Maxwell Model of Traffic Flows

TL;DR

This paper models traffic flow dynamics using kinetic Boltzmann equations with Maxwell collision integrals, revealing complex relaxation behaviors and a distinctive cluster size distribution with algebraic and exponential regimes.

Contribution

It introduces an analytical Maxwell model for traffic flows that characterizes transient behaviors and cluster size distributions, advancing understanding of traffic dynamics.

Findings

Relaxation times vary with velocity, spanning a wide range.

Steady state cluster size distribution exhibits a unique scaling form.

Distribution is algebraic for small cluster sizes and exponential for larger ones.

Abstract

We investigate traffic flows using the kinetic Boltzmann equations with a Maxwell collision integral. This approach allows analytical determination of the transient behavior and the size distributions. The relaxation of the car and cluster velocity distributions towards steady state is characterized by a wide range of velocity dependent relaxation scales, $R^{1/2}<\tau(v)<R$, with $R$ the ratio of the passing and the collision rates. Furthermore, these relaxation time scales decrease with the velocity, with the smallest scale corresponding to the decay of the overall density. The steady state cluster size distribution follows an unusual scaling form $P_m \sim < m>^{-4} \Psi(m/< m>^2)$. This distribution is primarily algebraic, $P_m\sim m^{-3/2}$, for $m\ll < m>^2$, and is exponential otherwise.