# Kardar-Parisi-Zhang Universality of the Nagel-Schreckenberg Model

## TL;DR

This paper proves that the Nagel-Schreckenberg traffic model belongs to the KPZ universality class for all maximum velocities, using hydrodynamics theory and large-scale simulations to confirm the theoretical predictions.

## Contribution

It establishes the KPZ universality class for the NaSch model with arbitrary maximum velocities, extending previous results limited to the case v_max=1.

## Key findings

- NaSch model belongs to KPZ class for v_max>1
- Simulation results match KPZ asymptotic solutions
- Early-time effects influence numerical exponent determination

## Abstract

Dynamical universality classes are distinguished by their dynamical exponent $z$ and unique scaling functions encoding space-time asymmetry for, e.g. slow-relaxation modes or the distribution of time-integrated currents. So far the universality class of the Nagel-Schreckenberg (NaSch) model, which is a paradigmatic model for traffic flow on highways, was not known except for the special case $v_{\text{max}}=1$. Here the model corresponds to the TASEP (totally asymmetric simple exclusion process) that is known to belong to the superdiffusive Kardar-Parisi-Zhang (KPZ) class with $z=3/2$. In this paper, we show that the NaSch model also belongs to the KPZ class \cite{KPZ} for general maximum velocities $v_{\text{max}}>1$. Using nonlinear fluctuating hydrodynamics theory we calculate the nonuniversal coefficients, fixing the exact asymptotic solutions for the dynamical structure function and the distribution of time-integrated currents. Performing large-scale Monte-Carlo simulations we show that the simulation results match the exact asymptotic KPZ solutions without any fitting parameter left. Additionally, we find that nonuniversal early-time effects or the choice of initial conditions might have a strong impact on the numerical determination of the dynamical exponent and therefore lead to inconclusive results. We also show that the universality class is not changed by extending the model to a two-lane NaSch model with dynamical lane changing rules.