Theoretical foundation of macroscopic traffic models

On the basis of the elementary laws of individual driver behavior concerning the acceleration and interaction of vehicles a gas-kinetic traffic model is constructed. This yields theoretical relations for the fundamental diagram and the equilibrium variance-density relation, but above all it allows the derivation of macroscopic traffic equations. These build a hierarchy of non-closed equations which can be closed by different zeroth-order approximations. In this way one obtains the traffic equations of Lighthill and Whitham or those of Phillips. Alternatively one can derive Euler-like traffic equations which are proposed here since they include additional dynamic equations for the velocity variance. The new variance equation is able to describe the empirically observed increase of variance directly before a traffic jam develops.