Freezing in Ising ferromagnets

Spontaneous symmetry breaking of cooperation between species

In mutualistic associations, two species cooperate by exchanging goods or services with members of another species for their mutual benefit. At the same time, competition for reproduction primarily continues with members of their own species. In intra-species interactions, the prisoner's dilemma is the leading mathematical metaphor to study the evolution of cooperation. Here we consider inter-species interactions in the spatial prisoner's dilemma, where members of each species reside on one lattice layer. Cooperators provide benefits to neighbouring members of the other species at a cost to themselves. Hence, interactions occur across layers but competition remains within layers. We show that rich and complex dynamics unfold when varying the cost-to-benefit ratio of cooperation, r. Four distinct dynamical domains emerge that are separated by critical phase transitions, each characterized by diverging fluctuations in the frequency of cooperation: (i) for large r cooperation is too costly and defection dominates; (ii) for lower r cooperators survive at equal frequencies in both species; (iii) lowering r further results in an intriguing, spontaneous symmetry breaking of cooperation between species with increasing asymmetry for decreasing r; (iv) finally, for small r, bursts of mutual defection appear that increase in size with decreasing r and eventually drive the populations into absorbing states. Typically, one species is cooperating and the other defecting and hence establish perfect asymmetry. Intriguingly and despite the symmetrical model set-up, natural selection can nevertheless favour the spontaneous emergence of asymmetric evolutionary outcomes where, on average, one species exploits the other in a dynamical equilibrium.

Opinion inertia and coarsening in the persistent voter model

We consider the persistent voter model (PVM), a variant of the voter model (VM) that includes transient, dynamically induced zealots. Due to peer reinforcement, the internal confidence η_{i} of a normal voter increases in steps of size Δη. Once it surpasses a given threshold, it becomes a zealot. Its opinion remains frozen until enough interactions with the opposing opinion occur, resetting its confidence. No longer a zealot, the regular voter may change opinion once again. This mechanism of opinion inertia, though simplified, is responsible for an effective surface tension, and the PVM exhibits a crossover from a fluctuation-driven dynamics, as in the VM, to a curvature-driven one, akin to the Ising model at low temperature. The average time τ to attain consensus is nonmonotonic with respect to Δη and reaches a minimum at Δη_{min}. In this paper we elucidate the mechanisms that accelerate the system towards consensus close to Δη_{min}. Near the crossover at Δη_{min}, the intermediate region around the domains where the regular voters accumulate (the active region, AR) is large. The surface tension, albeit small, is sufficient to maintain the shape and reduce the domain fragmentation. The large size of the AR in the region of Δη_{min} has two important effects that accelerate the dynamics. First, it dislodges the zealots within the bulk of the domains. Secondly, it maximally suppresses the formation of slowly evolving stripes typical in Ising-like models. This suggests the importance of understanding the role of the AR, where opinion changes are facilitated, and the interplay between regular voters and zealots in disrupting polarized states.

Superdiffusion-like behavior in zero-temperature coarsening of the [Formula: see text] Ising model

One key aspect of coarsening following a quench below the critical temperature is domain growth. For the non-conserved Ising model a power-law growth of domains of like spins with exponent [Formula: see text] is predicted. Including recent work, it was not possible to clearly observe this growth law in the special case of a zero-temperature quench in the three-dimensional model. Instead a slower growth with [Formula: see text] was reported. We attempt to clarify this discrepancy by running large-scale Monte Carlo simulations on simple-cubic lattices with linear lattice sizes up to [Formula: see text] employing an efficient GPU implementation. Indeed, at late times we measure domain sizes compatible with the expected growth law-but surprisingly, at still later times domains even grow superdiffusively, i.e., with [Formula: see text]. We argue that this new problem is possibly caused by sponge-like structures emerging at early times.

Metastable states in the J_{1}-J_{2} Ising model

We study the J_{1}-J_{2} Ising model on the square lattice using the random local field approximation (RLFA) and Monte Carlo (MC) simulations for various values of the ratio p=J_{2}/|J_{1}| with antiferromagnetic coupling J_{2}, ensuring spin frustration. RLFA predicts metastable states with zero order parameter (polarization) at low temperature for p∈(0,1). This is supported by our MC simulations, in which the system relaxes into metastable states with not only zero, but also with arbitrary polarization, depending on its initial value, external field, and temperature. We support our findings by calculating the energy barriers of these states at the level of individual spin flips relevant to the MC calculation. We discuss experimental conditions and compounds appropriate for experimental verification of our predictions.

