Multivalued fundamental diagrams of traffic flow in the kinetic   Fokker-Planck limit
  Starting from interaction rules based on two levels of stochasticity we study the influence of the microscopic dynamics on the macroscopic properties of vehicular flow. In particular, we study the qualitative structure of the resulting flux-density and speed-density diagrams for different choices of the desired speeds. We are able to recover multivalued diagrams as a result of the existence of a one-parameter family of stationary distributions, whose expression is analytically found by means of a Fokker-Planck approximation of the initial Boltzmann-type model. 
Giuseppe Visconti, Michael Herty, Gabriella Puppo, Andrea Tosin
cond-mat.stat-mech math.NA physics.soc-ph