Condensation in a Zero-Range Process with Fast Rates
Abstract
A Zero-range process is a mathematical model used to understand how a group of things, such as particles, move on a graph over time. In a Zero-range process, the rate at which the particles move from one place to another is determined by the number of particles present at the current location.
We study the Zero-range process as a random process of configurations on a lattice and how to understand this process in terms of its generator (see section Zero-Range Processes) and examine the unique stationary distribution of the Zero-range process and its behavior on large lattices with respect to the number of particles by lattice size quotient, . We found that the particles distribute significantly differently if exceeds 1⁄2, than in the case 1⁄2 where in the thermodynamic limit, as discussed in section Zero-Range Processes in the theorem Equivalence of Ensembles in the Thermodynamic Limit and thereafter and explored those concepts in each case numerically.
We found that clusters only form for 1⁄2, and examined the distribution of their size in sections Cluster Count, Cluster Size Tail and Additional Results for Cluster Count and Bulk. We found that the fraction of sites without clusters, , tends to 1 in the thermodynamic limit, meaning the number of sites with clusters on the lattice goes to zero.
To prove conjectured results on the cluster size distribution etc. involves a detailed analysis of the canonical measure . More generally it is interesting to consider, jump rates,
with a constant and power and see how this affects the typical cluster scale and distribution in analogy to results in [3], where similar generalizations led to different speeds of the dynamics within the clusters and in the bulk than in the specialized case.