For a given protein, I know that the NMR spectrometer magnet generates a field $B_0$ and that the interactions with the spins in the local environment generates a much smaller field $B_\mathrm{loc}$ (not necessary aligned with $B_0$).
Owing to Brownian collisions between solvent molecules and the protein, the atoms associated with these spins move, hence making $B_\mathrm{loc}$ a function of time ($B_0$ is constant).
We define a time correlation function$$G(t,\tau) = \overline{B_\mathrm{loc}(t)B_\mathrm{loc}(t+\tau)},$$ where $G(t,\tau)$ is a stationary random function (see Time evolution of correlation functions (specifically Onsager's hypothesis) in time correlation link)
Hence $G(t,\tau)$ only depends on $\tau$, the delay in measuring $B_\mathrm{loc}$.
So, for simplicity we set $t = 0$ and note that
$$G(t,\tau) = \overline{B_\mathrm{loc}(0)^2} \mathrm{e}^{-\tau/\tau_\mathrm{c}},$$
where $\tau_\mathrm{c}$ is the correlation time, the time it takes for the whole molecule to rotate by 1 radian in a process called rotational diffusion.
How do I derive the second equation from the first?