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

TL;DR

This paper analyzes how the structure of scale-free networks affects particle condensation in a zero range process, revealing conditions for complete condensation and its impact on relaxation dynamics.

Contribution

It provides an analytical condition for complete condensation in zero range processes on scale-free networks based on the degree exponent.

Findings

Complete condensation occurs when δ ≤ 1/(γ-1).

Nodes with degree above a threshold host macroscopic particles.

Relaxation time scales as a power law with network size.

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 $p(n)=n^\delta$. We show analytically that a complete condensation occurs when $\delta \leq \delta_c \equiv 1/(\gamma-1)$ where $\gamma$ 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 $\tau \sim L^z$ with the network size $L$ and a dynamic exponent $z$ in the condensed phase.