Interchanging between cluster movement and collision

From Sec. 4.2 and Sec. 4.6, properties of a cluster from a microscopic perspective are better understood. These properties are vital in adapting the algorithm for greater efficiencies. Firstly, we note that the mass of clusters are largely preserved until collision. Secondly,

the rate of entire clusters moving across lattice spaces can be accurately predicted for both asymmetric and symmetric systems (see Sec. 4.2.1), where clusters with massmtransports all its mass to a neighbouring site with rate ∼ mγ for the totally asymmetric case, and

∼mγ−1for the symmetric case. This property is used to improve numerical efficiencies for later parts of the dynamics, namely the coarsening and the explosive condensation stage.

For the later parts of system dynamics, where interactions are dominated by clus- ters, transportation of an entire cluster can be simplified as one step by using the rates of cluster movement. In this case, the scenario in Fig. 4.4 and Fig. 4.5 can be replaced by a single numerical step. Clusters are therefore regarded as independent entities each with non-zero rates, which characterises a separate continuous-time model. Updates on the con- figuration space would involve the movement of entire clusters instead of a unit of mass. The interactions thereby enters a purely cluster-centric phase, in which the computationally intensive process of particle-particle movement of clusters can be substituted by a single step.

Cut-offand cluster-cluster interactions

There remain two questions. Firstly, switching to a completely cluster-centric model is only accurate if clusters are well separated. During the nucleation phase, occupants on the con- figuration are of sizem∼O(1). The estimation of cluster speeds are only accurate for when the dynamics are dominated by the movement of large clusters of sizem∼O(L). Therefore, a cut-offtime for when large clusters dominate should be identified. Our algorithm should therefore switch from a particle-particle interaction to cluster-cluster interaction upon a good cut-off time. Secondly, it is true that the movement of clusters over empty lattice sites can be easily predicted by the said method above. However, the stochasticity of mass exchange in cluster-cluster interactions is hard to predict (c.f. Sec. 4.2.2).

We first consider the question of the switch-time. In the explosive condensation model, there is no clear cut-offbetween cluster nucleation and its subsequent stages. The simple solution is to use a stricter cut-offcondition. The most straightforward method is to switch methods when there are no two non-zero clusters neighbouring each other. This is the case when nucleation has strictly finished, such that

X

x∈Λ

ηxηx+1=0 . (5.4)

As explained in Sec. 4.2.2, the transfer of mass∆m(m₁,m₂) between clusters with sizem1andm2has no simple solution. While form1 m2in the totally asymmetric case, ∆m ∼ 0.5 form₂γ d, there [59]. For the symmetric case, we know that∆m ∼ m, but it is difficult to determine the exact solution, as there is a great element of stochasticity.

Therefore, to preserve the numerical accuracy of∆mexchanges between clusters, it is best to preserve interactions between clusters to the full simulation method.

Switching numerical methods and back

The numerical simulation is required to switch from the cluster-movement phase to a cluster- exchange phase, and revert back during cluster-cluster collision, such that the dynamics would begin normally; until condition Eq. (5.4) is reached. As this is another continuous- time Markov process, the Gillespie method can be implemented but with a different set of rates.

If there are k clusters on a symmetric lattice, each would have ratesri = mγ

−1

i d

for the model characterized by Eq. (4.2). For Waclaw’s original model this would beri = mγ_i−1dγ. However, if there is another cluster that shares the destination site as neighbour, the rate for that jump should be given byri =mγ_idandmγ

−1

i d

γ_{respectively. This is because}

that step would trigger another set of dynamics that is reverted back to the original set of algorithms. Therefore the rates for that specific jump should be the same as original rates as well. The same idea would apply for the totally asymmetric case. The time update is given by the same method using the Gillespie time update outlined in Eq. (5.1), but with R=Pk_i=1ri.

If a cluster jumps to a space that is neighboured by another cluster, it would enter another phase where the interactions between the two clusters are considered independently from the rest of the dynamics. Note that towards the end of the dynamics, system size is m∼ O(L). Consequently, the exchange of mass between two clusters greatly exceeds the rate of clusters moving into empty spaces, asm2γ mγd. Therefore, the diffusivity of clusters can be ignored when there are two clusters exchanging mass.

The interaction between the two colliding clusters can be mapped on the diagram in Fig. 5.2, where cluster-cluster interaction terminates when either the two clusters merge, or mass on both sites are conserved. In this process, some mass might be transferred from one to another. The entire algorithm can be found in Appendix B.3.