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arXiv:cond-mat/0107053 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Persistence in one-dimensional Ising models with parallel dynamics

Gautam I. Menon, Purusattam Ray, Prabodh Shukla

Published Sep 17, 2001• 14 citations
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Abstract

We study persistence in one-dimensional ferromagnetic and antiferromagnetic nearest-neighbor Ising models with parallel dynamics. The probability P(t) that a given spin has not flipped up to time t, when the system evolves from an initial random configuration, decays as P(t) approximately 1/t(straight theta(p)) with straight theta(p) approximately 0.75 numerically. A mapping to the dynamics of two decoupled A+A-->0 models yields straight theta(p)=3/4 exactly. A finite size scaling analysis clarifies the nature of dynamical scaling in the distribution of persistent sites obtained under this dynamics.

DOI
10.1103/physreve.64.046102

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We study persistence in one-dimensional ferromagnetic and antiferromagnetic nearest-neighbor Ising models with parallel dynamics. The probability P(t) that a given spin has not flipped up to time t, when the system evolves from an initial random configuration, decays as P(t) approximately 1/t(straight theta(p)) with straight theta(p) approximately 0.75 numerically. A mapping to the dynamics of two decoupled A+A-->0 models yields straight theta(p)=3/4 exactly. A finite size scaling analysis clarifies the nature of dynamical scaling in the distribution of persistent sites obtained under this dynamics.
We consider a periodic Ising chain with nearest-neighbor and rth neighbor interaction and quench it from infinite temperature to zero temperature. The persistence probability P(t) , measured as the probability that a spin remains unflipped up to time t , is studied by computer simulation for suitable values of r . We observe that as time progresses, P(t) first decays as t(-0.22) (the first regime), then the P(t)-t curve has a small slope (in log-log scale) for some time (the second regime) and at last it decays nearly as t(-3/8) (the third regime). We argue that in the first regime, the persistence behavior is the usual one for a two-dimensional system, in the second regime it is like that of a noninteracting ("zero-dimensional") system, and in the third regime the persistence behavior is like that of a one-dimensional Ising model. We also provide explanations for such behavior.
The nonequilibrium dynamics of the strongly diluted random-bond Ising model in two dimensions (2D) is investigated numerically. The persistence probability, P(t), of spins which do not flip by time t is found to decay to a nonzero, dilution-dependent value P(infinity). We find that p(t)=P(t)-P(infinity) decays exponentially to zero at large times. Furthermore, the fraction of spins which never flip is a monotonically increasing function over the range of bond-dilution considered. Our findings, which are consistent with a recent result of Newman and Stein [Phys. Rev. Lett. 82, 3944 (1999)], suggest that persistence in diluted and pure systems falls into different classes. Furthermore, its behavior would also appear to depend crucially on the strength of the dilution present.
We obtain θ_p(q) = 2θ_s(q) for one-dimensional q-state ferromagnetic Potts models evolving under parallel dynamics at zero temperature from an initially disordered state, where θ_p(q) is the persistence exponent for parallel dynamics and θ_s(q) = -{1/8}+ \frac{2}{π^2}[cos^{-1}{(2-q)/q\sqrt{2}}]^2 [PRL, {\bf 75}, 751, (1995)], the persistence exponent under serial dynamics. This result is a consequence of an exact, albeit non-trivial, mapping of the evolution of configurations of Potts spins under parallel dynamics to the dynamics of two decoupled reaction diffusion systems.
We investigate the dynamics of a two-dimensional axial next-nearest-neighbor Ising model following a quench to zero temperature. The Hamiltonian is given by H= -J_(0) summation operator(L)_(i,j=1)S_(i,j)S_(i+1,j)-J_(1)summation operator_(i,j=1)(S_{i,j}S_{i,j+1}-kappaS_{i,j}S_{i,j+2}) . For kappa<1 , the system does not reach the equilibrium ground state but slowly evolves to a metastable state. For kappa>1 , the system shows a behavior similar to that of the two-dimensional ferromagnetic Ising model in the sense that it freezes to a striped state with a finite probability. The persistence probability shows algebraic decay here with an exponent theta=0.235+/-0.001 while the dynamical exponent of growth z=2.08+/-0.01 . For kappa=1 , the system belongs to a completely different dynamical class; it always evolves to the true ground state with the persistence and dynamical exponent having unique values. Much of the dynamical phenomena can be understood by studying the dynamics and distribution of the number of domain walls. We also compare the dynamical behavior to that of a Ising model in which both the nearest and next-nearest-neighbor interactions are ferromagnetic.
We present very accurate numerical estimates of the time and size dependence of the zero-temperature local persistence in the 2d ferromagnetic Ising model. We show that the effective exponent decays algebraically to an asymptotic value θ that depends upon the initial condition. More precisely, we find that θ takes one universal value 0.199(2) for initial conditions with short-range spatial correlations as in a paramagnetic state, and the value 0.033(1) for initial conditions with the long-range spatial correlations of the critical Ising state. We checked universality by working with a square and a triangular lattice, and by imposing free and periodic boundary conditions. We found that the effective exponent suffers from stronger finite size effects in the former case.
We study the dynamics of ordering in ferromagnets via Monte Carlo simulations of the Ising model, employing the Glauber spin-flip mechanism, in space dimensions d=2d=2d=2 and 333. Results for the persistence probability and the domain growth are discussed for quenches to various temperatures (TfT_fTf​) below the critical one (TcT_{c}Tc​), from different initial temperatures Ti≥TcT_{i} \geq T_{c}Ti​≥Tc​. In long time limit, for Ti>TcT_{i} > T_{c}Ti​>Tc​, the persistence probability exhibits power-law decay with exponents θ≃0.22θ\simeq 0.22θ≃0.22 and ≃0.18\simeq 0.18≃0.18 in d=2d=2d=2 and 333, respectively. For finite TiT_iTi​, the early time behavior is a different power-law whose life-time diverges and exponent decreases as Ti→TcT_{i} \rightarrow T_{c}Ti​→Tc​. The crossover length between the two steps diverges as the equilibrium correlation length. Ti=TcT_i=T_cTi​=Tc​ is expected to provide a {\it{new universality class}} for which we obtain θ≃0.035θ\simeq 0.035θ≃0.035 in d=2d=2d=2 and ≃0.10\simeq 0.10≃0.10 in d=3d=3d=3. The time dependence of the average domain size ℓ\ellℓ, however, is observed to be rather insensitive to the choice of TiT_iTi​.