Anomalous relaxation of density waves in a ring-exchange system

We present the analysis of the slowing down exhibited by stochastic dynamics of a ring-exchange model on a square lattice, by means of numerical simulations. We find the preservation of coarse-grained memory of initial state of density-wave types for unexpectedly long times. This behavior is inconsistent with the prediction from a low frequency continuum theory developed by assuming a mean-field solution. Through a detailed analysis of correlation functions of the dynamically active regions, we exhibit an unconventional transient long ranged structure formation in a direction which is featureless for the initial condition, and argue that its slow melting plays a crucial role in the slowing-down mechanism. We expect our results to be relevant also for the dynamics of quantum ring-exchange dynamics of hard-core bosons and more generally for dipole moment conserving models.

Nonequilibrium dynamics in a three-state opinion-formation model with stochastic extreme switches

We investigate the nonequilibrium dynamics of a three-state kinetic exchange model of opinion formation, where switches between extreme states are possible, depending on the value of a parameter q. The mean field dynamical equations are derived and analyzed for any q. The fate of the system under the evolutionary rules used in S. Biswas et al. [Physica A 391, 3257 (2012)0378-437110.1016/j.physa.2012.01.046] shows that it is dependent on the value of q and the initial state in general. For q=1, which allows the extreme switches maximally, a quasiconservation in the dynamics is obtained which renders it equivalent to the voter model. For general q values, a "frozen" disordered fixed point is obtained which acts as an attractor for all initially disordered states. For other initial states, the order parameter grows with time t as exp[α(q)t] where α=1-q/3-q for q≠1 and follows a power law behavior for q=1. Numerical simulations using a fully connected agent-based model provide additional results like the system size dependence of the exit probability and consensus times that further accentuate the different behavior of the model for q=1 and q≠1. The results are compared with the nonequilibrium phenomena in other well-known dynamical systems.

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.

Curvature-driven growth and interfacial noise in the voter model with self-induced zealots

We introduce a variant of the voter model in which agents may have different degrees of confidence in their opinions. Those with low confidence are normal voters whose state can change upon a single contact with a different neighboring opinion. However, confidence increases with opinion reinforcement, and above a certain threshold, these agents become zealots, irreducible agents who do not change their opinion. We show that both strategies, normal voters and zealots, may coexist (in the thermodynamical limit), leading to competition between two different kinetic mechanisms: curvature-driven growth and interfacial noise. The kinetically constrained zealots are formed well inside the clusters, away from the different opinions at the surfaces that help limit their confidence. Normal voters concentrate in a region around the interfaces, and their number, which is related to the distance between the surface and the zealotry bulk, depends on the rate at which the confidence changes. Despite this interface being rough and fragmented, typical of the voter model, the presence of zealots in the bulk of these domains induces a curvature-driven dynamics, similar to the low temperature coarsening behavior of the nonconserved Ising model after a temperature quench.

Influence of roughening transition on magnetic ordering

In the literature of magnetic phase transitions, in addition to a critical point, the existence of another special point has been discussed. This is related to the broadening of the interface between two different ordering phases and is referred to as the point of roughening transition. While the equilibrium properties associated with this transition are well understood, the influence of this on nonequilibrium dynamics still needs to be investigated. In this paper we present comprehensive results, from Monte Carlo simulations, on coarsening dynamics in a system, over a wide range of temperature, in space dimension d=3, for which there exists a roughening transition at a nonzero temperature T_{R}. An advanced analysis of the simulation data, on structure, growth, and aging, shows that the onset of unexpected glasslike slow dynamics in this system, that has received attention in recent times, for quenches to zero temperature, actually occurs at this transition point. This implies that the structure and aging depend upon the final temperature, when the latter lies between 0 and T_{R}. This is a very interesting exception to universality in coarsening dynamics. The results also demonstrate an important structure-dynamics connection in the phase-ordering dynamics. We compare the key results with those from d=2, for which there exists no nonzero roughening transition temperature. The absence of the above-mentioned anomalous features in the latter dimension places our conjecture on the role of the roughening transition on a firmer footing.

Asymptotic states of Ising ferromagnets with long-range interactions

It is known that, after a quench to zero temperature (T=0), two-dimensional (d=2) Ising ferromagnets with short-range interactions do not always relax to the ordered state. They can also fall in infinitely long-lived striped metastable states with a finite probability. In this paper, we study how the abundance of striped states is affected by long-range interactions. We investigate the relaxation of d=2 Ising ferromagnets with power-law interactions by means of Monte Carlo simulations at both T=0 and T≠0. For T=0 and the finite system size, the striped metastable states are suppressed by long-range interactions. In the thermodynamic limit, their occurrence probabilities are consistent with the short-range case. For T≠0, the final state is always ordered. Further, the equilibration occurs at earlier times with an increase in the strength of the interactions.

