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.
Using a Hamiltonian formulation of the master equation, we carry out finite-size calculations for the dynamical critical exponent
z of the kinetic Ising model in one and two dimensions. In one dimension, different appropriately normalized transition probabilities give different values for
z; this result is based on both exact and finite-size calculations. We show that this "nonuniversal" behavior is due to the fact that the critical temperature for the one-dimensional Ising model is zero. In two dimensions we combine our finite-size calculations with finite-size scaling theory to calculate
z. Even in this case we find that
z depends on the form of the transition probability---a result that contradicts the universality hypothesis for dynamical critical behavior. We discuss the possible reasons for this contradiction.
The Ising model in two dimensions with the special boundary conditions of Brascamp and Kunz is analyzed. Leading and subdominant scaling behavior of the Fisher zeros are determined exactly. The exact finite-size scaling, with corrections, of the specific heat is determined both at critical and effective critical (pseudocritical) points. The shift exponents associated with the scaling of these effective critical points are not the same as the inverse correlation length critical exponent. All corrections to scaling are analytic.
The kinetic Ising model on a square lattice is studied by means of the finite-size scaling with the help of the Monte Carlo method. The results obtained are consistent with the universality hypothesis that the value of the dynamic critical expotent is universal for certain types of transition probabilities. The present results also suggest that the dynamic critical exponents for the relaxation times of the magnetization and energy are the same. Our estimation of the dynamic critical exponent z gives that z \cong2.2 ±0.1.