Noise-driven dynamic phase transition in a one-dimensional Ising-like model (original) (raw)
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Noise driven dynamic phase transition in a simple Ising-like model
2009
The dynamical evolution of a recently introduced model in [1] (henceforth referred to as model I), has been made stochastic by introducing a parameter β such that β = 0 corresponds to the Ising model and β → ∞ to the original model I. The equilibrium behaviour for any value of β is identical: a homogeneous state. We argue, from the behaviour of the dynamical exponent z, that for any β = 0, the system belongs to the dynamical class of model I indicating a dynamic phase transition at β = 0. On the other hand, the persistence probabilities in a system of L spins saturate at a value Psat(β, L) = (β/L) α f (β), where α remains constant for all β = 0 supporting the existence of the dynamic phase transition at β = 0. The scaling function f (β) shows a crossover behaviour with f (β) = constant for β << 1 and f (β) ∝ β −α for β >> 1.
Physical Review E, 2008
We study the dynamical response of a two-dimensional Ising model subject to a square-wave oscillating external field. In contrast to earlier studies, the system evolves under a so-called soft Glauber dynamic [P. A. Rikvold and M. Kolesik, J. Phys. A: Math. Gen. 35, L117 (2002)], for which both nucleation and interface propagation are slower and the interfaces smoother than for the standard Glauber dynamic. We choose the temperature and magnitude of the external field such that the metastable decay of the system following field reversal occurs through nucleation and growth of many droplets of the stable phase, i.e., the multidroplet regime. Using kinetic Monte Carlo simulations, we find that the system undergoes a nonequilibrium phase transition, in which the symmetry-broken dynamic phase corresponds to an asymmetric stationary limit cycle for the time-dependent magnetization. The critical point is located where the half-period of the external field is approximately equal to the metastable lifetime of the system. We employ finite-size scaling analysis to investigate the characteristics of this dynamical phase transition. The critical exponents and the fixed-point value of the fourth-order cumulant are found to be consistent with the universality class of the two-dimensional equilibrium Ising model. As this universality class has previously been established for the same nonequilibrium model evolving under the standard Glauber dynamic, our results indicate that this far-from-equilibrium phase transition is universal with respect to the choice of the stochastic dynamics.
Minority-spin dynamics in the nonhomogeneous Ising model: Diverging time scales and exponents
Physical Review E, 2016
We investigate the dynamical behaviour 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 behaviour Pmin(t) ∼ t −γ exp(−(t/τ) δ) with time t, while for the down (majority) spins, Pmaj (t) approaches a finite value. We find that the timescale τ diverges as (xc − x) −λ , where xc = 0.5 and λ ≃ 2.31. The exponent γ varies as θ 2d + c0(xc − 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−xc xc)L 1/ν , with ν ≈ 1.47. This result suggests that τ ∼ Lz, wherez = λ ν = 1.57 ± 0.11 is an exponent not explored earlier.
Non-equilibrium phase transitions in one-dimensional kinetic Ising models
Journal of Physics A: Mathematical and General, 1995
A family of nonequilibrium kinetic Ising models, introduced earlier, evolving under the competing effect of spin flips at zero temperature and nearest neighbour random spin exchanges is further investigated here. By increasing the range of spin exchanges and/or their strength the nature of the phase transition 'Ising-to-active' becomes of (dynamic) mean-field type and a first order tricitical point is located at the Glauber (δ = 0) limit. Corrections to mean-field theory are evaluated up to sixth order in a cluster approximation and found to give good results concerning the phase boundary and the critical exponent β of the order parameter which is obtained as β ≃ 1.0.
Journal of Physics A: Mathematical and General, 1997
One-dimensional non-equilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and nearest neighbour spin exchanges exhibiting a parity-conserving (PC) phase transition on the level of kinks are investigated here numerically from the point of view of the underlying spin system. The dynamical persistency exponent Θ and the exponent λ characterising the two-time autocorrelation function of the total magnetization under nonequilibrium conditions are reported. It is found that the PC transition has strong effect: the process becomes non-Markovian and the above exponents exhibit drastic changes as compared to the Glauber-Ising case.
Nonequilibrium dynamics in Ising-like models with biased initial condition
Physical Review E, 2021
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 generalised 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 behaviour of the exit probability E(x0), defined as the probability that a configuration ends up with all spins up starting with x0 fraction of up spins. An interesting result is E(x0) = x0 in the mean field approximation when z = 2, which is consistent with the conserved magnetisation in the system. For larger values of z, E(x0) shows the usual finite size dependent non linear behaviour both in the mean field model and in Ising model with nearest neighbour interaction on different two dimensional lattices. For such a behaviour, a data collapse of E(x0) is obtained using y = (x 0 −xc) xc 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.
Nonequilibrium kinetic Ising models: phase transitions and universality classes in one dimension
Brazilian Journal of Physics, 2000
Nonequilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and Kawasaki-type spin-exchange kinetics at infinite temperature T are investigated here in one dimension from the point of view of phase transition and critical behaviour. Branching annihilating random walks with an even number of offspring (on the part of the ferromagnetic domain boundaries), is a decisive process in forming the steady state of the system for a range of parameters, in the family of models considered. A wide variety of quantities characterize the critical behaviour of the system.Results of computer simulations and of a generalized mean field theory are presented and discussed.
Journal of Physics A: Mathematical and General, 1996
One-dimensional non-equilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and nearest neighbour spin exchanges exhibiting a parity-conserving (PC) phase transition on the level of kinks are now further investigated, numerically, from the point of view of the underlying spin system. Critical exponents characterising its statics and dynamics are reported. It is found that the influence of the PC transition on the critical exponents of the spins is strong and the origin of drastic changes as compared to the Glauber-Ising case can be traced back to the hyperscaling law stemming from directed percolation. Effect of an external magnetic field, leading to directed percolation type behaviour on the level of kinks, is also studied, mainly via the generalized mean field approximation.
Journal of Statistical Physics, 1997
The dynamic behavior of a spin-I lsing system with arbitrary bilinear and biquadratic pair interactions is studied by using the path probability method, and approaches of the system toward the stable or metastable equilibrium states according to the ratio of interaction parameters and rate constants are presented. In particular, we investigate the relaxation of the order parameters for temperatures less than, equal to, and greater than the second-order and first-order phase transitions. From this investigation, the "flatness" property of metastable states is seen explicitly. We also show how a system freezes in a metastable state as well as how it escapes from one metastable state to the other.
The phase transition in a general class of Ising-type models is sharp
Journal of Statistical Physics, 1987
For a family of translation-invariant, ferromagnetic, one-component spin systems--which includes Ising and (r 4 models--we prove that (i)the phase transition is sharp in the sense that at zero magnetic field the high-and lowtemperature phases extend up to a common critical point, and (ii)the critical exponent fl obeys the mean field bound fl ~< 1/2. The present derivation of these nonperturbative statements is not restricted to "regular" systems, and is based on a new differential inequality whose Ising model version is M <~ flh/. + M 3 + tiM z OM/~fl. The significance of the inequality was recognized in a recent work on related problems for percolation models, while the inequality itself is related to previous results, by a number of authors, on ferromagnetic and percolation models.