Nucleation in scale-free networks

Abstract

We have studied nucleation dynamics of the Ising model in scale-free networks with degree distribution P (k) ∼ k −γ by using forward flux sampling method, focusing on how the network topology would influence the nucleation rate and pathway. For homogeneous nucleation, the new phase clusters grow from those nodes with smaller degree, while the cluster sizes follow a power-law distribution. Interestingly, we find that the nucleation rate RHom decays exponentially with the network size N , and accordingly the critical nucleus size increases linearly with N , implying that homogeneous nucleation is not relevant in the thermodynamic limit. These observations are robust to the change of γ and also present in random networks. In addition, we have also studied the dynamics of heterogeneous nucleation, wherein w impurities are initially added, either to randomly selected nodes or to targeted ones with largest degrees. We find that targeted impurities can enhance the nucleation rate RHet much more sharply than random ones. Moreover, ln(RHet/RHom) scales as w γ−2/γ−1 and w for targeted and random impurities, respectively. A simple mean field analysis is also present to qualitatively illustrate above simulation results.PACS numbers: 89.75. Hc, 05.50.+q

Studying rare events using forward-flux sampling: Recent breakthroughs and future outlook

“…In recent years, FFS has been extensively utilized for studying the sensitivity of the nucleation kinetics in the Ising and lattice gas models to the existence and energetic properties of impurities, 34,36,41 mechanical deformations, 37,38 internal and external interfaces, 35,36,[41][42][43] and the topological properties of the underlying network. 39,40,44,45 Since nucleation will only occur if the underlying transition is first-order, all these simulations were conducted in the presence of an external magnetic field or chemical potential, and the starting basin A is always set to the respective metastable phase in each system. Due to the simplicity of the Ising model, it is possible to conduct FFS using both global and local order parameters, namely the total number of up spins (or occupied sites), and the size of the largest cluster of the new phase, respectively.…”

“…In three subsequent papers 40,44,45 from the same group, nucleation was studied in Barabási-Albert (BA) 191 scale-free random networks.…”

“…For instance, Chen et al 40 showed that the rate of homogeneous nucleation decays exponentially with the number of nodes in the BA network, implying that homogeneous nucleation is not possible in the thermodynamic limit (i.e., when the number of nodes goes to infinity). Adding a few impurities (i.e., fixed up-spin nodes), however, resulted in a significant increase in nucleation rate.…”

“…Therefore, developing more robust and efficient variants and implementations of FFS has become an intense focus of research, with earlier methodological efforts covered in several comprehensive reviews published shortly after its development. 32,33 In recent years, FFS has been extensively utilized for studying a wide variety of rare events, such as phase transitions in the Ising [34][35][36][37][38][39][40][41][42][43][44][45] and Potts 46-49 models, crystal nucleation, 50-80 evaporation, [81][82][83][84][85][86] phase separation, 87,88 coalescence, 89 wetting, 90,91 protein folding, rupture and aggregation, [92][93][94][95][96][97][98] DNA hybridization, [99][100][101][102][103][104][105] polymer relaxation and translocation, [106][107][108][109][110][111] ion transport 112 and genetic [113]…”

Nucleation of a three-state spin model on complex networks

“…In addition, a size-effect of the nucleation rate on mean-field-type networks [34] and a nonmonotonic dependences of the nucleation rate on the modularity of networks [35] and on the degree heterogeneity [36] were reported. However, many real systems possess complicated free-energy landscape with several local minima where phase transition happens usually via these intermediate metastable states [37].…”