ABSTRACT
A theoretical study is undertaken of the dynamics of a ball which is bouncing inelastically on a randomly vibrating platform. Of interest are the distributions of the number of flights nf and the total time tauc until the ball has effectively "collapsed," i.e., coalesced with the platform. In the strictly elastic case both distributions have power law tails characterized by exponents that are universal, i.e., independent of the detail of the platform noise distribution. However, in the inelastic case both distributions have exponential tails: P(nf) approximately exp[-theta1nf] and P(tauc) approximately exp[-theta2tauc]. The decay exponents theta1 and theta2 depend continuously on the coefficient of restitution and are nonuniversal; however, as one approaches the elastic limit, they vanish in a manner which turns out to be universal. An explicit expression for theta1 is provided for a particular case of the platform noise distribution.
ABSTRACT
A detailed study is undertaken of the v{max}=1 limit of the cellular automaton traffic model proposed by Nagel and Paczuski [Phys. Rev. E 51, 2909 (1995)]. The model allows one to analyze the behavior of a traffic jam initiated in an otherwise freely flowing stream of traffic. By mapping onto a discrete-time queueing system, itself related to various problems encountered in lattice combinatorics, exact results are presented in relation to the jam lifetime, the maximum jam length, and the jam mass (the space-time cluster size or integrated vehicle waiting time), both in terms of the critical and the off-critical behavior. This sets existing scaling results in their natural context and also provides several other interesting results in addition.
