Semi-deterministic processes with applications in random billiards

TL;DR

The paper advances the theory of semi-deterministic dynamics by characterizing ergodicity for two linked models: an alternating random walk in an elliptical region and a disk-in-strip billiard with rough collision laws. It introduces a dual, coordinate-based framework where ergodicity is transferred between angular and chording components via reversible and $\\dag$-reversible Markov kernels, and leverages the Abel transform to prove uniqueness of ergodicity in the underlying integral equations. A key contribution is the explicit construction of rough microstructures producing ergodic dynamics and the demonstration of a robust isomorphism between the two systems, revealing a deep connection between microscopic roughness and macroscopic ergodicity. The results illuminate how singular transition kernels can nonetheless exhibit strong ergodic behavior, situating these semi-deterministic processes between diffusion-like and fully deterministic dynamics with practical implications for modeling physically realistic systems with microstructure-induced randomness.

Abstract

We study the ergodic properties of two classes of random dynamical systems: a type of Markov chain which we call the \textit{alternating random walk} and a certain stochastic billiard system which describes the motion of a free-moving rough disk bouncing between two parallel rough walls. Our main results characterize the types of Markov transition kernels which make each system ergodic -- in the first case, with respect to uniform measure on the state space, and in the second case, with respect to Lambertian measure (a classic measure from geometric optics). In addition, building on results from \cite{rudzis2022}, we give explicit examples of rough microstructures which produce ergodic dynamics in the second system. Both systems have the property that the transition kernel governing the dynamics is singular with respect to uniform measure on the state space. As a result, these systems occupy a kind of mean in the problem space between diffusive processes, where establishing ergodicity is relatively easy, and physically realistic deterministic systems, where questions of ergodicity are far less approachable.

Figures (7)

• Figure 1: The alternating random walk alternates between taking steps along vertical and horizontal chords of the ellipse $E$. We make the process into a Markov process by introducing an extra "sign" variable $s \in \{\pm1\}$ to keep track of whether the next step will be in the horizontal or vertical direction.
• Figure 2: Above is shown the kinetic energy unit sphere $\mathbb{S}^2$, oriented so that the unit vector $n_2 = (0,0,1)$, points "out of the board." The vectors $\chi = \sqrt{2(J+m)^{-1}}(1,-1,0)$ and $\chi' = R(\chi) = \sqrt{2(J+m)^{-1}}(1,1,0)$ determine the locally conserved quantities \ref{['eq:conserved1']} and \ref{['eq:conserved2']}, respectively. The angle between $\chi$ and $\chi'$, with respect to the inner product $\langle \cdot, \cdot \rangle$, is $\pi - \gamma$, where $\gamma$ is defined by \ref{['eq:gammadef']}. The velocity process alternates between points in the hemispheres $\mathbb{S}^2_-$ and $\mathbb{S}^2_+$, represented above by the points labeled $\ominus$ and $\oplus$, respectively. Under the no-slip collision law, the projected images of the velocities alternate between reflections through the vector $\chi$ and through the vector $\chi'$. Consequently, if $v^-$ is the initial velocity, the subsequent velocities will be concentrated in the pair of circles $\mathcal{C}_{v^-} := \{v \in \mathbb{S}^2 : \langle v, n_2\rangle = \pm \langle v^-,n_2\rangle\}$ for all time. If $\gamma/\pi$ is irrational, then uniform measure on $\mathcal{C}_{v^-}$ is ergodic for any $v^- \in \hat{\mathbb{S}}^2$, whereas if $\gamma/\pi$ is rational, then any ergodic measure will be uniform over a finite subset of $\mathcal{C}_{v^-}$.
• Figure 3: (A) Rectangular teeth microstructure. (B) Derivation of the reflection law for the rectangular teeth microstructure: Starting from the initial condition $(x,\theta)$, if the unfolded trajectory between two teeth ends at a point $x'$ in a white region, then the angle of exit $\theta'$ will equal $\pi - \theta$. Otherwise, if $x'$ lies in a gray region, then $\theta' = \theta$. (C) Focusing circular arc microstructure.
• Figure 4: (A) Original periodic microstructure $W$. (B) New microstructure $W^\delta$. (C) Detail of a $\delta$-nub.
• Figure 5: Reflection from a nub.

Theorems & Definitions (46)

• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 2.4
• Theorem 2.5
• Remark 2.6
• Remark 2.7
• Remark 2.8
• Proposition 3.1
• Remark 3.2