Finite Random Domino Automaton

Finite version of Random Domino Automaton (FRDA) - recently proposed a toy model of earthquakes - is investigated. Respective set of equations describing stationary state of the FRDA is derived and compared with infinite case. It is shown that for the system of big size, these equations are coincident with RDA equations. We demonstrate a non-existence of exact equations for size N bigger then 4 and propose appropriate approximations, the quality of which is studied in examples obtained within Markov chains framework. We derive several exact formulas describing properties of the automaton, including time aspects. In particular, a way to achieve a quasi-periodic like behaviour of RDA is presented. Thus, based on the same microscopic rule - which produces exponential and inverse-power like distributions - we extend applicability of the model to quasi-periodic phenomena.

💡 Research Summary

The paper introduces the Finite Random Domino Automaton (FRDA), a finite‑size version of the Random Domino Automaton (RDA) previously proposed as a toy model of earthquakes. The system consists of a one‑dimensional periodic lattice of N cells, each of which can be empty or occupied by a single “ball”. At each discrete time step a ball is thrown at a uniformly chosen cell. If the cell is empty it becomes occupied with probability ν; if it is already occupied, an “avalanche” occurs with probability µ_i (which may depend on the size i of the occupied cluster) and removes the ball from the hit cell together with all adjacent occupied cells. The authors focus on the stationary (steady‑state) regime and derive exact balance equations for the average density ρ, the total number of clusters n_R, and the distribution n_i of clusters of size i.

The density balance (Eq. 4) equates the gain ν(1−ρ) (empty cell becoming occupied) with the loss due to avalanches, ∑_i µ_i n_i i /N. The total‑cluster balance (Eq. 9) accounts for creation of new 1‑clusters inside empty clusters, merging of two clusters across a single empty cell, and loss by avalanches. These equations are exact; no mean‑field approximations are made at this stage.

A crucial observation is that for N ≥ 5 the probabilities governing the creation and merging processes—denoted α_E(i) (probability that an empty cluster adjacent to a size‑i cluster has length > 1) and γ_E(i) (probability that two clusters of sizes k and i−1−k are separated by a single empty cell)—cannot be expressed solely in terms of the macroscopic variables n_i and n_i^0. The authors prove in the Appendix that exact closed forms for α_E(i) and γ_E(i) do not exist beyond N = 4. Consequently, they propose mean‑field approximations:

• α_A(i) ≈ 1 − n_0^1 / ∑_{k=1}^{N−i} n_0^k,
• γ_A(i) ≈ ∑_{k=1}^{i−2}