dimensional board becomes a model for one-dimensional mass transport along the direction of the external field. This is a purely diffusive process which admits fractal nonequilibrium stationary states under flux boundary conditions. Analytical results are obtained for the statistics of multibaker maps modeling such a nonuniform diffusion process. A correspondence is established between the local phase-space statistics and their macroscopic counterparts. The fractality of the invariant state is shown to be responsible for the positiveness of the entropy production rate.

Figure 1

Cylindrical Galton board with a nonvanishing external field and the energy $E=0$. A trajectory is released at zero velocity at $x=0$ and falls along the external field until it collides a first time with a disk. It then wanders around, coming back close to $x=0$ once, after which it moves further along the channel until it reaches the border at $x=L$. To compute the successive collision events, the numerical integration scheme uses an exact quartic equation solver based on the Galois formula [24].

Figure 2

(Color online) Difference between the generalized angle coordinate ${\psi }_n\left(\varphi \right)$ and $\varphi$, here computed for $\sigma =0.44l$, $n_0=0$, and $n=1,\dots ,20$. Larger differences occur at smaller $n$, where the effect of the external field is most noticeable.

Figure 3

Invariant density associated to a closed Galton board of size $L=10l$, with reflection at the boundaries $x=0$ and $x=10l$ and energy $E=l/2$. This is an equilibrium system as reflected by the uniformity of the phase portraits.

Figure 4

(Color online) Equilibrium stationary density of the closed Galton board obtained for a channel of length $L=1$, with $2N+1$ disks, $N=25$. The solid line is the constant equilibrium density $\mathcal{P}\left(X\right)=1$.

Figure 5

(Color online) Nonequilibrium stationary density of the Galton board obtained for a channel of length $L=1$, with 51 disks $\left(N=25\right)$. Two solid lines are shown which are barely distinguishable, corresponding to Eq. (15) (red) and Eq. (29) (green) with $E=l/2$ and thus $n_0=0$.

Figure 6

Nonequilibrium phase-space densities of the open Galton channel with a geometry similar to that shown in Fig. 1, with absorbing boundaries at $x=0$ and $x=1$, and stochastic injection of particles at $x=0$ only. The plots are histograms of the phase space attached to a given disk, computed over grids of $500\times 500$ cells, and counting the average collision rates of many trajectories on that disk. Trajectories are computed one after the other, from their injections, here at the left boundary only, to their subsequent absorption upon their first passage to either of the left and right boundaries. The corresponding disk labels $n=25,\dots ,50$ are indicated in the respective figures. Disk 50 is the one before last. Black areas correspond to absorption at the nearby boundary. The color white is associated to injection from the left boundary. Thus hues of gray correspond to phase-space regions with mixtures of phase-space points which are mapped backward to the left and right borders. The corresponding overall densities are shown in Fig. 5.

Figure 7

(Color online) Forced multibaker map [Eq. (39)]. The cells have areas $a_n{l}^2$, with coordinates $\left(x,y\right)$, and labeled by a positive integer $n$ ($n_0=0$ here). The map divides each cell into three vertical rectangles. The left rectangle is mapped to the bottom horizontal rectangle in the left neighboring cell. Likewise the right rectangle is mapped to the top horizontal rectangle in the right neighboring cell. The middle vertical rectangle is mapped to the middle horizontal rectangle in the same cell but are so tiny they are barely visible already for $n=1$.

Figure 8

(Color online) The external field induces a bifurcation such that the simple periodic orbit bouncing off two neighboring disks at normal angles is replaced by two such orbits. In this situation where $E=0$, which corresponds to a vanishing kinetic energy on the left border, one of these two periodic orbits is stable (to the left) and the other one unstable (to the right). The stability of the periodic orbit is quickly lost as $E$ is increased.

Figure 9

Elliptic islands around the stable periodic orbit shown in Fig. 8: (a) disk on the upper left corner; (b) central disk. The phase-space coordinates used here (not the appropriate Birkhoff coordinates) are $\varphi$, the angle around the corresponding disk, and $\xi$ the sinus of the velocity angle measures with respect to the normal to the disk.

Figure 10

(Color online) Comparison between the graphs of $F_n\left(x_k\right)$ vs $y_n\left(x_k\right)$ (red and slightly shifted to the right) and the corresponding incomplete Takagi functions $T_n\left(x_k\right)$ (blue). Each curve is computed at $3^{10}+1$ different points $x$, uniformly spread between 0 and 1. Only $2^{10}+1$ correspond to different points in the graphs of $T_n$.