  We investigate the dynamical critical behavior of the two- and
three-dimensional Ising model with Glauber dynamics in equilibrium. In contrast
to the usual standing, we focus on the mean-squared deviation of the
magnetization $M$, MSD$_M$, 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 MSD$_M$ features a first crossover at time $\tau_1 \sim
L^{z_{1}}$, from ordinary diffusion with MSD$_M$ $\sim t$, to anomalous
diffusion with MSD$_M$ $\sim t^\alpha$. Purely on numerical grounds, we obtain
the values $z_1=0.45(5)$ and $\alpha=0.752(5)$ 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 $\tau_2 \sim L^{z_{2}}$. Here,
MSD$_M$ saturates to its late-time value $\sim L^{2+\gamma/\nu}$, while the
autocorrelation function crosses over from stretched-exponential decay to
simple exponential one. We also confirm numerically the value
$z_{2}=2.1665(12)$, earlier reported as the single dynamic exponent. Continuity
of MSD$_M$ requires that $\alpha(z_{2}-z_{1})=\gamma/\nu-z_1$. We speculate
that $z_{1} = 1/2$ and $\alpha = 3/4$, values that indeed lead to the expected
$z_{2} = 13/6$ result. A complementary analysis for the three-dimensional Ising
model provides the estimates $z_{1} = 1.35(2)$, $\alpha=0.90(2)$, and $z_{2} =
2.032(3)$. While $z_{2}$ has attracted significant attention in the literature,
we argue that for all practical purposes $z_{1}$ is more important, as it
determines the number of statistically independent measurements during a long
simulation.
