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Sakarya University Journal of Science

On the Loci of Relaxation Time and Magnetic Dispersion Maxima in the Mean-Field Ising Model

Songül Özüm

Published Sep 29, 2020• 0 citations
Publisher

Abstract

Based on the phenomenological approach, loci of relaxation time and magnetic dispersion maxima near the critical regime in a spin-1/2 mean-field Ising model were performed. The shift in temperature (T) of relaxation time (τ) maximum was detected and its behavior near the second-order transition points are presented at different magnetic field values (h) and different lattice coordination numbers (q). An expression for the dynamic (or complex) susceptibility (χ = χ1 - iχ2) is also derived. The temperature dependence of the magnetic dispersion (χ1) and magnetic absorption (χ2) factors have been studied near the critical regime. It is found that the maximum of χ1 as a function of frequency (ω) and kinetic coefficient (L) obeying an approximately exponential increases and decreases in T- ω and T-L planes near the critical region.

DOI
10.16984/saufenbilder.780082

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Based on the phenomenological approach, loci of relaxation time and magnetic dispersion maxima near the critical regime in a spin-1/2 mean-field Ising model were performed. The shift in temperature (T) of relaxation time (τ) maximum was detected and its behavior near the second-order transition points are presented at different magnetic field values (h) and different lattice coordination numbers (q). An expression for the dynamic (or complex) susceptibility (χ = χ1 - iχ2) is also derived. The temperature dependence of the magnetic dispersion (χ1) and magnetic absorption (χ2) factors have been studied near the critical regime. It is found that the maximum of χ1 as a function of frequency (ω) and kinetic coefficient (L) obeying an approximately exponential increases and decreases in T- ω and T-L planes near the critical region.
Magnetization relaxation and the steady state response of the S = 1 Ising model with random crystal field to a time varying magnetic field with a frequency ω is modelled and studied here by a method that combines the statistical equilibrium theory with the theory of irreversible thermodynamics. The method offers information on the relaxation time (τ) of the system as well as the temperature (θ) and ω dependencies of the complex (AC or dynamical) susceptibility (i.e., χ(ω) = χ′(ω) − iχ″(ω)). The so-called low- and high-frequency regions are separated by τ because τ−1 → 0 as θ approaches the critical temperatures (θc). One can choose to keep the frequency ω fixed and observe the low-frequency behaviors followed by the high-frequency behaviors when θ → θc. It is shown that χ(ω) exhibits different behaviors in low- and high-frequency regimes that are separated by the quantity ωτ: χ′(ω) converges to static susceptibility and χ″(ω) → 0 for ωτ ≪ 1. However, in the high-frequency region where ωτ ≫ 1, χ′(ω) vanishes and χ″(ω) displays a peak at the critical temperature (θc). Besides the above, the logarithm of the susceptibility components versus log(ω) is also plotted. From these plots, one plateau (a step-like) region and a shifted peak with rising temperature is observed for the real and imaginary parts, respectively.
As a continuation of our previously published work, we propose a theoretical framework for the determination of steady-state kinetics in a spin-1 Ising model by the path probability method. The framework is based on the principles of non-equilibrium statistical physics and is quite different from the phenomenological approach. We construct a set of linear kinetic equations for the order parameters using the non-linear dynamic (or rate) equations in the presence of external magnetic field. From the steady-state solutions of the linear equations, an expression for the complex (or dynamic) magnetic susceptibility [Formula: see text] is derived. The temperature dependence of the magnetic dispersion relation [Formula: see text] and magnetic absorption factor [Formula: see text] has been studied in the ferromagnetic (FM) and paramagnetic (PM) phases as well as near the critical regime.
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.
A simple dynamical model of interacting Ising spins is discussed near the critical point. It is shown that all the moments are positive and never vanish even at the critical point. Lower and upper bounds of the susceptibility are obtained, which yield the non-vanishing of the real part of the susceptibility. This gives a simple proof for the poly-dispersive nature of our system. The method of high temperature expansions for the dynamical susceptibility is presented and is applied to the two-dimensional Ising model up to the ninth order, the results of which show that the critical index of slowing down (the relaxation time τ (1) ∝( T - T c ) - Δ ; Δ =2±0.05) is different from that of the static susceptibility.
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.