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arXiv:cond-mat/9611181 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Nonequilibrium phase transition in the kinetic Ising model: Critical slowing down and the specific-heat singularity

Muktish Acharyya

Published Sep 1, 1997• 96 citations
arXivHTMLPDF

Abstract

The nonequilibrium dynamic phase transition, in the kinetic Ising model in the presence of an oscillating magnetic field has been studied both by Monte Carlo simulation and by solving numerically the mean-field dynamic equation of motion for the average magnetization. In both cases, the Debye ``relaxation'' behavior of the dynamic order parameter has been observed and the ``relaxation time'' is found to diverge near the dynamic transition point. The Debye relaxation of the dynamic order parameter and the power law divergence of the relaxation time have been obtained from a very approximate solution of the mean-field dynamic equation. The temperature variation of appropriately defined ``specific heat'' is studied by the Monte Carlo simulation near the transition point. The specific heat has been observed to diverge near the dynamic transition point.

DOI
10.1103/physreve.56.2407

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The nonequilibrium dynamic phase transition, in the kinetic Ising model in the presence of an oscillating magnetic field has been studied both by Monte Carlo simulation and by solving numerically the mean-field dynamic equation of motion for the average magnetization. In both cases, the Debye relaxation'' behavior of the dynamic order parameter has been observed and the relaxation time'' is found to diverge near the dynamic transition point. The Debye relaxation of the dynamic order parameter and the power law divergence of the relaxation time have been obtained from a very approximate solution of the mean-field dynamic equation. The temperature variation of appropriately defined ``specific heat'' is studied by the Monte Carlo simulation near the transition point. The specific heat has been observed to diverge near the dynamic transition point.
The nonequilibrium dynamic phase transition, in the kinetic Ising model in presence of an oscillating magnetic field, has been studied by Monte Carlo simulation. The fluctuation of dynamic order parameter has been studied as a function of temperature near the dynamic transition point. The temperature variation of appropriately defined `susceptibility' has also been studied near the dynamic transition point. Similarly, fluctuation of energy and appropriately defined `specific-heat' have been studied as a function of temperature near the dynamic transition point. In both the cases, the fluctuations (of dynamic order parameter and energy) and the corresponding responses diverge (as power law fashion) near the dynamic transition point with similar critical behavior (with identical exponent values).
We studied the dynamical phase transition in kinetic Ising ferromagnets driven by oscillating magnetic field in meanfield approximation. The meanfield differential equation was solved by sixth-order Runge–Kutta–Felberg method. The time-averaged magnetization plays the role of the dynamic order parameter. We studied the relaxation behavior of the dynamic order parameter close to the transition temperature, which depends on the amplitude of the applied magnetic field. We observed the critical slowing down along the dynamic phase boundary. We proposed a power law divergence of the relaxation time and estimated the exponent. We also found its dependence on the field amplitude and compared the result with the exact value in limiting case.
The two-dimensional ferromagnetic Ising model in the presence of a propagating magnetic field wave (with a well-defined frequency and wavelength) was studied by Monte Carlo simulation. This study differs from similar earlier studies, because in earlier studies the oscillating magnetic field was considered to be uniform in space. The time averaged magnetization over a full cycle (the time period) of the propagating magnetic field acted as the dynamic order parameter. The dynamical phase transition was observed. The temperature variation of the dynamic order parameter, the mean square deviation of the dynamic order parameter, the dynamic specific heat and the derivative of the dynamic order parameter were studied. The mean square deviation of the dynamic order parameter and the dynamic specific heat show sharp maxima near the transition point. The derivative of the dynamic order parameter shows a sharp minimum near the transition point. The transition temperature is found to depend also on the speed of propagation of the magnetic field wave.
Using Monte Carlo simulations based on the Metropolis algorithm, we investigate the dynamic phase transition properties of kinetic Ising model driven by a sinusoidally oscillating magnetic field in the presence of additive white noise. We calculate equilibrium and dynamic properties such as the temperature dependence of average magnetization and magnetic specific heat, as well as the period dependence of dynamic order parameter and scaled variance. After determining the critical period at which order-disorder transition takes place, we perform finite size scaling analysis to extract the exponent ratios, and discuss the variation of these properties in the presence of noisy magnetic field. As a general result, we show that for a noisy system, DPT does not fall into a universality class of the conventional dynamic (and also equilibrium) universality class of the Ising model.
We report the results of mean field and the Monte Carlo study of the dynamic magnetization-reversal transition in the Ising model, brought about by the application of an external field pulse applied in opposition to the existing order before the application of the pulse. The parameters controlling the transition are the strength hp{h}_{p}hp​ and the duration $\ensuremath{\Delta}t$ of the pulse. In the mean field case, an approximate analytical expression is obtained for the phase boundary which agrees well with that obtained numerically in the small $\ensuremath{\Delta}t$ and large TTT limit. The order parameter of the transition has been identified, and is observed to vary continuously near the transition. The order parameter exponent $\ensuremath{\beta}$ was estimated both for the mean field $(\ensuremath{\beta}=1)$ and the Monte Carlo $(\ensuremath{\beta}=0.90\ifmmode\pm\else\textpm\fi{}0.02$ in two dimensions) cases. The transition shows a ``critical slowing down'' type behavior near the phase boundary with diverging relaxation time. The divergence was found to be logarithmic in the mean field case and exponential in the Monte Carlo case. The finite size scaling technique was employed to estimate the correlation length exponent $\ensuremath{\nu}$ $(=1.5\ifmmode\pm\else\textpm\fi{}0.3$ in two dimensions) in the Monte Carlo case.
We have analyzed the dynamic phase transition of the kinetic Ising model in mean-field approximation by means of an analytical approach. Specifically, we study the evolution of the system under the simultaneous influence of time-dependent and time-independent magnetic fields. We demonstrate that within the approximate analytical treatment of our approach, the dynamic phase transition exhibits power-law dependencies for the order parameter that have the same critical exponents as the mean-field equilibrium case. Moreover we have obtained an equation of state, with which we can prove that the time-independent field component is effectively the conjugate field of the order parameter. Our analysis is limited to the parameter range, in which only second-order phase transitions occur, i.e., for small applied field amplitudes and temperatures close to the Curie point. In order to ensure the reliability of our analytical results we have corroborated them by comparison to numerical evaluations of the same model.