One-dimensional kinetic ising model with one impurity

The effect of one impurity with arbitrary temperature independent relaxation time on the one-dimensional kinetic Ising model is exactly studied. The dynamical susceptibility for the low concentration of defect is also obtained.Work partially by CNPq and FINEP (Brazilian Agencies). A preliminary version of this paper has been presented at the 15th International Conference on Thermodynamics and Statistical Mechanics, Edinburgh, 1983     

Analytical and Numerical Studies of the One-Dimensional Spin Facilitated Kinetic Ising Model

The one-dimensional spin facilitated kinetic Ising model is studied analytically using the master equation and by simulations. The local state of the spins (corresponding to mobile and immobile cells) can change depending on the state of the neighbored spins, which reflects the high cooperativity inherent in glassy materials. The short-time behavior is analyzed using a Fock space representation for the master equation. The hierarchy of evolution equations for the averaged spin state and the time dependence of the spin autocorrelation function are calculated with different methods (mean-field theory, expansion in powers of the time, partial summation) and compared with numerical simulations. The long-time behavior can be obtained by mapping the one-dimensional spin facilitated kinetic Ising model onto a one-dimensional diffusion model containing birth and death processes. The resulting master equation is solved by van Kampen's size expansion, which leads to a Langevin equation with Gaussian noise. The predicted autocorrelation function and the global memory offer in the long-time limit a screened algebraic decay and a stretched exponential decay, respectively, consistent with numerical simulations.     

Magnetic relaxation in a spin-1 Ising model near the second-order phase transition point

The magnetic relaxation of a spin-1 Ising model with bilinear and biquadratic interactions is formulated within the framework of statistical equilibrium theory and the thermodynamics of irreversible processes. Using a molecular-field expression for the magnetic Gibbs energy, the magnetic Gibbs energy produced in the irreversible process is calculated and time derivatives of the dipolar and quadrupolar order parameters are treated as fluxes conjugate to their appropriate generalized forces in the sense of Onsager theory. The kinetic equations are obtained by introducing kinetic coefficients that satisfy the Onsager relation. By solving these equations an expression is derived for the dynamic or complex magnetic susceptibility. From the real and imaginary parts of this expression, magnetic dispersion and absorption factor are calculated and analyzed near the second-order phase transition.     

The one-dimensional kinetic Ising model: A series expansion study

Exact power series expansions (through eight terms) in the time are derived for relaxation in the one-dimensional Ising model with nearest-neighbor interactions for a general rate parameter where the activation energy is a variable fraction of the energy required to break nearest-neighbor bonds. It is found that the qualitative nature of the relaxation is very dependent on this parameter, varying from nearly simple exponential decay (as with Glauber dynamics) for an intermediate value of this parameter, to an initial rate of change that is either much slower or faster than a simple exponential at the extremes of the range of variation of the parameter. The rate equations for the limit of rapid internal diffusion (internal equilibration) are integrated for several special values of the rate parameter. In general the internal equilibration approximation is not a good representation of the relaxation except when the relaxation is similar to Glauber dynamics.     

Spectra of quantum chains without the Yang-Baxter equation

We study one-dimensional reaction-diffusion models described by master equations and their associated two-state quantum hamiltonians. By choosing appropriate rates, the equations of motion decouple into certain subsets. We solve the first subset which has a close relation to the problem of lattice electrons in an electric field. In this way we obtain L(L − 1) + 1 energy levels of a quantum chain with L sites. The corresponding hamiltonian depends on seven parameters and does not look integrable using conventional methods. As an application, we compute the dynamical critical exponent of a new type of kinetic Ising model.     

Relaxation theory of spin-3/2 Ising system near phase transition temperatures

<正>Dynamics of a spin-3/2 Ising system Hamiltonian with bilinear and biquadratic nearest-neighbour exchange interactions is studied by a simple method in which the statistical equilibrium theory is combined with the Onsager's theory of irreversible thermodynamics.First,the equilibrium behaviour of the model in the molecular-field approximation is given briefly in order to obtain the phase transition temperatures,i.e.the first- and second-order and the tricritical points.Then,the Onsager theory is applied to the model and the kinetic or rate equations are obtained.By solving these equations three relaxation times are calculated and their behaviours are examined for temperatures near the phase transition points.Moreover,the z dynamic critical exponent is calculated and compared with the z values obtained for different systems experimentally and theoretically,and they are found to be in good agrement.     

