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

After a sudden quench from the disordered high-temperature T_{0}→∞ phase to a final temperature well below the critical point T_{F}≪T_{c}, the nonconserved order parameter dynamics of the two-dimensional ferromagnetic Ising model on a square lattice initially approaches the critical percolation state before entering the coarsening regime. This approach involves two timescales associated with the first appearance (at time t_{p_{1}}>0) and stabilization (at time t_{p}>t_{p_{1}}) of a giant percolation cluster, as previously reported. However, the microscopic mechanisms that control such timescales are not yet fully understood. In this paper, to study their role on each time regime after the quench (T_{F}=0), we distinguish between spin flips that decrease the total energy of the system from those that keep it constant, the latter being parametrized by the probability p. We show that observables such as the cluster size heterogeneity H(t,p) and the typical domain size ℓ(t,p) have no dependence on p in the first time regime up to t_{p_{1}}. Furthermore, when energy-decreasing flips are forbidden while allowing constant-energy flips, the kinetics is essentially frozen after the quench and there is no percolation event whatsoever. Taken together, these results indicate that the emergence of the first percolating cluster at t_{p_{1}} is completely driven by energy decreasing flips. However, the time for stabilizing a percolating cluster is controlled by the acceptance probability of constant-energy flips: t_{p}(p)∼p^{-1} for p≪1 (at p=0, the dynamics gets stuck in a metastable state). These flips are also the relevant ones in the later coarsening regime where dynamical scaling takes place. Because the phenomenology on the approach to the percolation point seems to be shared by many 2D systems with a nonconserved order parameter dynamics (and certain cases of conserved ones as well), our results may suggest a simple and effective way to set, through the dynamics itself, t_{p_{1}} and t_{p} in such systems.

Dynamical cluster size heterogeneity.

Only recently has the essential role of the percolation critical point 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, H_{eq}(T), a measure of how different the domains are in size. We extend this equilibrium measure here 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 timescales in the problem, namely the short percolative and the long coarsening one.

Domain-size heterogeneity in the Ising model: Geometrical and thermal transitions.

A measure of cluster size heterogeneity (H), introduced by Lee et al. [Phys. Rev. E 84, 020101 (2011)] in the context of explosive percolation, was recently applied to random percolation and to domains of parallel spins in the Ising and Potts models. It is defined as the average number of different domain sizes in a given configuration and a new exponent was introduced to explain its scaling with the size of the system. In thermal spin models, however, physical clusters take into account the temperature-dependent correlation between neighboring spins and encode the critical properties of the phase transition. We here extend the measure of H to these clusters and, moreover, present new results for the geometric domains for both d=2 and 3. We show that the heterogeneity associated with geometric domains has a previously unnoticed double peak, thus being able to detect both the thermal and percolative transitions. An alternative interpretation for the scaling of H that does not introduce a new exponent is also proposed.

Evolutionary model with genetics, aging, and knowledge.

We represent a process of learning by using bit strings, where 1 bits represent the knowledge acquired by individuals. Two ways of learning are considered: individual learning by trial and error, and social learning by copying knowledge from other individuals or from parents in the case of species with parental care. The age-structured bit string allows us to study how knowledge is accumulated during life and its influence over the genetic pool of a population after many generations. We use the Penna model to represent the genetic inheritance of each individual. In order to study how the accumulated knowledge influences the survival process, we include it to help individuals to avoid the various death situations. Modifications in the Verhulst factor do not show any special feature due to its random nature. However, by adding years to life as a function of the accumulated knowledge, we observe an improvement of the survival rates while the genetic fitness of the population becomes worse. In this latter case, knowledge becomes more important in the last years of life where individuals are threatened by diseases. Effects of offspring overprotection and differences between individual and social learning can also be observed. Sexual selection as a function of knowledge shows some effects when fidelity is imposed.

Nonequilibrium probabilistic dynamics of the logistic map at the edge of chaos.

We consider nonequilibrium probabilistic dynamics in logisticlike maps x(t+1)=1-a|x(t)|(z), (z>1) at their chaos threshold: We first introduce many initial conditions within one among W>>1 intervals partitioning the phase space and focus on the unique value q(sen)<1 for which the entropic form S(q) identical with (1- summation operator Wp(q)(i))/(q-1) linearly increases with time. We then verify that S(q(sen))(t)-S(q(sen))( infinity ) vanishes like t(-1/[q(rel)(W)-1]) [q(rel)(W)>1]. We finally exhibit a new finite-size scaling, q(rel)( infinity )-q(rel)(W) proportional, variant W(-|q(sen)|). This establishes quantitatively, for the first time, a long pursued relation between sensitivity to the initial conditions and relaxation, concepts which play central roles in nonextensive statistical mechanics.