Eli Ben-Naim

Coarsening and persistence in the voter model

We investigate coarsening and persistence in the voter model by introducing the quantity

TOWARDS GRANULAR HYDRODYNAMICS IN TWO DIMENSIONS

P_n(t)

Nucleation and growth in one dimension

, defined as the fraction of voters who changed their opinion n times up to time t. We show that

Aggregation with multiple conservation laws.

P_n(t)

Individual entanglements in a simulated polymer melt.

exhibits scaling behavior that strongly depends on the dimension as well as on the initial opinion concentrations. Exact results are obtained for the average number of opinion changes,, and the autocorrelation function,

Kinetics of aggregation-annihilation processes

A(t)equiv sum (-1)^n P_nsim t^{-d/2}

Reaction kinetics of cluster impurities

in arbitrary dimension d. These exact results are complemented by a mean-field theory, heuristic arguments and numerical simulations. For dimensions d>2, the system does not coarsen, and the opinion changes follow a nearly Poissonian distribution, in agreement with mean-field theory. For dimensions d<=2, the distribution is given by a different scaling form, which is characterized by nontrivial scaling exponents. For unequal opinion concentrations, an unusual situation occurs where different scaling functions correspond to the majority and the minority, as well as for even and odd n.

Two scales in asynchronous ballistic annihilation

We study steady-state properties of inelastic gases in two-dimensions in the presence of an energy source. We generalize previous hydrodynamic treatments to situations where high and low density regions coexist. The theoretical predictions compare well with numerical simulations in the nearly elastic limit. It is also seen that the system can achieve a nonequilibrium steady-state with asymmetric velocity distributions, and we discuss the conditions under which such situations occur.

Space covering by growing rays

We study statistical properties of the Kolmogorov-Avrami-Johnson-Mehl nucleation-and-growth model in one dimension. We obtain exact results for the gap density as well as the island distribution. When all nucleation events occur simultaneously, the island distribution has discontinuous derivatives on the rays x_n(t)=nt, n=1,2,3... We introduce an accelerated growth mechanism where the velocity increases linearly with the island size. We solve for the inter-island gap density and show that the system reaches complete coverage in a finite time and that the near-critical behavior of the system is robust, i.e., it is insensitive to details such as the nucleation mechanism.

Species segregation in a model of interacting populations

Aggregation processes with an arbitrary number of conserved quantities are investigated. On the mean-field level, an exact solution for the size distribution is obtained. The asymptotic form of this solution exhibits nontrivial ``double scaling. While processes with one conserved quantity are governed by a single scale, processes with multiple conservation laws exhibit an additional diffusion-like scale. The theory is applied to ballistic aggregation with mass and momentum conserving collisions and to diffusive aggregation with multiple species.