= Solution
For an exponentially correlated local field, a single-frequency illustrative relaxation rate is $T_1^{-1}=C\tau_c/(1+\omega_0^2\tau_c^2)$. Its derivative with respect to $\tau_c$ changes sign at $\omega_0\tau_c=1$. Slow motion gives $T_1\propto\tau_c$, and fast motion gives $T_1\propto1/\tau_c$; an <Arrhenius plot> consequently has a \b[minimum of $T_1$] between opposite-slope limits.
For identical spin-one-half nuclei with homonuclear dipolar relaxation, include both transition <frequencies>:
$$
\frac1{T_1}=C\left[\frac{\tau_c}{1+\omega_0^2\tau_c^2}+\frac{4\tau_c}{1+4\omega_0^2\tau_c^2}\right].
$$
Writing $x=\omega_0\tau_c$, the maximum of the rate satisfies
$$
\frac{1-x^2}{(1+x^2)^2}+4\frac{1-4x^2}{(1+4x^2)^2}=0,
\qquad x=x_*\simeq0.616.
$$
Let $\ell$ be jump length, $\tau_j$ the total waiting time, $\tau_c=c\tau_j$, and $f_D$ the <diffusion jump correlation factor>. In three dimensions $D=f_D\ell^2/(6\tau_j)=f_Dc\ell^2/(6\tau_c)$. Combining this with $D=D_0\exp[-E_D/(k_BT)]$ gives
$$
\boxed{D_0\exp\!\left[-\frac{E_D}{k_BT^*}\right]=\frac{f_Dc\ell^2\omega_0}{6x_*}},\qquad \omega_0=|\gamma|B_0.
$$
The numerical constant is one for the single-frequency illustration and 0.616 for the homonuclear dipolar model. Assumptions are a single activated process, exponential field correlations, a temperature-independent prefactor and coupling amplitude, and an established relation between local correlation and translational motion. Defect creation can enter the measured <activation energy>; it cannot automatically be identified with a bare migration barrier. The fluctuation-spectrum approach originates in the relaxation theory of <Nicolaas Bloembergen>, <Edward Mills Purcell> and <Robert Pound>.<Relaxation Effects in Nuclear Magnetic Resonance Absorption by Nicolaas Bloembergen and others>[Physical Review 73, pp. 679–712: dipolar relaxation and motional correlation]{ref}
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