Order Parameter as an Additional State Variable of Unstable Traffic Flow

Lubashevsky, Ihor; Mahnke, R. · 2000 · OpenAlex-citations

DOI: 10.1007/978-3-642-59751-0_37

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Summary

This paper addresses the challenge of modeling unstable traffic flow on multilane highways, specifically focusing on the phenomenon of "synchronized motion." Previous macroscopic models described traffic dynamics using only car density ($\rho$) and mean velocity ($v$), which proved insufficient for capturing the complex structure of synchronized flow. The authors aim to explain the observed sequence of phase transitions—free flow to synchronized motion to jam—and the associated hysteresis, which experimental data indicates are essentially multilane effects driven by correlations between lanes. To resolve this, the authors introduce a phenomenological approach that adds a new state variable called the "order parameter" ($h$), ranging from 0 to 1. This parameter accounts for internal correlations in vehicle motion across different lanes, representing "many-body" effects in car interactions that are ignored by standard density and velocity metrics. The authors formulate an evolution equation for $h$ that governs the lane-changing rate, linking it to driver behavior. Specifically, $h$ reflects the compromise between the danger of accidents during lane changes and drivers' desire to move faster. The model assumes that "platoons in speed"—groups of fast-moving cars—drive the value of $h$, while the mean velocity chosen by typical drivers depends on both density and the resulting danger from frequent lane changes. The analysis of the governing equations reveals that the steady-state dependence of the order parameter on density, $h(\rho)$, exhibits an "S"-shaped curve. This structure identifies regions of stability and instability. The lower branch of the curve corresponds to stable free flow ($h \ll 1$), while the upper branch corresponds to the synchronized mode ($h \approx 1$). The decreasing middle branch represents unstable traffic flow. This mathematical structure naturally produces first-order phase transitions with hysteresis. As density increases beyond a critical threshold, the system jumps from free flow to synchronized motion. Conversely, decreasing density requires a lower threshold to return to free flow, creating a metastable region. The model also predicts a second transition from synchronized motion to jam at very high densities, where lane-changing becomes too dangerous, causing $h$ to drop and lane flows to become independent. The significance of this work lies in its ability to describe complex traffic phenomena using a minimal set of variables. By treating the order parameter as an independent state variable, the model explains the sequence of phase transitions and hysteresis without additional ad-hoc assumptions. It clarifies that synchronized motion is characterized by highly correlated vehicle flows across lanes, whereas free flow and jams are characterized by depressed correlations. This approach provides a theoretical foundation for understanding the instability of homogeneous traffic flow and offers a framework for analyzing multilane highway dynamics that aligns with experimental observations of traffic phase transitions.

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• traffic density