Shocks in the Burgers Equation and the Asymmetric Simple Exclusion Process

Abstract

The Burgers equation is used as a model of transport in highways. The function u(r, t) represents the density of cars at r ∈ IR at time t. We assume that, in the absence of other cars, the velocity of a single car is θ. Due to the presence of other cars this velocity can be lower. The variation of density at r in an infinitesimal time interval dt is given by the number of cars entering in the infinitesimal interval (r, r + dr) that is θu (r - dr, t)(1 - u(r, t)) minus the number of cars exiting that interval: θu(r, t)(1- u(r + dr,t)) (note that 1 - u is the density of free space in the interval). Hence the density must satisfy the equation

$$ \frac{{\partial u}}{{\partial t}} = - \theta \frac{{\partial [u(1 - u)]}}{{\partial r}} $$

((1.1a))