Formularization of entropy and anticipation of metastable states using mutual information in one-dimensional traffic flow

Abstract

Both macro and micro states of Newtonian particles are used to define entropy. However, in previous research studies, which focused on traffic flow from the perspective of thermodynamics and information theory, researchers did not fully determine how entropy can be defined for self-driven particles that do not satisfy Newton’s laws. One attempt to formularize entropy in a one-dimensional traffic flow has been based on information theory. Entropy has been defined by focusing only on the location variation of the vehicles but not on the velocity variation of the vehicles. Moreover, no study has succeeded in considering the time evolution of entropy in a self-driven particle system. In this study, we redefine the entropy and mutual information in one-dimensional traffic flow from the information theoretical perspective, focusing on the velocity variation of vehicles. Moreover, we advance the thermodynamic analysis of one-dimensional traffic flow by examining the time evolution of the new formularized entropy. To observe the time evolution of the formularized entropy, we conducted numerical simulations using both stochastic optimal velocity (SOV) and zero-range process with a slow-start-rule (ZRP+SLS) models that can reproduce a metastable state, which occurs before the congestion begins at a high average velocity and short headway. Moreover, we used the mutual information to anticipate a metastable state – a sign of congestion – by calculating experimental data from three following vehicles on a real expressway to demonstrate the practical application of the formularized mutual information. Thus, we analyzed traffic flow both macroscopically and microscopically using the new formularized entropy to contribute to the studies of self-driven particle systems in both ways of thermodynamics and statistical mechanics or information theory.