Headway oscillations and phase transitions for diffusing particles with increased velocity

An asymmetric exclusion process with $N$ particles on $L$ sites is considered where particles can move one or two sites per infinitesimal time-step. An exact analysis for N=2 and a mean-field theory in comparison with simulations show even/odd oscillations in the headway distribution of particles. Oscillations become maximal if particles try to move as far as possible with regard to their maximum velocity and particle exclusion. A phase transition separates two density profiles around a generated perturbation that plays the role of a defect. The matrix-product ansatz is generalized to obtain the exact solution for finite $N$ and $L$. Thermodynamically, the headway distribution yields the mean-field result as $N^{-1}\to 0$ while it is not described generally by a product measure.