Velocity and cluster distributions in a bottleneck system

Velocity and cluster distributions for particles with unidirectional motion in one dimension are studied. The particles never pass each other, like cars on a narrow road that does not allow overtaking. As a result, particles cluster behind slow particles (queues are formed behind slow cars). Thus, the actual velocity of each particle is to a large extent determined by slow particles further ahead. Considering all possible permutations of N particles with initial velocities {v(i)}, the average number of particles with actual velocity v(i) is (N+1)/[i(i+1)] (in the sequence {v(i)}, the initial velocities are listed with monotonically increasing values). For i large and v(i)proportional to i the average number of actual velocities is thus a power law in v(i), even though the average cluster density is found to be independent of cluster size, L. On the other hand, the cluster density varies significantly with cluster velocity; we obtain [(N-i)!(N-L)!]/[N center dot N!(N-L-i+1)!]. The average velocity at a given position in the sequence of N particles, and the average global velocity are determined. Explicit results for several distributions of the initial velocities show that the global velocity depends sensitively on the form of this distribution.