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

TL;DR

This paper develops an analytical framework for a deterministic traffic model using a space-time geometric approach, revealing critical behavior and scaling laws similar to equilibrium statistical mechanics.

Contribution

It introduces a novel height function formulation that captures the dynamics of ECA184, enabling analytical derivation of macroscopic and microscopic observables.

Findings

Derives scaling forms and critical exponents matching numerical results.

Provides a geometric interpretation of space-time jamming transitions.

Connects deterministic traffic dynamics to equilibrium-like statistical mechanics.

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.