E. Ben-Naim

Kinetics of clustering in traffic flows.

We study a simple aggregation model that mimics the clustering of traffic on a one-lane roadway. In this model, each ``car moves ballistically at its initial velocity until it overtakes the preceding car or cluster. After this encounter, the incident car assumes the velocity of the cluster which it has just joined. The properties of the initial distribution of velocities in the small-velocity limit control the long-time properties of the aggregation process. For an initial velocity distribution with a power-law tail at small velocities,

Slow relaxation in granular compaction

{mathit{P}}_{0}

Scaling and multiscaling in models of fragmentation

(v)ensuremath{sim}

Inhomogeneous two-species annihilation in the steady state

{mathit{v}}^{mathrm{ensuremath{mu}}}

Stable distributions in stochastic fragmentation

as vensuremath{rightarrow}0, a simple scaling argument shows that the average cluster size grows as mensuremath{sim}

Scale invariance and lack of self-averaging in fragmentation

{mathit{t}}^{(mathrm{ensuremath{mu}}+1)/(mathrm{ensuremath{mu}}+2)}

Fragmentation with a steady source

and that the average velocity decays as vensuremath{sim}

Kinetics of heterogeneous single-species annihilation

{mathit{t}}^{mathrm{ensuremath{-}}1/(mathrm{ensuremath{mu}}+2)}

Multiscaling in fragmentation

as tensuremath{rightarrow}ensuremath{infty}. We derive an analytical solution for the survival probability of a single car and an asymptotically exact expression for the joint mass-velocity distribution function. We also consider the properties of spatially heterogeneous traffic and the kinetics of traffic clustering in the presence of an input of cars.

Cluster approximation for the contact process

Abstract Experimental studies show that the density of a vibrated granular material evolves from a low density initial state into a higher density final steady state. The relaxation towards the final density follows an inverse logarithmic law. As the system approaches its final state, a growing number of beads have to be rearranged to enable a local density increase. A free volume argument shows that this number grows as N = ϱ (1−ϱ) . The time scale associated with such events increases exponentially ∼ eN, and as a result a logarithmically slow approach to the final state is found ϱ ∞ − ϱ(t) ∼ 1 ln t . Furthermore, a one-dimensional toy model that captures this relaxation dynamics as well as the observed density fluctuations is discussed.