Condensation In A Zero Range Process On Scale-free Networks

The condensation is an intriguing phenomenon observed in nonequilibrium systems. In interacting particle systems, a finite fraction of particles may be condensed onto a single site. The condensation has been studied mostly on periodic lattices. On the other hand, Particularly, many real-world networks are found to be scale-free. Jae Dong Noh et al. has studied the condensation phenomenon in a zero range process on scale-free networks. They show that the stationary state property depends only on the degree distribution of underlying networks. The model displays a stationary state phase transition between a condensed phase and an uncondensed phase, and the phase diagram is obtained analytically.We study the condensation phenomenon in a zero range process on weighted scale-free networks in order to show how the weighted transport influences the particle condensation. Instead of the approach of grand canonical ensemble which is generally used in a zero range process, we introduce an alternate approach of the mean-field equations to study the dynamics of particle transport. It is recently shown that the weighted static network may seriously influence the particle condensation on it, where the weights on an edge are asymmetric. However, most of the realistic networks are weighted evolving networks where the weights on an edge are symmetric. For understanding how the structure of these networks influence the particle evolution we study the condensation phenomenon on two typical weighted evolving networks with symmetric weights on the edge, where the relationships between the strength and the degree of a node are linear and power law, respectively. By using of both the approach of grand canonical ensemble and the approach of mean field equation to describe the evolution of particles on a node, we find that the stationary state of particle is determined by the strength distribution, in contrary to the case of asymmetric weights on the edge where the stationary state is determined by the degree distribution. Moreover, we have shown that the relaxation time t follows a power law with the network size L in the condensed phase. Numerical simulations confirm the analytic results.