Nonequilibrium dynamics in Ising-like models with biased initial condition

We investigate the dynamical fixed points of the zero temperature Glauber dynamics in Ising-like models. The stability analysis of the fixed points in the mean field calculation shows the existence of an exponent that depends on the coordination number z in the Ising model. For the generalized voter model, a phase diagram is obtained based on this study. Numerical results for the Ising model for both the mean field case and short ranged models on lattices with different values of z are also obtained. A related study is the behavior of the exit probability E(x_{0}), defined as the probability that a configuration ends up with all spins up starting with x_{0} fraction of up spins. An interesting result is E(x_{0})=x_{0} in the mean field approximation when z=2, which is consistent with the conserved magnetization in the system. For larger values of z, E(x_{0}) shows the usual finite size dependent nonlinear behavior both in the mean field model and in the Ising model with nearest neighbor interaction on different two dimensional lattices. For such a behavior, a data collapse of E(x_{0}) is obtained using y=(x_{0}-x_{c})/x_{c}L^{1/ν} as the scaling variable and f(y)=1+tanh(λy)/2 appears as the scaling function. The universality of the exponent and the scaling factor is investigated.

Finite-size effects on the convergence time in continuous-opinion dynamics

We study finite-size effects on the convergence time in a continuous-opinion dynamics model. In the model, each individual's opinion is represented by a real number on a finite interval, e.g., [0,1], and a uniformly randomly chosen individual updates its opinion by partially mimicking the opinion of a uniformly randomly chosen neighbor. We numerically find that the characteristic time to the convergence increases as the system size increases according to a particular functional form in the case of lattice networks. In contrast, unless the individuals perfectly copy the opinion of their neighbors in each opinion updating, the convergence time is approximately independent of the system size in the case of regular random graphs, uncorrelated scale-free networks, and complete graphs. We also provide a mean-field analysis of the model to understand the case of the complete graph.

Zero-temperature coarsening in the two-dimensional long-range Ising model

We investigate the nonequilibrium dynamics following a quench to zero temperature of the nonconserved Ising model with power-law decaying long-range interactions ∝1/r^{d+σ} in d=2 spatial dimensions. The zero-temperature coarsening is always of special interest among nonequilibrium processes, because often peculiar behavior is observed. We provide estimates of the nonequilibrium exponents, viz., the growth exponent α, the persistence exponent θ, and the fractal dimension d_{f}. It is found that the growth exponent α≈3/4 is independent of σ and different from α=1/2, as expected for nearest-neighbor models. In the large σ regime of the tunable interactions only the fractal dimension d_{f} of the nearest-neighbor Ising model is recovered, while the other exponents differ significantly. For the persistence exponents θ this is a direct consequence of the different growth exponents α as can be understood from the relation d-d_{f}=θ/α; they just differ by the ratio of the growth exponents ≈3/2. This relation has been proposed for annihilation processes and later numerically tested for the d=2 nearest-neighbor Ising model. We confirm this relation for all σ studied, reinforcing its general validity.

High-degeneracy Potts coarsening

I examine the fate of a kinetic Potts ferromagnet with a high ground-state degeneracy that undergoes a deep quench to zero temperature. I consider single spin-flip dynamics on triangular lattices of linear dimension 8≤L≤128 and set the number of spin states q equal to the number of lattice sites L×L. The ground state is the most abundant final state, and is reached with probability ≈0.71. Three-hexagon states occur with probability ≈0.26, and hexagonal tessellations with more than three clusters form with probabilities of O(10^{-3}) or less. Spanning stripe states-where the domain walls run along one of the three lattice directions-appear with probability ≈0.03. "Blinker" configurations, which contain perpetually flippable spins, also emerge, but with a probability that is vanishingly small with the system size.

Field-induced freezing in the unfrustrated Ising antiferromagnet

We study instantaneous quenches from infinite temperature to well below T_{c} in the two-dimensional square lattice Ising antiferromagnet in the presence of a longitudinal external magnetic field. Under single-spin-flip Metropolis algorithm Monte Carlo dynamics, this protocol produces a pair of magnetization plateaus that prevent the system from reaching the equilibrium ground state except for some special values of the field. We explain the plateaus in terms of local spin configurations that are stable under the dynamics.

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.

Topology-controlled Potts coarsening

We uncover unusual topological features in the long-time relaxation of the q-state kinetic Potts ferromagnet on the triangular lattice that is instantaneously quenched to zero temperature from a zero-magnetization initial state. For q=3, the final state is either the ground state (frequency ≈0.75), a frozen three-hexagon state (frequency ≈0.16), a two-stripe state (frequency ≈0.09), or a three-stripe state (frequency <2×10^{-4}). Other final state topologies, such as states with more than three hexagons, occur with probability 10^{-5} or smaller, for q=3. The relaxation to the frozen three-hexagon state is governed by a time that scales as L^{2}lnL. We provide a heuristic argument for this anomalous scaling and present additional new features of Potts coarsening on the triangular lattice for q=3 and for q>3.

