On the kinetic theory of vehicular traffic flow: Chapman-Enskog expansion versus Grads moment method

Based on a Boltzmann-like traffic equation for aggressive drivers we construct in this paper a second-order continuum traffic model which is similar to the Navier-Stokes equations for viscous fluids by applying two well-known methods of gas-kinetic theory, namely: the Chapman-Enskog method and the method of moments of Grad. The viscosity coefficient appearing in our macroscopic traffic model is not introduced in an ad hoc way - as in other second-order traffic flow models - but comes into play through the derivation of a first-order constitutive relation for the traffic pressure. Numerical simulations show that our Navier-Stokes-like traffic model satisfies the anisotropy condition and produces numerical results which are consistent with our daily experiences in real traffic.

💡 Research Summary

The paper presents a rigorous derivation of a second‑order macroscopic traffic flow model that closely parallels the Navier‑Stokes equations for viscous fluids. Starting from a Boltzmann‑type kinetic equation specifically designed for “aggressive drivers,” the authors treat each vehicle as a particle whose interactions are modeled as inelastic collisions. The collision operator incorporates driver‑specific parameters such as reaction time, acceleration/deceleration capability, and a sensitivity factor that captures the tendency of aggressive drivers to close gaps quickly. Consequently, the single‑particle distribution function (f(x,v,t)) depends on position, speed, and time, and its moments give the macroscopic fields: vehicle density (\rho(x,t)), average speed (u(x,t)), and a traffic “temperature” (\theta) (the variance of speeds).

Two classical gas‑kinetic techniques are applied to close the hierarchy of moment equations: the Chapman‑Enskog expansion and Grad’s moment method. In the Chapman‑Enskog approach the distribution is expanded in powers of a small parameter (\epsilon) representing the mean free path (or mean collision time). The zeroth‑order term is a local equilibrium (Maxwell‑Boltzmann‑like) distribution, while the first‑order correction introduces a term proportional to the spatial gradient of the mean speed. Substituting the resulting expressions for the moments into the conservation laws yields a continuity equation and a momentum equation that contain a pressure tensor (P). Crucially, the pressure tensor includes a viscous contribution (-\nu \partial u/\partial x) where the viscosity coefficient (\nu) is expressed analytically in terms of microscopic quantities (average collision time (\tau) and the aggressiveness parameter (\alpha)). Thus, (\nu) is not an ad‑hoc constant but a physically derived parameter.

Grad’s method proceeds by representing (f) as a Gaussian core plus weighted contributions of higher‑order moments (the second‑order pressure tensor and the third‑order heat flux). By truncating the hierarchy at the second order and approximating the third‑order moment through the lower‑order fields, a closed set of equations identical in structure to those obtained via Chapman‑Enskog is derived. Again, the viscous term emerges naturally, and the same functional dependence of (\nu) on (\tau) and (\alpha) is recovered.

Both derivations lead to the following macroscopic system (in one spatial dimension):

1. Continuity: (\displaystyle \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x}=0).
2. Momentum: (\displaystyle \frac{\partial (\rho u)}{\partial t} + \frac{\partial (\rho u^{2}+P)}{\partial x}=0), with
   \