Complete Condensation in a Zero Range Process on Scale-Free Networks

Abstract

We study a zero range process on scale-free networks in order to investigate how network structure influences particle dynamics. The zero range process is defined with the particle jumping rate function $p(n)=n^δ$. We show analytically that a complete condensation occurs when $δ\leq δ_c \equiv 1/(γ-1)$ where $γ$ is the degree distribution exponent of the underlying networks. In the complete condensation, those nodes whose degree is higher than a threshold are occupied by macroscopic numbers of particles, while the other nodes are occupied by negligible numbers of particles. We also show numerically that the relaxation time follows a power-law scaling $τ\sim L^z$ with the network size $L$ and a dynamic exponent $z$ in the condensed phase.