Decelerating Microdynamics Can Accelerate Macrodynamics in the Voter Model

For the voter model, we study the effect of a memory-dependent transition rate. We assume that the transition of a spin into the opposite state decreases with the time it has been in its current state. Counterintuitively, we find that the time to reach a macroscopically ordered state can be accelerated by slowing down the microscopic dynamics in this way. This holds for different network topologies, including fully connected ones. We find that the ordering dynamics is governed by two competing processes which either stabilize the majority or the minority state. If the first one dominates, it accelerates the ordering of the system. The conclusions of this Letter are not restricted to the voter model, but remain valid to many other spin systems as well.

Figure 1

Average consensus times $T_{\kappa }$ for varying values of the inertia slope $\mu$ and fixed saturation value ${\nu }_s=0.9$. Sample sizes vary between ${10}^3–{10}^4$ simulation runs. Filled, black symbols always indicate the values of $T_{\kappa }$ at $\mu =0$. (a) $2d$ regular lattices ($k_i=4$) with system sizes: (○) $N=100$, (△) $N=400$, (□) $N=900$. The inset shows how consensus time scales with system size in regular lattices at $\mu ={\mu }^{*}$: (◇) $1d$, (×) $2d$, (⊲) $3d$, (★) $4d$. (b) Small-world networks obtained by randomly rewiring a $2d$ regular lattice with probability: (○) $p_r=0$, (△) $p_r=0.001$, (□) $p_r=0.01$, (◇) $p_r=0.1$, (★) $p_r=1$. The system size is $N=900$. (c) Fully connected networks (mean-field case, $k_i=N-1$) with system sizes: (○) $N=100$, (□) $N=900$, (◇) $N=2500$, (★) $N={10}^4$. Lines represent the numerical solutions of Eqs. (5, 6, 7) with the specifications in the text. The inset shows the collapse of the simulation curves by scaling $\mu$ and $T_{\kappa }$ as explained in the text.