Coarsening and percolation in the kinetic 2d Ising model with spin exchange updates and the voter model

Abstract

We study the early time dynamics of bimodal spin systems on 2d lattices evolving with different microscopic stochastic updates. We treat the ferromagnetic Ising model with locally conserved order parameter (Kawasaki dynamics), the same model with globally conserved order parameter (nonlocal spin exchanges), and the voter model. As already observed for non-conserved order parameter dynamics (Glauber dynamics), in all the cases in which the stochastic dynamics satisfy detailed balance, the critical percolation state persists over a long period of time before usual coarsening of domains takes over and eventually takes the system to equilibrium. By studying the geometrical and statistical properties of time-evolving spin clusters we are able to identify a characteristic length p (t), different from the usual length d (t) ∼ t 1/z d that describes the late time coarsening, that is involved in all scaling relations in the approach to the critical percolation regime. We find that this characteristic length depends on the particular microscopic dynamics and the lattice geometry. In the case of the voter model, we observe that the system briefly passes through a critical percolation state, to later approach a dynamical regime in which the scaling behaviour of the geometrical properties of the ordered domains can be ascribed to a different criticality.arXiv:1805.05775v2 [cond-mat.stat-mech]

Dynamical cluster size heterogeneity

“…As shown in Ref. [30], the timescales are all larger in the VM, nonetheless, the overall behavior of H(t) is similar, Fig. 6.…”

“…Such mechanism, that breaks the degenerescency of areas depending on the number of sides is absent in the Ising model. Another interesting cases are the Ising model with conserved order parameter [30,35,36] or disorder [37][38][39]. Although we focused here on geometric domains, the heterogeneity associated with the Coniglio-Klein clusters [1] would also be of interest [15], along with the heterogeneity of perimeters.…”

“…6. Moreover, defining t p1 and t p as above (even if the critical properties of the percolating cluster do not correspond to critical percolation [30], we can see in Fig. 6 that they are related, respectivelly, to the end of the precursor peak and the end of the plateau.…”

High-degeneracy Potts coarsening

“…Once a percolating domain structure has formed, the fate of the zerotemperature Ising model is sealed [7][8][9]. Percolation and the domain growth in bi-imensional coarsening have been readily studied [7][8][9][10][11][12][13][14][15][16][17][18].…”

Anomalous Ising freezing times

“…The probability of observing each of these topologically distinct behaviours is seemingly identical to the equivalent spanning probability in continuum percolation, and correspondingly explains how the "fate" of the system is sealed [6][7][8][9][10][11]. The role of percolation in the coarsening of 2D Ising ferromagnets has since been studied extensively [12][13][14][15][16][17][18][19][20][21]. In three-dimensions, the system is essentially always trapped in non-static topologically complex configurations at iso-energy ad infinitum [22][23][24][25].…”