Kinetic Theory of Traffic Flows
E. Ben-Naim and P.L. Krapivsky
We describe traffic flows in one lane roadways using kinetic theory,
with special emphasis on the role of quenched randomness in the
velocity distributions. When passing is forbidden, growing clusters
are formed behind slow cars and the cluster velocity distribution is
governed by an exact Boltzmann equation which is linear and has an
infinite memory. The distributions of the cluster size and the
cluster velocity exhibit scaling behaviors, with exponents dominated
solely by extremal characteristics of the intrinsic velocity
distribution. When passing is allowed, the system approaches a steady
state, whose nature is determined by a single dimensionless number,
the ratio of the passing time to the collision time, the two time
scales in the problem. The flow exhibits two regimes, a laminar flow
regime, and a congested regime where large slow clusters dominate the
flow. A phase transition separates these two regimes when only the
next-to-leading car can pass.
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