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First-passage processes in a deterministic one-dimensional cellular automaton model of traffic flow
Ofer Biham1, Gilad Hertzberg Rabinovich1, Eytan Katzav1
1The Hebrew University, Racah Institute of Physics, Jerusalem 9190401, Israel.
Physical Review. E
|July 24, 2026
Summary
This study analyzes traffic flow using a cellular automaton (CA) model, revealing insights into car stopping times and congestion dynamics. Analytical results provide a deeper understanding of traffic flow relaxation processes.
Area of Science:
- Physics
- Complex Systems
- Statistical Mechanics
Background:
- Traffic flow can be modeled using deterministic one-dimensional cellular automata (CA).
- CA rule 184 exhibits a continuous dynamical phase transition at car density p=1/2, separating free-flowing and congested phases.
Purpose of the Study:
- To present analytical results for first-passage processes in a deterministic CA traffic flow model.
- To derive closed-form expressions for first-stopping (FS) and stopping probabilities.
- To analyze last-stopping (LS) times and the number of stopping events in the low-density phase.
Main Methods:
- Utilizing the framework of first-passage processes.
- Deriving closed-form expressions for various time distributions (FS, stopping, LS).
- Analyzing the relationship between LS time and the cumulative number of stopping events (N_S).
Main Results:
- Closed-form expressions for P(T_FS=t) and P_S(t) were derived.
- In the 0
- Joint and conditional distributions P(T_LS=t, N_S=n) were presented.
Conclusions:
- The study provides insights into the timescales of congestion and relaxation in deterministic traffic flow from an individual car's perspective.
- The findings offer broader implications for understanding complex relaxation processes in many-particle systems, including deterministic surface growth.
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