A finite-size scaling approach is used to show numerically that dynamical scaling occurs for short and long times independently of the initial conditions. Its main idea is to construct particular quantities scaling as
L0 in the thermodynamic limit L\ensuremath{\rightarrow}\ensuremath{\infty}, L being the linear size of the system. These are the quantities for which the dynamic scaling occurs for short and long times. This approach is applied to obtain the critical dynamical behavior of two- and three-dimensional ferromagnetic Ising models, subjected to Glauber dynamics.
We obtain the exact critical relaxation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{L}}$(\ensuremath{\xi}), where \ensuremath{\xi} is the bulk correlation length, for the Glauber kinetic Ising model of spins on a one-dimensional lattice of finite length L for both periodic and free boundary conditions (BC's). We show that, independent of the BC's, the dynamic critical exponent has the well-known value z=2, and we comment on a recent claim that z=1 for this model. The ratio ${\mathrm{\ensuremath{\tau}}}_{\mathit{L}}$(\ensuremath{\xi})/${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\infty}}}$(\ensuremath{\xi}), in the double limit L,\ensuremath{\xi}\ensuremath{\rightarrow}\ensuremath{\infty} for fixed x=L/\ensuremath{\xi}, approaches a limiting functional form, ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(L/\ensuremath{\xi}), the finite-size scaling function. For free BC's we derive the exact scaling function ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)=[1+(\ensuremath{\omega}(x)/x
)2${]}^{\mathrm{\ensuremath{-}}1}$, where \ensuremath{\omega}(x) is the smallest root of the transcendental equation \ensuremath{\omega}tan(\ensuremath{\omega}/2)=x. We provide expansions of \ensuremath{\omega}(x) in powers of x and ${\mathit{x}}^{\mathrm{\ensuremath{-}}1}$ for the regimes of small and large x, respectively, and establish their radii of convergence. The scaling function shows anomalous behavior at small x, ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)\ensuremath{\approxeq}x, instead of the usual ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)\ensuremath{\approxeq}
xz, as x\ensuremath{\rightarrow}0. This is because, even for finite L, the lifetime of the slowest dynamical mode diverges for T\ensuremath{\rightarrow} 0 K. For periodic BC's, with the exception of one system, ${\mathrm{\ensuremath{\tau}}}_{\mathit{L}}$ is independent of L, and hence ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$=1. The exceptional system, that with an odd number of spins and antiferromagnetic couplings, exhibits frustration at T=0 K, and the scaling function is given by ${\mathit{f}}_{\mathrm{\ensuremath{\tau}}}$(x)=[1+(\ensuremath{\pi}/x
)2${]}^{\mathrm{\ensuremath{-}}1}$. \textcopyright{} 1996 The American Physical Society.
The dynamical critical exponent z of the two-dimensional Ising model with competing Glauber and Kawasaki processes is evaluated by Monte Carlo simulations. This model exhibits the phenomenon of self-organization when the Kawasaki dynamics is the dominant one. We have calculated the exponent z as a function of the parameter that controls the competition between the two processes. Our calculations are performed at the phase boundary that separates the nonequilibrium ordered and disordered phases. The values of z we obtain are around z=2 for all values of the competing parameter. When the Glauber dynamics is the only one present the value of z is in agreement with most recent calculations. \textcopyright{} 1996 The American Physical Society.
We investigate the dynamical critical behavior of the two- and three-dimensional Ising models with Glauber dynamics in equilibrium. In contrast to the usual standing, we focus on the mean-squared deviation of the magnetization
M,MSDM, as a function of time, as well as on the autocorrelation function of
M. These two functions are distinct but closely related. We find that
MSDM features a first crossover at time ${\ensuremath{\tau}}_{1}\ensuremath{\sim}{L}^{{z}_{1}}$, from ordinary diffusion with ${\mathrm{MSD}}_{M}\ensuremath{\sim}t$, to anomalous diffusion with ${\mathrm{MSD}}_{M}\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$. Purely on numerical grounds, we obtain the values
z1=0.45(5) and $\ensuremath{\alpha}=0.752(5)$ for the two-dimensional Ising ferromagnet. Related to this, the magnetization autocorrelation function crosses over from an exponential decay to a stretched-exponential decay. At later times, we find a second crossover at time ${\ensuremath{\tau}}_{2}\ensuremath{\sim}{L}^{{z}_{2}}$. Here,
MSDM saturates to its late-time value $\ensuremath{\sim}{L}^{2+\ensuremath{\gamma}/\ensuremath{\nu}}$, while the autocorrelation function crosses over from stretched-exponential decay to simple exponential one. We also confirm numerically the value
z2=2.1665(12), earlier reported as the single dynamic exponent. Continuity of
MSDM requires that $\ensuremath{\alpha}({z}_{2}\ensuremath{-}{z}_{1})=\ensuremath{\gamma}/\ensuremath{\nu}\ensuremath{-}{z}_{1}$. We speculate that
z1=1/2 and $\ensuremath{\alpha}=3/4$, values that indeed lead to the expected
z2=13/6 result. A complementary analysis for the three-dimensional Ising model provides the estimates ${z}_{1}=1.35(2), \ensuremath{\alpha}=0.90(2)$, and
z2=2.032(3). While
z2 has attracted significant attention in the literature, we argue that for all practical purposes
z1 is more important, as it determines the number of statistically independent measurements during a long simulation.
We study the dynamical critical behaviour of a double chain of spins where we take into account short-range four-spin interactions. This system is exactly soluble only in the thermodynamical equilibrium. We use the initial response rate of the order parameter to establish a lower bound to the dynamical critical exponent z. We show that, for the one- and two-spin-flip Glauber transition rates, the exponent z depends on the microscopic details of the Hamiltonian, that is, the dynamical critical exponent appears non-universal. This type of non-universal behaviour has already been seen in the one-dimensional Ising model with non-uniform exchange interactions.