The condensation phenomena of conserve-mass aggregation model with mass-dependent fragmentation

We study a conserved mass aggregation model with mass-dependent fragmentation in regular lattice and scale-free networks. In the model, the whole mass $m$ of a site isotropically diffuse with unit rate. With rate $\omega$, a mass $m^{\lambda}$ is fragmented from the site and moves to a randomly selected nearest neighbor site. Since the fragmented mass is smaller than the whole mass $m$ of a site for $\lambda < 1$, the on-site attractive interaction exists for the case. For $\lambda = 0$, the model is known to undergo the condensation phase transitions as the density of total masses ($\rho$) increases beyond a critical density $\rho_c$. For $0< \lambda <1$, we numerically confirm for several values of $\omega$ that $\rho_c$ diverges with the system size $L$. Hence in thermodynamic limit, the condensed phase disappears and no transitions take place in one dimension. We also explain that there are no transitions in any dimension. On scale-free networks with degree distribution $P(k) \sim k^{-\gamma}$, we numerically confirm for $\gamma >3$ that the condensation transitions occurs at $\rho_c >0$ and its nature is the same as that in regular lattice. However, for $\gamma \leq 3$, the condensation always takes place for $\lambda < \lambda_c$ and masses distribute uniformly without aggregation for $\lambda \geq lambda_c$. We derive $\lambda_c = 1/{\gamma-1}$ via mean-field argument.