Condensation phase transitions of symmetric conserved-mass aggregation model on complex networks

TL;DR

This paper studies how symmetric conserved-mass aggregation models undergo phase transitions on complex networks, revealing the influence of network topology on condensation phenomena and identifying conditions for their occurrence.

Contribution

It demonstrates the effects of network structure on condensation transitions in the SCA model, especially highlighting differences between networks with degree exponent $oldsymbol{eta}$ greater than or less than 3.

Findings

Condensation transitions occur on RNs and SFNs with $oldsymbol{eta > 3}$, similar to regular lattices.

No phase transition occurs on SFNs with $oldsymbol{eta extless 3}$; instead, condensation always occurs.

Lamb survival analysis indicates indefinite survival on RNs and SFNs with $oldsymbol{eta > 3}$, but not on SFNs with $oldsymbol{eta extless 3}$.

Abstract

We investigate condensation phase transitions of symmetric conserved-mass aggregation (SCA) model on random networks (RNs) and scale-free networks (SFNs) with degree distribution $P(k) \sim k^{-\gamma}$. In SCA model, masses diffuse with unite rate, and unit mass chips off from mass with rate $\omega$. The dynamics conserves total mass density $\rho$. In the steady state, on RNs and SFNs with $\gamma>3$ for $\omega \neq \infty$, we numerically show that SCA model undergoes the same type condensation transitions as those on regular lattices. However the critical line $\rho_c (\omega)$ depends on network structures. On SFNs with $\gamma \leq 3$, the fluid phase of exponential mass distribution completely disappears and no phase transitions occurs. Instead, the condensation with exponentially decaying background mass distribution always takes place for any non-zero density. For the existence of the condensed phase for $\gamma \leq 3$ at the zero density limit, we investigate one lamb-lion problem on RNs and SFNs. We numerically show that a lamb survives indefinitely with finite survival probability on RNs and SFNs with $\gamma >3$, and dies out exponentially on SFNs with $\gamma \leq 3$. The finite life time of a lamb on SFNs with $\gamma \leq 3$ ensures the existence of the condensation at the zero density limit on SFNs with $\gamma \leq 3$ at which direct numerical simulations are practically impossible. At $\omega = \infty$, we numerically confirm that complete condensation takes place for any $\rho > 0$ on RNs. Together with the recent study on SFNs, the complete condensation always occurs on both RNs and SFNs in zero range process with constant hopping rate.