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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical dynamics of the Gaussian model with multispin transitions

Xiang-Mu Kong, Zhao‐Di Yang

Published May 23, 2003• 0 citations
Publisher

Abstract

In this paper, we present a multispin transition mechanism, which is an extension of the Glauber one, to investigate critical dynamics. By exactly solving the master equation, the influence of the multispin transition mechanism on the dynamic critical behavior is studied for the Gaussian model with nearest-neighbor interactions on d-dimensional lattices (d=1, 2, and 3). The time evolution of magnetization is exactly calculated, and the exact results of relaxation time and dynamic critical exponent are obtained. Our models are divided into two kinds: one is the spin-cluster transition and the other is the arbitrary multispin transition. It is found that there are different relaxation times, but the same dynamical critical exponent for different kinds of multispin transitions. The results show that the dynamical critical exponents are independent of spatial dimensions and configurations of transitional spins, and that the dynamical critical exponent is the same as that of the Glauber dynamics, and thus give a strong support to the simple single-spin-transition dynamics. Finally, we give a brief discussion on the results.

DOI
10.1103/physreve.67.056121

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In this paper, we present a multispin transition mechanism, which is an extension of the Glauber one, to investigate critical dynamics. By exactly solving the master equation, the influence of the multispin transition mechanism on the dynamic critical behavior is studied for the Gaussian model with nearest-neighbor interactions on d-dimensional lattices (d=1, 2, and 3). The time evolution of magnetization is exactly calculated, and the exact results of relaxation time and dynamic critical exponent are obtained. Our models are divided into two kinds: one is the spin-cluster transition and the other is the arbitrary multispin transition. It is found that there are different relaxation times, but the same dynamical critical exponent for different kinds of multispin transitions. The results show that the dynamical critical exponents are independent of spatial dimensions and configurations of transitional spins, and that the dynamical critical exponent is the same as that of the Glauber dynamics, and thus give a strong support to the simple single-spin-transition dynamics. Finally, we give a brief discussion on the results.
In this paper, we have exactly solved Glauber's critical dynamics of the Gaussian model in three dimensions. Of course, it is much easier to apply in the low-dimensional case. The key steps are that we generalize the spin change mechanism from Glauber's single-spin flipping to single-spin transition and give a normalized version of the transition probability. We have also investigated the dynamical critical exponent and found surprisingly that the dynamical critical exponent is highly universal; that is, for one, two, and three dimensions they have the same value independent of spatial dimensionality in contrast to static (equilibrium) critical exponents.
In this paper dynamical critical phenomena of the Gaussian model with long-range interactions decaying as on d-dimensional hypercubic lattices (d = 1, 2, and 3) are studied. First, the critical points are exactly calculated, and it is found that the critical points depend on the value of δ and the range of interactions. Then the critical dynamics is considered. We calculate the time evolutions of the local magnetizations and the spin-spin correlation functions, and further the dynamic critical exponents are obtained. For one-, two- and three-dimensional lattices, it is found that the dynamic critical exponents are all which agrees with the result when only considering nearest neighboring interactions, and that they are all δ if It shows that the dynamic critical exponents are independent of the spatial dimensionality but depend on the value of δ.
In this paper dynamical critical phenomena of the Gaussian model with long-range interactions decaying as 1/rd+δ (δ＞ 0) on d-dimensional hypercubic lattices (d = 1, 2, and 3) are studied. First, the critical points are exactly calculated, and it is found that the critical points depend on the value of δ and the range of interactions. Then the critical dynamics is considered. We calculate the time evolutions of the local magnetizations and the spin-spin correlation functions, and further the dynamic critical exponents are obtained. For one-, two- and three-dimensional lattices, it is found that the dynamic critical exponents are all z = 2 if δ＞ 2, which agrees with the result when only considering nearest neighboring interactions, and that they are all δ if 0 ＜δ＜ 2. It shows that the dynamic critical exponents are independent of the spatial dimensionality but depend on the value of δ.
In this work we study the Glauber dynamics of the one-dimensional Ising model with nearest-neighbor and next-nearest-neighbor interactions, for which an approximate solution of the magnetization per site is obtained. When the dynamical critical exponent z is investigated following the treatment of Cordery, Sarker, and Tobochnik [Phys. Rev. B 24, 5402 (1981)], our observation shows that its upper-bound value is the same as the known value, thus implying that z is independent of the range of the interaction. We also suggest a high-temperature expansion approximation which is then used to solve the two-dimensional Glauber dynamics governed by a master equation; this solution is compared with that of the decoupling method. The time-delayed correlation function is also calculated.
We explore the critical properties of the recently discovered finite-time dynamical phase transition in the nonequilibrium relaxation of Ising magnets after a temperature quench. The transition is characterized by a sudden switch in the relaxation dynamics and it occurs at a sharp critical time. While previous works have focused either on mean-field interactions or on investigating the critical time, we analyze the critical fluctuations at the phase transition in the nearest-neighbor Ising model on a square lattice using Monte Carlo simulations. By means of a finite-size scaling analysis, we extract the critical exponents for the transition. In two spatial dimensions, we find that the exponents are consistent with those of the two-dimensional Ising universality class when the system is initially in the vicinity of the critical point. For initial temperatures below the critical one, however, the critical exponents differ from the Ising-exponents, indicating a distinct dynamical critical phenomenon.
The critical dynamics of the kinetic Glauber-Ising model is studied on a family of the diamond-type hierarchical lattices with various branches. By carrying out the time-dependent real-space renormalization-group transformation to the master equation of the systems considered, the dynamic exponent is calculated. We find that the dynamic exponent depends on fractal dimension d(f) or the branch number m in a generator, and that it increases with the increase of d(f) or m. We notice that for the case of m=1 (one-dimensional spin chain, d(f)=1) our result z=2 is the same as the exact result obtained by Glauber, and for the case of m=2 (the simplest one in the diamond-type hierarchical lattices, d(f)=2) the exponent z=2.626 is higher than those of the two-dimensional regular lattice and the triangular lattice.