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arXiv:1006.3150

Evidence of invariance of time scale at critical point in the Ising meanfield equilibrium equation of state

Muktish Acharyya, Ajanta Bhowal Acharyya

Published Jun 16, 2010• 0 citations
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Abstract

We solved the equilibrium meanfield equation of state of Ising ferromagnet (obtained from Bragg-Williams theory) by Newton-Raphson method. The number of iterations required to get a convergent solution (within a specified accuracy) of equilibrium magnetisation, at any particular temperature, is observed to diverge in a power law fashion as the temperature approaches the critical value. This was identified as the critical slowing down. The exponent is also estimated. This value of the exponent is compared with that obtained from analytic solution. Besides this, the numerical results are also compared with some experimental results exhibiting satisfactory degree of agreement. It is observed from this study that the information of the invariance of time scale at the critical point is present in the meanfield equilibrium equation of state of Ising ferromagnet.

DOI
10.48550/arxiv.1006.3150

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We solved the equilibrium meanfield equation of state of Ising ferromagnet (obtained from Bragg-Williams theory) by Newton-Raphson method. The number of iterations required to get a convergent solution (within a specified accuracy) of equilibrium magnetisation, at any particular temperature, is observed to diverge in a power law fashion as the temperature approaches the critical value. This was identified as the critical slowing down. The exponent is also estimated. This value of the exponent is compared with that obtained from analytic solution. Besides this, the numerical results are also compared with some experimental results exhibiting satisfactory degree of agreement. It is observed from this study that the information of the invariance of time scale at the critical point is present in the meanfield equilibrium equation of state of Ising ferromagnet.
We solve the equilibrium meanfield equation of state of Ising ferromagnet (obtained from Bragg—Williams theory) by Newton—Raphson method. The number of iterations required to get a convergent solution (within a specified accuracy) of equilibrium magnetisation, at any particular temperature, is observed to diverge in a power law fashion as the temperature approaches the critical value. This is identified as the critical slowing down. The exponent is also estimated. This value of the exponent is compared with that obtained from analytic solution. Besides this, the numerical results are also compared with some experimental results exhibiting satisfactory degree of agreement. It is observed from this study that the information of the invariance of time scale at the critical point is present in the meanfield equilibrium equation of state of Ising ferromagnet.
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.
We present a new procedure that can identify and measure the critical temperature. This method is based on the divergence of the relaxation time approaching the critical point in quenches from infinite temperature. We introduce a dimensionless quantity that turns out to be time independent at the critical temperature. The procedure does not need equilibration and allows for a relatively fast identification of the critical temperature. The method is first tested in the ferromagnetic Ising model and in the two-dimensional EA model and then applied to the one-dimensional Ising spin glass with power law interactions. Here we always find a finite critical temperature also in the presence of a uniform external field, in agreement with the mean-field picture for the low-temperature phase of spin glasses.
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.
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
Cooperative behaviors near the disorder-induced critical point in a random-field Ising model are numerically investigated by analyzing time-dependent magnetization in ordering processes from a special initial condition. We find that the intensity of fluctuations of time-dependent magnetization, chi(t) , attains a maximum value at a time t=tau in a normal phase and that chi(tau) and tau exhibit divergences near the disorder-induced critical point. Furthermore, spin configurations around the time tau are characterized by a length scale, which also exhibits a divergence near the critical point. We estimate the critical exponents that characterize these power-law divergences by using a finite-size scaling method.