Effects of Passive Phospholipid Flip-Flop and Asymmetric External Fields on Bilayer Phase Equilibria

Compositional asymmetry between the leaflets of bilayer membranes modifies their phase behavior and is thought to influence other important features such as mechanical properties and protein activity. We address here how phase behavior is affected by passive phospholipid flip-flop, such that the compositional asymmetry is not fixed. We predict transitions from "pre-flip-flop" behavior to a restricted set of phase equilibria that can persist in the presence of passive flip-flop. Surprisingly, such states are not necessarily symmetric. We further account for external symmetry breaking, such as a preferential substrate interaction, and show how this can stabilize strongly asymmetric equilibrium states. Our theory explains several experimental observations of flip-flop-mediated changes in phase behavior and shows how domain formation and compositional asymmetry can be controlled in concert, by manipulating passive flip-flop rates and applying external fields.

Phase transition and power-law coarsening in an Ising-doped voter model

We examine an opinion formation model, which is a mixture of Voter and Ising agents. Numerical simulations show that even a very small fraction (∼1%) of the Ising agents drastically changes the behavior of the Voter model. The Voter agents act as a medium, which correlates sparsely dispersed Ising agents, and the resulting ferromagnetic ordering persists up to a certain temperature. Upon addition of the Ising agents, a logarithmically slow coarsening of the Voter model (d=2), or its active steady state (d=3), change into an Ising-type power-law coarsening.

Tricriticality in the q-neighbor Ising model on a partially duplex clique

We analyze a modified kinetic Ising model, a so-called q-neighbor Ising model, with Metropolis dynamics [Phys. Rev. E 92, 052105 (2015)PLEEE81539-375510.1103/PhysRevE.92.052105] on a duplex clique and a partially duplex clique. In the q-neighbor Ising model each spin interacts only with q spins randomly chosen from its whole neighborhood. In the case of a duplex clique the change of a spin is allowed only if both levels simultaneously induce this change. Due to the mean-field-like nature of the model we are able to derive the analytic form of transition probabilities and solve the corresponding master equation. The existence of the second level changes dramatically the character of the phase transition. In the case of the monoplex clique, the q-neighbor Ising model exhibits a continuous phase transition for q=3, discontinuous phase transition for q≥4, and for q=1 and q=2 the phase transition is not observed. On the other hand, in the case of the duplex clique continuous phase transitions are observed for all values of q, even for q=1 and q=2. Subsequently we introduce a partially duplex clique, parametrized by r∈[0,1], which allows us to tune the network from monoplex (r=0) to duplex (r=1). Such a generalized topology, in which a fraction r of all nodes appear on both levels, allows us to obtain the critical value of r=r^{*}(q) at which a tricriticality (switch from continuous to discontinuous phase transition) appears.

Kinetic Ising models with various single-spin-flip dynamics on quenched and annealed random regular graphs

We investigate a kinetic Ising model with several single-spin-flip dynamics (including Metropolis and heat bath) on quenched and annealed random regular graphs. As expected, on the quenched structures all proposed algorithms reproduce the same results since the conditions for the detailed balance and the Boltzmann distribution in an equilibrium are satisfied. However, on the annealed graphs the situation is far less clear-the network annealing disturbs the equilibrium moving the system away from it. Consequently, distinct dynamics lead to different steady states. We show that some algorithms are more resistant to the annealed disorder, which causes only small quantitative changes in the model behavior. On the other hand, there are dynamics for which the influence of annealing on the system is significant, and qualitative changes arise like switching the type of phase transition from a continuous to a discontinuous one. We try to identify features of the proposed dynamics which are responsible for the above phenomenon.

Zero-temperature coarsening in the Ising model with asymmetric second-neighbor interactions in two dimensions

We consider the zero-temperature coarsening in the Ising model in two dimensions where the spins interact within the Moore neighborhood. The Hamiltonian is given by H=-∑_{〈i,j〉}S_{i}S_{j}-κ∑_{〈i,j^{'}〉}S_{i}S_{j^{'}}, where the two terms are for the first neighbors and second neighbors, respectively, and κ≥0. The freezing phenomenon, already noted in two dimensions for κ=0, is seen to be present for any κ. However, the frozen states show more complicated structure as κ is increased; e.g., local antiferromagnetic motifs can exist for κ>2. Finite-sized systems also show the existence of an isoenergetic active phase for κ>2, which vanishes in the thermodynamic limit. The persistence probability shows universal behavior for κ>0; however, it is clearly different from the κ=0 results when a nonhomogeneous initial condition is considered. Exit probability shows universal behavior for all κ≥0. The results are compared with other models in two dimensions having interactions beyond the first neighbor.