Dynamics of the generalized Glauber-Ising chain in a magnetic field

A one-dimensional kinetic Ising model with nearest neighbor interactionJ and magnetic fieldH 0 is treated in both linear and nonlinear response, using the most general single spin-flip transition probabilities that depend on nearest neighbor states only. The dynamics is reformulated in terms of kinetic equations for the concentration n_l ⁺(t) [@#@ n_l(t) of clusters containingl up- [or down-] spins, which is exact in the homogeneous case. The initial relaxation time ^* of the magnetization is obtained rigorously for arbitraryJ, H, and temperatureT. The relaxation function is found by numerical integration forJ/T < 2. It is shown that coagulation of minus-clusters becomes negligible for bothJ/T andH/T large, and the resulting set of equations is solved exactly in terms of an eigenvalue problem. A perturbation theory is developed to take into account the neglected coagulation terms. The relaxation function is found to be non-Lorentzian in general, in contrast to the Glauber results atH = 0, which are recovered as a special case. In addition, nonlinear and linear relaxation functions differ forH 0. Consequences for the application to biopolymers are briefly mentioned.Supported in part by the Deutsche Forschungsgemeinschaft (SFB 130).     

Kinetics of isothermal creep in metallic glasses including the statistical distribution of activation parameters

A generalized theoretical model is proposed for the structural relaxation of metallic glasses under load. Structural relaxation is treated as a set of irreversible, uncorrelated, two-stage atomic displacements in some regions of the structure, the “relaxation centers.” In loaded samples structural relaxation acquires a directional character, leading to the buildup of plastic deformation in accordance with the magnitude and orientation of the applied mechanical stress. General equations are obtained for creep kinetics including a continuous statistical distribution of the principal activation parameters. These equations are compared with the results of a special experiment. The model is found to provide an adequate interpretation of the observed creep kinetics, except for the first 10¹–10² seconds after loading. It is argued that the initial stage of creep is determined by reversible atomic realignments in relaxation centers having symmetric two-well potential. Fiz. Tverd. Tela (St. Petersburg) 39, 2008–2015 (November 1997)     

On the nonergodicity of the transverse magnetization in the transverse Ising model

The nonergodic behavior exhibited by the transverse spin correlation function _q=0 ^xx (t) the transverse Ising model obtained as the solution of approximate kinetic equations (derived on the basis of Résibois and De Leener's method), is shown to be an intrinsic property of the model and not the result of the approximations made in the derivation of the kinetic equations.Chargé de Recherches au Fonds National Belge de la Recherche Scientifique.     

Monte Carlo computer experiments on critical phenomena and metastable states

Ising and Heisenberg models are studied by the Monte Carlo method. Several hundred up to 60 000 spins located at two- and three-dimensional lattices are treated and various boundary conditions used to elucidate various aspects of phase transitions. Using free boundaries the finite size scaling theory is tested and surface properties are derived, while the periodic boundary condition or the effective field-like ‘self-consistent’ boundary condition are used to derive bulk critical properties. Since Monte Carlo averages can be interpreted as time averages of a stochastic model, ‘critical slowing down of convergence’ occurs. The critical dynamics is investigated in the case of the single spin-flip kinetic Ising model. Also non-equilibrium relaxation processes are treated, e.g. switching on small negative fields the magnetization reversal and nucleation processes are studied. The metastable states found can be understood in terms of a scaling theory and the droplet model. Using a spin exchange model the phase separation kinetics of a binary alloy is simulated.     

Solvable real space renormalization group for the kinetic Ising chain

A real-space renormalization group for the one-dimensional kinetic Ising model is established. The parameter space of the model must be enlarged to include non-Markovian kernels in the equation of motion. The recursion relations for these kernels can be iterated analytically so that the global flow under the renormalization group can be traced exactly. The resulting fixed-point equation is non-Markovian.     

One Dimensional Phase-Ordering in the Ising Model with Space Decaying Interactions

Journal of Statistical Physics - The study of the phase ordering kinetics of the ferromagnetic one-dimensional Ising model dates back to 1963 (R. J. Glauber, J. Math. Phys. 4, 294) for non...