# Multi-class fundamental diagrams from the Prigogine-Herman-Boltzmann equation

## TL;DR

This paper develops a theoretical multi-class traffic flow fundamental diagram using a gas-kinetic model, identifying a critical density influenced by vehicle class proportions and validating it against empirical data.

## Contribution

It introduces a multi-class generalization of the Prigogine-Herman-Boltzmann equation to derive a fundamental flow-density relation for mixed vehicle types.

## Key findings

- Existence of a critical density depending on vehicle class proportions
- Derived flow-density relation for two-class vehicle mixture
- Comparison with empirical traffic data

## Abstract

Our aim in this paper is to establish a theoretical fundamental diagram for a multi-class traffic flow from a gas-kinetic-like traffic model. We start with a multi-class generalization of the Prigogine-Herman-Boltzmann equation to construct the fundamental relation for this system. We show that there exists a critical density which depends on the relative concentration of slow and fast users and describe a procedure to find the threshold value. Finally, our flow-density relation for a two-class mixture of vehicles is contrasted with empirical data in the literature.