Long-time predictability in disordered spin systems following a deep quench

We study the problem of predictability, or "nature vs nurture," in several disordered Ising spin systems evolving at zero temperature from a random initial state: How much does the final state depend on the information contained in the initial state, and how much depends on the detailed history of the system? Our numerical studies of the "dynamical order parameter" in Edwards-Anderson Ising spin glasses and random ferromagnets indicate that the influence of the initial state decays as dimension increases. Similarly, this same order parameter for the Sherrington-Kirkpatrick infinite-range spin glass indicates that this information decays as the number of spins increases. Based on these results, we conjecture that the influence of the initial state on the final state decays to zero in finite-dimensional random-bond spin systems as dimension goes to infinity, regardless of the presence of frustration. We also study the rate at which spins "freeze out" to a final state as a function of dimensionality and number of spins; here the results indicate that the number of "active" spins at long times increases with dimension (for short-range systems) or number of spins (for infinite-range systems). We provide theoretical arguments to support these conjectures, and also study analytically several mean-field models: the random energy model, the uniform Curie-Weiss ferromagnet, and the disordered Curie-Weiss ferromagnet. We find that for these models, the information contained in the initial state does not decay in the thermodynamic limit-in fact, it fully determines the final state. Unlike in short-range models, the presence of frustration in mean-field models dramatically alters the dynamical behavior with respect to the issue of predictability.

Interplay of interfacial noise and curvature-driven dynamics in two dimensions

We explore the effect of interplay of interfacial noise and curvature-driven dynamics in a binary spin system. An appropriate model is the generalized two-dimensional voter model proposed earlier [M. J. de Oliveira, J. F. F. Mendes, and M. A. Santos, J. Phys. A: Math. Gen. 26, 2317 (1993)JPHAC50305-447010.1088/0305-4470/26/10/006], where the flipping probability of a spin depends on the state of its neighbors and is given in terms of two parameters, x and y. x=0.5andy=1 correspond to the conventional voter model which is purely interfacial noise driven, while x=1 and y=1 correspond to the Ising model, where coarsening is fully curvature driven. The coarsening phenomena for 0.5<x<1 keeping y=1 is studied in detail. The dynamical behavior of the relevant quantities show characteristic differences from both x=0.5 and 1. The most remarkable result is the existence of two time scales for x≥x_{c} where x_{c}≈0.7. On the other hand, we have studied the exit probability which shows Ising-like behavior with a universal exponent for any value of x>0.5; the effect of x appears in altering the value of the parameter occurring in the scaling function only.

Minority-spin dynamics in the nonhomogeneous Ising model: Diverging time scales and exponents

We investigate the dynamical behavior of the Ising model under a zero-temperature quench with the initial fraction of up spins 0≤x≤1. In one dimension, the known results for persistence probability are verified; it shows algebraic decay for both up and down spins asymptotically with different exponents. It is found that the conventional finite-size scaling is valid here. In two dimensions, however, the persistence probabilities are no longer algebraic; in particular for x≤0.5, persistence for the up (minority) spins shows the behavior P_{min}(t)∼t^{-γ}exp[-(t/τ)^{δ}] with time t, while for the down (majority) spins, P_{maj}(t) approaches a finite value. We find that the timescale τ diverges as (x_{c}-x)^{-λ}, where x_{c}=0.5 and λ≃2.31. The exponent γ varies as θ_{2d}+c_{0}(x_{c}-x)^{β} where θ_{2d}≃0.215 is very close to the persistence exponent in two dimensions; β≃1. The results in two dimensions can be understood qualitatively by studying the exit probability, which for different system size is found to have the form E(x)=f[(x-x_{c}/x_{c})L^{1/ν}], with ν≈1.47. This result suggests that τ∼L^{z[over ̃]}, where z[over ̃]=λ/ν=1.57±0.11 is an exponent not explored earlier.

Oscillating hysteresis in the q-neighbor Ising model

We modify the kinetic Ising model with Metropolis dynamics, allowing each spin to interact only with q spins randomly chosen from the whole system, which corresponds to the topology of a complete graph. We show that the model with q≥3 exhibits a phase transition between ferromagnetic and paramagnetic phases at temperature T*, which linearly increases with q. Moreover, we show that for q=3 the phase transition is continuous and that it is discontinuous for larger values of q. For q>3, the hysteresis exhibits oscillatory behavior-expanding for even values of q and shrinking for odd values of q. Due to the mean-field-like nature of the model, we are able to derive the analytical form of transition probabilities and, therefore, calculate not only the probability density function of the order parameter but also precisely determine the hysteresis and the effective potential showing stable, unstable, and metastable steady states. Our results show that a seemingly small modification of the kinetic Ising model leads not only to the switch from a continuous to a discontinuous phase transition, but also to an unexpected oscillating behavior of the hysteresis and a puzzling phenomenon for q=5, which might be taken as evidence for the so-called mixed-order phase transition.

Percolation and coarsening in the bidimensional voter model

We study the bidimensional voter model on a square lattice with numerical simulations. We demonstrate that the evolution takes place in two distinct dynamic regimes; a first approach towards critical site percolation and a further approach towards full consensus. We calculate the time dependence of the two growing lengths, finding that they are both algebraic but with different exponents (apart from possible logarithmic corrections). We analyze the morphology and statistics of clusters of voters with the same opinion. We compare these results to the ones for curvature driven two-dimensional coarsening.

