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TURKISH JOURNAL OF PHYSICS

Dynamic Behavior of a Spin-1 Ising Model. Relaxation of Order Parameters and the `Flatness' Property of Metastable States

Mustafa KESKİN, Rıza Erdem

Published Jan 1, 1997• 0 citations
Publisher

Abstract

Dynamic behavior of a spin-1 Ising model [1] with arbitrary bilinear (J) and biquadratic (K) pair interactions is studied by using the path probability method [2]. First the equilibrium behavior of the model is given briefly in order to understand the nonequilibrium behavior. Then the path probability method is applied to the spin-1 Ising model with J and K Hamiltonian and the system of nonlinear differential equations, which are also called the dynamic equations or rate equations, is found. The dynamic equations are solved using the Runga-Kutta method and the solutions, namely relaxation of order parameters are examined to see the "flatness" property of metastable states. It is found that the system relaxes to either stable or metastable states depending upon the initial values of the order parameters. Moreover, if there are more than one metastable states, the relaxation occurs according to the initial values of the order parameters or free energy values. Finally, the results are compared with the flow diagrams which are the solutions of the dynamic equations in two-dimensional phase space [3] and it is seen that the same results are found except for the "flatness" property of metastable states.

DOI
10.55730/1300-0101.2456

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Dynamic behavior of a spin-1 Ising model [1] with arbitrary bilinear (J) and biquadratic (K) pair interactions is studied by using the path probability method [2]. First the equilibrium behavior of the model is given briefly in order to understand the nonequilibrium behavior. Then the path probability method is applied to the spin-1 Ising model with J and K Hamiltonian and the system of nonlinear differential equations, which are also called the dynamic equations or rate equations, is found. The dynamic equations are solved using the Runga-Kutta method and the solutions, namely relaxation of order parameters are examined to see the "flatness" property of metastable states. It is found that the system relaxes to either stable or metastable states depending upon the initial values of the order parameters. Moreover, if there are more than one metastable states, the relaxation occurs according to the initial values of the order parameters or free energy values. Finally, the results are compared with the flow diagrams which are the solutions of the dynamic equations in two-dimensional phase space [3] and it is seen that the same results are found except for the "flatness" property of metastable states.
The equilibrium properties of the spin-1 Ising system Hamiltonian with arbitrary bilinear (J), biquadratic (K) and odd (L), which is also called dipolar-quadrupolar, interactions is studied for zero magnetic field in the lowest approximation of the cluster variation method. The odd interaction is combined with the bilinear (dipolar) and biquadratic (quadrupolar) exchange interactions by the geometric mean. In this system, phase transitions depend on the ratio of the coupling parameters, alpha = J/K; therefore, the dependence of the nature of the phase transition on alpha is investigated extensively and it is found that for alpha less than or equal to 1 and alpha greater than or equal to 2000 a second-order phase transition occurs, and for 1 < alpha < 2000 a first-order phase transition occurs. The critical temperatures in the case of a second-order phase transition and the upper and lower limits of stability temperature in the case of a first-order phase transition are obtained for different values of alpha calculated using the Hessian determinant. The first-order phase transition temperatures are found by using the free energy values while increasing and decreasing the temperature. Besides the stable branches of the order parameters, we establish also the metastable and unstable parts of these curves and the thermal variations of these solutions as a function of the reduced temperature are investigated. The unstable solutions for the first-order phase transitions are obtained by displaying the free energy surfaces in the form of a contour map. Results are compared with the spin-1 Ising system Hamiltonian with the bilinear and biquadratic interactions and it is found that the odd interaction greatly influences the phase transitions.
We have presented the time-dependent behaviors of a spin-1 Ising model (with both bilinear and biquadratic interactions) in the neighborhood of the equilibrium states via the path probability method. The rate equations as introduced by Keskin and co-workers (1989) have been linearized to obtain the characteristic relaxation times. The temperature variations of these times were produced in accordance with the Tanaka and Mannari’s work (1976) on the equilibrium properties. Results are compared to one using Onsager’s phenomenological theory and a good agreement is achieved.
A spin-1 or three state system will undergo a first or second order phase transition depending on the ratio of coupling parameters α. Using the pair correlation approximation, the transition temperature is determined in order to establish the unstable, the metastable, as well as the stable regions of this cooperative system. The dynamics of the system is studied by means of the most probable path method and the flow lines and fixed points of the system are given for zero field. The choice of possible initial conditions is discussed. The role of the unstable points in the phase diagram, as separators between the stable and metastable points, is described, and they are computed for a number of cases.
The nonlinear rate equations based on the path probability method is used to derive the dynamic (or complex) susceptibility ( ? ) for a spin-1 Ising model with bilinear and biquadratic interactions when the external magnetic field exists. Firstly, the rate equations are linearized. Then, these are solved in the presence of an oscillating external magnetic field. From the real and imaginary parts of ? , the magnetic dispersion and absorption factors are found. Temperature dependence of these quantities in the ferromagnetic and paramagnetic phase regions (as well as their critical properties) are presented. Finally, the theoretical results are compared with those obtained using irreversible thermodynamics
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.
We apply a recently developed theory for metastability in open quantum systems to a one-dimensional dissipative quantum Ising model. Earlier results suggest this model features either a nonequilibrium phase transition or a smooth but sharp crossover, where the stationary state changes from paramagnetic to ferromagnetic, accompanied by strongly intermittent emission dynamics characteristic of first-order coexistence between dynamical phases. We show that for a range of parameters close to this transition or crossover point the dynamics of the finite system displays pronounced metastability, i.e., the system relaxes first to long-lived metastable states before eventual relaxation to the true stationary state. From the spectral properties of the quantum master operator we characterize the low-dimensional manifold of metastable states, which are shown to be probability mixtures of two, paramagnetic and ferromagnetic, metastable phases. We also show that for long times the dynamics can be approximated by a classical stochastic dynamics between the metastable phases that is directly related to the intermittent dynamics observed in quantum trajectories and thus the dynamical phases.