Billiards with Spatial Memory

Many classes of active matter develop spatial memory by encoding information in space. We present a framework based on mathematical billiards, wherein particles remember their past trajectories. Despite its deterministic rules, such a system is strongly nonergodic and exhibits intermittent statistics and complex pattern formation. We show how these features emerge from the dynamic change of topology. Our work illustrates how the dynamics of a single-body system can dramatically change with spatial memory, laying the groundwork to further explore systems with complex memory kernels.

Figure 1

(a) Underlying principle of self-avoiding billiards. A particle moves ballistically from the initial condition $({\mathbf{x}}_0,{\alpha }_0)$ where ${\mathbf{x}}_0$ is the initial position vector and ${\alpha }_0$ is the initial angle. The particle elastically reflects on the boundaries and its own trajectories, creating a line segment of length $l_i$, where $i\in [1,\infty )$ is the number of segments. The particle moves until it self-traps in a singular point, where the total length of the trajectory is $\mathcal{L}=\mathrm{\Sigma }{l}_i$ (see Supplemental Material [54], Video 1). (b) Two particles at initially close distance $r_0=|{\mathbf{x}}_0^2-{\mathbf{x}}_0^1|$ and identical initial angle ${\alpha }_0$ diverge significantly from their path and self-trap on distinct locations at a distance $r_f$ (see Supplemental Material, Video 2). (c) A self-avoiding particle changes the effective geometry of the billiard ($\mathrm{\Omega }$, highlighted in blue) over time. ${\mathcal{M}}_{p_i}$ denotes the incident vector of the particle where I, II, and III are the left, bottom, and right edges of the triangle, respectively, and the digits refer to the segment numbers of the trajectory (e.g., 1 for $l_1$). (d) The dynamics of effective billiard area $\tilde{\mathrm{\Omega }}$ and the distance between the two particles $r$. The inset shows the ensemble average of the effective area versus time normalized by the trapping time $t_f$. (e) Distribution of $r_f$ for polygons with $\mathcal{N}\in [3,8]$. In all cases, $r_0={10}^{-3}$ and the distributions are calculated with ${10}^5$ samples.

Figure 2

The arrested state of triangular self-avoiding billiard. (a) Probability density ($\rho$) of final self-trapping locations. (b) The total length distribution. Inset: semilogarithmic representation to highlight the tail. (c) Distribution of self-trapped positions in regions I–V, shown in (b), and the examples of trajectories of various lengths. The inset on the right shows a magnified view of the zigzag motion in region V. For other polygons, see Supplemental Material Sec. E [54].

Figure 3

The arrested state of polygon billiards. (a) Probability density ($\rho$) of final self-trapping locations in a square, pentagon, hexagon, heptagon, octagon, and nonagon as well as enneadecagon ($\mathcal{N}=19$) and icosagon ($\mathcal{N}=20$). Note that the color bars for each polygon are adjusted to better highlight the distributions. (b) The logarithmic total length distribution for regular polygons of different shapes.