Slicing the three-dimensional Ising model: Critical equilibrium and coarsening dynamics

We study the evolution of spin clusters on two-dimensional slices of the three-dimensional Ising model in contact with a heat bath after a sudden quench to a subcritical temperature. We analyze the evolution of some simple initial configurations, such as a sphere and a torus, of one phase embedded into the other, to confirm that their area disappears linearly with time and to establish the temperature dependence of the prefactor in each case. Two generic kinds of initial states are later used: equilibrium configurations either at infinite temperature or at the paramagnetic-ferromagnetic phase transition. We investigate the morphological domain structure of the coarsening configurations on two-dimensional slices of the three-dimensional system, compared with the behavior of the bidimensional model.

Low-temperature Glauber dynamics under weak competing interactions

We consider the low but nonzero-temperature regimes of the Glauber dynamics in a chain of Ising spins with first- and second-neighbor interactions J1,J2. For 0<-J2/|J1|<1 it is known that at T=0 the dynamics is both metastable and noncoarsening, while being always ergodic and coarsening in the limit of T→0+. Based on finite-size scaling analyses of relaxation times, here we argue that in that latter situation the asymptotic kinetics of small or weakly frustrated -J2/|J1| ratios is characterized by an almost ballistic dynamic exponent z≃1.03(2) and arbitrarily slow velocities of growth. By contrast, for noncompeting interactions the coarsening length scales are estimated to be almost diffusive.

Opinion dynamics and influencing on random geometric graphs

We investigate the two-word Naming Game on two-dimensional random geometric graphs. Studying this model advances our understanding of the spatial distribution and propagation of opinions in social dynamics. A main feature of this model is the spontaneous emergence of spatial structures called opinion domains which are geographic regions with clear boundaries within which all individuals share the same opinion. We provide the mean-field equation for the underlying dynamics and discuss several properties of the equation such as the stationary solutions and two-time-scale separation. For the evolution of the opinion domains we find that the opinion domain boundary propagates at a speed proportional to its curvature. Finally we investigate the impact of committed agents on opinion domains and find the scaling of consensus time.

Nature versus nurture: predictability in low-temperature Ising dynamics

Consider a dynamical many-body system with a random initial state subsequently evolving through stochastic dynamics. What is the relative importance of the initial state ("nature") versus the realization of the stochastic dynamics ("nurture") in predicting the final state? We examined this question for the two-dimensional Ising ferromagnet following an initial deep quench from T=∞ to T=0. We performed Monte Carlo studies on the overlap between "identical twins" raised in independent dynamical environments, up to size L=500. Our results suggest an overlap decaying with time as t(-θ)(h) with θ(h)=0.22 ± 0.02; the same exponent holds for a quench to low but nonzero temperature. This "heritability exponent" may equal the persistence exponent for the two-dimensional Ising ferromagnet, but the two differ more generally.

Frozen into stripes: fate of the critical Ising model after a quench

In this article we study numerically the final state of the two-dimensional ferromagnetic critical Ising model after a quench to zero temperature. Beginning from equilibrium at T_{c}, the system can be blocked in a variety of infinitely long lived stripe states in addition to the ground state. Similar results have already been obtained for an infinite temperature initial condition and an interesting connection to exact percolation crossing probabilities has emerged. Here we complete this picture by providing an example of stripe states precisely related to initial crossing probabilities for various boundary conditions. We thus show that this is not specific to percolation but rather that it depends on the properties of spanning clusters in the initial state.

Limiting shapes in two-dimensional Ising ferromagnets

We consider an Ising model on a square lattice with ferromagnetic spin-spin interactions spanning beyond nearest neighbors. Starting from initial states with a single unbounded interface separating ordered phases, we investigate the evolution of the interface subject to zero-temperature spin-flip dynamics. We consider an interface which is initially (i) the boundary of the quadrant or (ii) the boundary of a semi-infinite bar. In the former case the interface recedes from its original location in a self-similar diffusive manner. After a rescaling by √[t], the shape of the interface becomes more and more deterministic; we determine this limiting shape analytically and verify our predictions numerically. The semi-infinite bar acquires a stationary shape resembling a finger, and this finger translates along its axis. We compute the limiting shape and the velocity of the Ising finger.

Fate of 2D kinetic ferromagnets and critical percolation crossing probabilities

We present evidence for a deep connection between the zero-temperature coarsening of both the two-dimensional time-dependent Ginzburg-Landau equation and the kinetic Ising model with critical continuum percolation. In addition to reaching the ground state, the time-dependent Ginzburg-Landau equation and kinetic Ising model can fall into a variety of topologically distinct metastable stripe states. The probability to reach a stripe state that winds a times horizontally and b times vertically on a square lattice with periodic boundary conditions equals the corresponding exactly solved critical percolation crossing probability P(a,b) for a spanning path with winding numbers a and b.

