Dynamical cluster size heterogeneity

Abstract

Only recently the essential role of the percolation critical point has been considered on the dynamical properties of connected regions of aligned spins (domains) after a sudden temperature quench. In equilibrium, it is possible to resolve the contribution to criticality by the thermal and percolative effects (on finite lattices, while in the thermodynamic limit they merge at a single critical temperature) by studying the cluster size heterogeneity, Heq(T ), a measure of how different the domains are in size. We here extend this equilibrium measure and study its temporal evolution, H(t), after driving the system out of equilibrium by a sudden quench in temperature. We show that this single parameter is able to detect and well separate the different time regimes, related to the two time scales in the problem, the short, percolative and the long, coarsening one.

Phase-ordering kinetics in the Allen-Cahn (Model A) class: universal aspects elucidated by electrically-induced transition in liquid crystals

“…The traditional road includes the continued interest on spinodal decomposition of alloys and glasses [12][13][14], evolving patterns of foams [15,16], growing crystalline films [17], not to mention the evergreen simulations of Ising (e.g. [18][19][20][21][22]), Potts [23,24] and voter [25] kinetics. The second path is directed to a quantitative understanding of emergent behaviours in biological systems, which range from the intra-cellular assembly kinetics and cellular functionalization -notably liquid-liquid phase separations involved in the formation of membraneless cell compartments - [11,26,27], to the segregation [28] and phase separation [29] within uni-cellular (bacteria) communities, pattern formation in groups of complex multi-cellular organisms [30,31], besides experiments [32][33][34] and theoretical [10] accounts for artificial active matter undergoing motilityinduced phase separation.…”

Energy-lowering and constant-energy spin flips: Emergence of the percolating cluster in the kinetic Ising model

“…These two competing mechanisms explain the peak of H(t) occurring slightly before t p1 , the time when the largest cluster percolates [3,4]. Despite the longer timescales, an analogous precursor behavior was also observed for the Voter model [20], whose dynamics, without surface tension, is much slower [22,23]. The dynamical heterogeneity H(t) presents three well separated regimes: a peak immediately after the quench in temperature, followed by an incipient plateau and, eventually, the powerlaw decaying.…”

“…The dynamical heterogeneity H(t) presents three well separated regimes: a peak immediately after the quench in temperature, followed by an incipient plateau and, eventually, the powerlaw decaying. These correspond, respectively, to the appearance of the first percolating cluster, the stabilization of the largest cluster (and the approach to the percolation critical point) and, finally, the coarsening, curvature driven growth [20]. The approximated analytical results of Ref.…”

“…The non homogeneity of the clusters may be quantified by the so-called cluster size heterogeneity H, the average number of distinct cluster sizes occurring in a finite sample, irrespective of the number of domains that are equally large. This observable was studied in percolation [14][15][16], spin equilibrium models [17][18][19] and recently extended to out-ofequilibrium [20] and more general contexts [21]. The rich behavior of the time-dependent heterogeneity, H(L, t), confirms that it may be useful to understand the interplay between percolation and coarsening dynamics.…”

“…Quenching the system from T 0 → ∞, the dynamical heterogeneity H(t) presents a very pronounced peak at the beginning of the dynamics [20]. After the quench, correlations start building larger clusters, the domainsize distribution widens and the growth of H(t) is a consequence of the larger size diversity.…”