Phase transition in a difference equation model of traffic flow

A difference equation is presented to describe traffic flow on a highway. The difference equation model is derived from the optimal velocity models formulated in terms of the differential equations. It is compared with the differential equation models. We investigate phase transitions among the freely moving phase, the coexisting phase in which jams appear, and the uniform congested phase. The linear stability theory is applied and the neutral stability line is obtained. We find the critical point below which no jams appear. To derive the modified Korteweg–de Vries equation near the critical point we apply the reductive perturbation method. We also compare the nonlinear analysis result with that of the optimal velocity model. It is shown that the critical point and the amplitude of the jam are different from those of the optimal velocity model.