Phase diagram for a zero-temperature Glauber dynamics under partially synchronous updates

We consider generalized zero-temperature Glauber dynamics under a partially synchronous updating mode for a one-dimensional system. Using Monte Carlo simulations, we calculate the phase diagram and show that the system exhibits phase transition between the ferromagnetic and active antiferromagnetic phases. Moreover, we provide analytical calculations that allow us to understand the origin of the phase transition and confirm simulation results obtained earlier for synchronous updates.

Absorbing states of zero-temperature Glauber dynamics in random networks

We study zero-temperature Glauber dynamics for Ising-like spin variable models in quenched random networks with random zero-magnetization initial conditions. In particular, we focus on the absorbing states of finite systems. While it has quite often been observed that Glauber dynamics lets the system be stuck into an absorbing state distinct from its ground state in the thermodynamic limit, very little is known about the likelihood of each absorbing state. In order to explore the variety of absorbing states, we investigate the probability distribution profile of the active link density after saturation as the system size N and (k) vary. As a result, we find that the distribution of absorbing states can be split into two self-averaging peaks whose positions are determined by (k), one slightly above the ground state and the other farther away. Moreover, we suggest that the latter peak accounts for a nonvanishing portion of samples when N goes to infinity while (k) stays fixed. Finally, we discuss the possible implications of our results on opinion dynamics models.

Sequences of phase transitions in Ising models on correlated networks

Using the generic example of Ising spins on scale-free networks, we demonstrate that degree-degree correlations can induce a large number of thermodynamically stable states in networks that otherwise exhibit only the two completely ordered states. The additional stable states are related to the layered network structure. As one increases the temperature, a cascade of first-order phase transitions is found, at which some layers of the network become disordered, while others remain ordered. Negative degree-degree correlations are found to stabilize ordered layers against thermal fluctuations. Positively correlated networks can exhibit an infinite number of ground states and phase transitions, while in negatively correlated networks both numbers are finite.

Zero-temperature relaxation of three-dimensional Ising ferromagnets

We investigate the properties of the Ising-Glauber model on a periodic cubic lattice of linear dimension L after a quench to zero temperature. The resulting evolution is extremely slow, with long periods of wandering on constant energy plateaus, punctuated by occasional energy-decreasing spin-flip events. The characteristic time scale τ for this relaxation grows exponentially with the system size; we provide a heuristic and numerical evidence that τ~exp(L(2)). For all but the smallest-size systems, the long-time state is almost never static. Instead, the system contains a small number of "blinker" spins that continue to flip forever with no energy cost. Thus, the system wanders ad infinitum on a connected set of equal-energy blinker states. These states are composed of two topologically complex interwoven domains of opposite phases. The average genus g(L) of the domains scales as L(γ), with γ≈1.7; thus, domains typically have many holes, leading to a "plumber's nightmare" geometry.

Zero-temperature freezing in the three-dimensional kinetic Ising model

We investigate the relaxation of the Ising-Glauber model on a periodic cubic lattice after a quench to zero temperature. In contrast to the conventional picture from phase-ordering kinetics, we find the following: (i) domains at long time are highly interpenetrating and topologically complex, with average genus growing algebraically with system size; (ii) the long-time state is almost never static, but rather contains "blinker" spins that can flip ad infinitum with no energy cost. (iii) The energy relaxation is extremely slow, with a characteristic time that grows exponentially with system size.

Competing with oneself: introducing self-interaction in a model of competitive learning

A competitive learning model was introduced in Mehta and Luck (Phys Rev E 60, 5:5218-5230, 1999), in which the learning is outcome-related. Every individual chooses between a pair of existing strategies or types, guided by a combination of two factors: tendency to conform to the local majority, and a preference for the type with higher perceived success among its neighbors, based on their relative outcomes. Here, an extension of the interfacial model of Mehta and Luck (Phys Rev E 60, 5:5218-5230, 1999) is proposed, in which individuals additionally take into account their own outcomes in arriving at their outcome-based choices. Three possible update rules for handling bulk sites are considered. The corresponding phase diagrams, obtained at coexistence, show systematic departures from the original interfacial model. Possible relationships of these variants with the cooperative model of Mehta and Luck (Phys Rev E 60, 5:5218-5230, 1999) are also touched upon.

Freezing into stripe states in two-dimensional ferromagnets and crossing probabilities in critical percolation

When a two-dimensional Ising ferromagnet is quenched from above the critical temperature to zero temperature, the system eventually converges to either a ground state or an infinitely long-lived metastable stripe state. By applying results from percolation theory, we analytically determine the probability to reach the stripe state as a function of the aspect ratio and the form of the boundary conditions. These predictions agree with simulation results. Our approach generally applies to coarsening dynamics of nonconserved scalar fields in two dimensions.

