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Communications in Theoretical Physics

Kawasaki Dynamics on Square Lattice

Shiming Wu, Zhao‐Di Yang, Qin Yong

Published Feb 14, 1996• 0 citations
Publisher

Abstract

The bond-moving and decimation renormalization group techniques are applied to obtaining the critical dynamic exponent of the Kawasaki-Ising model on square lattice. The result is found to be , where , is the critical dynamic exponent of the Glauber model on the same lattice. We guess that the relation may be valid for all -dimensional systems with .

DOI
10.1088/0253-6102/25/1/39

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The bond-moving and decimation renormalization group techniques are applied to obtaining the critical dynamic exponent of the Kawasaki-Ising model on square lattice. The result is found to be , where , is the critical dynamic exponent of the Glauber model on the same lattice. We guess that the relation may be valid for all -dimensional systems with .
The dynamic Monte Carlo renormalization group is applied to the two-dimensional Kawasaki and three-dimensional Glauber models. Our best "matching" results for the Kawasaki model yield a value of z=3.80z=3.80z=3.80 for the dynamical exponent, as compared with the exact value z=3.75z=3.75z=3.75. This provides strong confirmation for the validity of this method. The results for the Glauber model are less accurate, but our estimate of $z\ensuremath{\simeq}2.08$ is in reasonable agreement with the $\ensuremath{\epsilon}$ expansion.
The critical temperature of the paramagnetic-ferromagnetic phase transition on the Ising model of the square lattice is studied by means of the Monte Carlo method.The Glauber and Kawasaki dynamics are considered in the calculation for the stochastic and sequent processes and their effects on the critical temperatures are discussed.It's shown that the critical temperature can be calculated in the stochastic and sequent processes when the Glauber dynamic is in the lead.But only in the stochastic process the critical temperature can be calculated when the Kawasaki dynamic is in the lead.
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.
The two-dimensional Ising model with competing Glauber and Kawasaki dynamics is studied by Monte Carlo simulations. We show that the model exhibits the phenomenon of self-organization when the Kawasaki dynamics is the dominant one. For this model we show that the values of the critical exponents calculated at the stationary states are in accordance with the exact ones known for the equilibrium Ising model. These results give support to the idea that the equilibrium and nonequilibrium Ising models, which exhibit up-down symmetry, belong to the same universality class. \textcopyright{} 1996 The American Physical Society.