Equilibrium-like statistical mechanics in space-time for a deterministic traffic model far from equilibrium

TL;DR

The paper addresses the transient, nonequilibrium dynamics of the deterministic traffic rule ECA184 and its space-time percolation-like transition. It develops an analytic framework by mapping initial lattice configurations to a height function $H(X)$, yielding an equilibrium-like measure over space-time and allowing observables to be computed from the geometry of $H(X)$. Through a drift-diffusion (Fokker–Planck) description and a path-integral formalism, it derives scaling forms and critical exponents that reproduce previously observed numerics, linking macroscopic jam delay and relaxation time to height-function geometry and first-return statistics with an inverse-Gaussian form. The work suggests a principled space-time thermodynamics for deterministic, nonequilibrium systems and points to connections with percolation theory and phenomenological jam dynamics, thereby broadening the conceptual bridge between equilibrium ideas and space-time behavior in driven systems.

Abstract

Motivated by earlier numerical evidence for a percolation-like transition in space-time jamming, we present an analytic description of the transient dynamics of the deterministic traffic model elementary cellular automaton rule 184 (ECA184). By exploiting the deterministic structure of the dynamics, we reformulate the problem in terms of a height function constructed directly from the initial condition, and obtain an equilibrium statistical mechanics-like description over the lattice configurations. This formulation allows macroscopic observables in space-time, such as the total jam delay and jam relaxation time, as well as microscopic jam statistics, to be expressed in terms of geometric properties of the height function. We thereby derive the associated scaling forms and recover the critical exponents previously observed in numerical studies. We discuss the physical implications of this space-time geometric approach.

Figures (9)

• Figure 1: (Left) Spacetime plot of a system of size $L = 12$ and filling density $\rho = 0.5$, showing two jam clusters. The cluster areas $a_i$ and lifetimes $\theta_i$ are indicated by the arrows. (Right) Decomposition of the largest cluster into elementary jams of lengths $m_x, m_y$ and $m_z$.
• Figure 2: Observables: (a) (macroscopic) order parameter $\phi$ for different system sizes $L$ (b) (microscopic) cluster lifetime distribution $p'(\theta)$ for different $L$
• Figure 3: a) $H(X)$ for a system of size $L = 10$ for initial configuration $[0 0 0 1 1 0 1 0 1 1]$, the FCT are shown visually as horizontal line segments with labels $\delta X_i$. b) The space-time plot (with X-axis flipped) for the given initial condition with the elementary jams with lengths $m_i$ colored similarly to the corresponding FCT $\delta X_i$.
• Figure 4: Example of a space–time plot showing three arbitrary subsequences $S1$, $S2$, and $S3$ of the initial condition. Each subsequence contains an equal number of 0s and 1s, and the colored diagonal arrows indicate the corresponding features on the space-time plot. $S1$, $S2$ and $S3$ corresponds to elementary jams, while $S1$ and $S2$ also correspond to jam lifetimes.
• Figure 5: The total delay $\mathcal{A}[H(X)]$ (sum of the colored cluster areas in the space-time plots) and its relationship to the area elements of $H(X)$ (via matching colors) : a) For $\rho_c = 0.5$, $\mathcal{A}[H(X)]$ is given by half the area under the curve $H(X)$. b) For $\rho<0.5$, $\mathcal{A}[H(X)]$ is given by half the area under the "triangular tents" of $H(X)$. c) For $\rho > 0.5$, the area of the terminating jam clusters is given by the "triangular tents" portions, and the non-terminating clusters contribute to infinitely extending horizontal area elements as shown. Note: the space-time plots are presented with the X-axis flipped.