Nonequilibrium structures and slow dynamics in a two-dimensional spin system with competing long-range and short-range interactions

We present a lattice spin model that mimics a system of interacting particles through a short-range repulsive potential and a long-range attractive power-law decaying potential. We perform a detailed analysis of the general equilibrium phase diagram of the model at finite temperature, showing that the only possible equilibrium phases are the ferromagnetic and the antiferromagnetic ones. We then study the nonequilibrium behavior of the model after a quench to subcritical temperatures, in the antiferromagnetic region of the phase diagram region, where the pair interaction potential behaves in the same qualitative way as in a Lennard-Jones gas. We find that even in the absence of quenched disorder or geometric frustration, the competition between interactions gives rise to nonequilibrium disordered structures at low enough temperatures that strongly slow down the relaxation of the system. This nonequilibrium state presents several features characteristic of glassy systems such as subaging, nontrivial fuctuation dissipation relations, and possible logarithmic growth of free-energy barriers to coarsening.

Percolation framework in Ising-spin relaxation

We introduce a framework based on the percolation idea to investigate the relaxation under zero-temperature Glauber and outflow dynamics on L x L square and triangular lattices. This helps us to understand the appearance of a double time regime in the survival probability. We show that the first, short-time, regime corresponds to relaxation through droplets and the second, long-time, regime corresponds to relaxation through stripes. For both dynamics the probability that the system becomes ordered through droplets (which indicates fast relaxation) is about 2/3 .

The organization of intrinsic computation: complexity-entropy diagrams and the diversity of natural information processing

Intrinsic computation refers to how dynamical systems store, structure, and transform historical and spatial information. By graphing a measure of structural complexity against a measure of randomness, complexity-entropy diagrams display the different kinds of intrinsic computation across an entire class of systems. Here, we use complexity-entropy diagrams to analyze intrinsic computation in a broad array of deterministic nonlinear and linear stochastic processes, including maps of the interval, cellular automata, and Ising spin systems in one and two dimensions, Markov chains, and probabilistic minimal finite-state machines. Since complexity-entropy diagrams are a function only of observed configurations, they can be used to compare systems without reference to system coordinates or parameters. It has been known for some time that in special cases complexity-entropy diagrams reveal that high degrees of information processing are associated with phase transitions in the underlying process space, the so-called "edge of chaos." Generally, though, complexity-entropy diagrams differ substantially in character, demonstrating a genuine diversity of distinct kinds of intrinsic computation.

Three types of outflow dynamics on square and triangular lattices and universal scaling

In this paper we propose a generalization of the one-dimensional outflow dynamics (KD). The rule is introduced as a simplification of Galam dynamics (GD) proposed in an earlier paper. We simulate three types of outflow dynamics, GD, Stauffer dynamics, and KD, both on the square and triangular lattices and show whether the outflow dynamics is sensitive to the lattice structure. Moreover, we took into account several types of initial configuration -- random, "stripes," and "circle." We investigate the dependence between the mean relaxation time and the initial density p of up-spins for each type of initial conditions, as well as dependence between the mean relaxation time and the size of the system. As a result, we show differences and similarities between three types of the outflow dynamics.

Long-term ordering kinetics of the two-dimensional q-state Potts model

We studied the nonequilibrium dynamics of the q-state Potts model in the square lattice, after a quench to subcritical temperatures. By means of a continuous time Monte Carlo algorithm (nonconserved order parameter dynamics) we analyzed the long term behavior of the energy and relaxation time for a wide range of quench temperatures and system sizes. For q>4 we found the existence of different dynamical regimes, according to quench temperature range. At low (but finite) temperatures and very long times the Lifshitz-Allen-Cahn domain growth behavior is interrupted with finite probability when the system gets stuck in highly symmetric nonequilibrium metastable states, which induce activation in the domain growth, in agreement with early predictions of Lifshitz [JETP 42, 1354 (1962)]. Moreover, if the temperature is very low, the system always gets stuck at short times in highly disordered metastable states with finite lifetime, which have been recently identified as glassy states. The finite size scaling properties of the different relaxation times involved, as well as their temperature dependency, are analyzed in detail.

Effect of initial conditions on Glauber dynamics in complex networks

The effect of initial spin configurations on zero-temperature Glauber spin dynamics in complex networks is investigated. In a system in which the initial spins are defined by centrality measures at the vertices of a network, a variety of nontrivial diffusive behaviors arise, particularly in relation to functional relationships between the initial and final fractions of positive spins, some of which exhibit a critical point. Notably, the majority spin in the initial state is not always dominant in the final state and the phenomena that occur as a result of the dynamics differ according to the initial condition, even for the same network. It is thus concluded that the initial condition of a complex network exerts an influence on spin dynamics that is equally as strong as that exerted by the network structure.