Kawasaki-type dynamics: diffusion in the kinetic Gaussian model

Temporal evolution of self-organization of gelatin molecules and clusters on quartz surface

Aqueous gelatin solutions when spread on hydrophilic substrates form self-organized structures where the gelatin molecules and clusters are arranged as self-similar objects giving a mass fractal dimension df=1.67 and 1.72 for solutions made with KCl and NaCl salts as estimated from atomic force microscopic studies. The dehydration driven self-organization of particles changed the df values to 1.78 and 1.81, respectively, after 24 h. This further changed to 1.83 and 1.85 after a time lapse of 10 days. The dynamics of formation of these structures are modeled through spin-exchange kinetics in the nonequilibrium steady state regime in order to understand their complex behavior. Kawasaki spin exchange dynamics has been applied to a diffusion limited aggregation type fractal object, and the growth of the domains was observed by minimizing the free energy. The fractal dimension of such a system changed from 1.70 to 1.82 which inferred the loss of fractal behavior and the generation of a more compact object. The experimentally observed temporal evolution of these complex structures could be adequately described through the results obtained from the computer simulation data.

One-dimensional Ising model built on small-world networks: competing dynamics

In this paper, we offer a competing dynamic analysis of the one-dimensional Ising model built on the small-world network (SWN). Adding-type SWNs are investigated in detail using a simplified Hamiltonian of mean-field nature, and the result of rewiring-type is given because of the similarities of these two typical networks. We study the dynamical processes with competing Glauber mechanism and Kawasaki mechanism. The Glauber-type single-spin transition mechanism with probability p simulates the contact of the system with a heat bath and the Kawasaki-type dynamics with probability 1-p simulates an external energy flux. By studying the phase diagram obtained in the present work, we can realize some dynamical properties influenced by the small-world effect.

Phase transformation in a lattice system in the presence of spin-exchange dynamics

A joint action of the Glauber single-spin-flip and the Kawasaki spin-exchange mechanisms upon the processes of phase transformation is examined in the framework of the one-dimensional kinetic Ising model. It is shown that the addition of the Kawasaki dynamics to that of Glauber accelerates the process of phase transformation in the initial stage, but slows it down in later stages. For the truncated form of Glauber dynamics, which excludes the processes of splitting and coagulation of clusters, the addition of the Kawasaki dynamics always accelerates the phase transformation process. Acting alone, the Kawasaki mechanism provides a cluster growth proportional to t(1/2) (where t is the time) in the initial stage and proportional to t(1/3) (Lifshitz-Slyozov-Wagner law) in the intermediate stage. In the final stage, a cluster size approaches exponentially its equilibrium value.

Critical dynamics of the Gaussian model with multispin transitions

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.

Introducing small-world network effects to critical dynamics

We analytically investigate the kinetic Gaussian model and the one-dimensional kinetic Ising model of two typical small-world networks (SWN), the adding type and the rewiring type. The general approaches and some basic equations are systematically formulated. The rigorous investigation of the Glauber-type kinetic Gaussian model shows the mean-field-like global influence on the dynamic evolution of the individual spins. Accordingly a simplified method is presented and tested, which is believed to be a good choice for the mean-field transition widely (in fact, without exception so far) observed for SWN. It yields the evolving equation of the Kawasaki-type Gaussian model. In the one-dimensional Ising model, the p dependence of the critical point is analytically obtained and the nonexistence of such a threshold p(c), for a finite-temperature transition, is confirmed. The static critical exponents gamma and beta are in accordance with the results of the recent Monte Carlo simulations, and also with the mean-field critical behavior of the system. We also prove that the SWN effect does not change the dynamic critical exponent z=2 for this model. The observed influence of the long-range randomness on the critical point indicates two obviously different hidden mechanisms.

Generalized competing Glauber-type dynamics and Kawasaki-type dynamics

In this paper, we have given a systematic formulation of a generalized competing mechanism: The Glauber-type single-spin transition mechanism, with probability p, simulates the contact of the system with the heat bath, and the Kawasaki-type spin-pair redistribution mechanism, with probability 1-p, simulates an external energy flux. These two mechanisms are natural generalizations of Glauber's single-spin flipping mechanism and Kawasaki's spin-pair exchange mechanism respectively. On the one hand, the proposed mechanism is, in principle, applicable to arbitrary systems, while on the other hand, our formulation is able to contain a mechanism that just directly combines single-spin flipping and spin-pair exchange in their original form. Compared with the conventional mechanism, the proposed mechanism does not assume the simplified version and leads to a greater influence of temperature. The fact, order for lower temperature and disorder for higher temperature, will be universally true. In order to exemplify this difference, we applied the mechanism to the one-dimensional Ising model and obtained analytical results. We also applied this mechanism to the kinetic Gaussian model and found that above the critical point there will be only paramagnetic phase, while below the critical point, the self-organization as a result of the energy flux will lead the system to an interesting heterophase, instead of the initially guessed antiferromagnetic phase. We studied this process in details.