For a given protein, I know that the NMR Spectroscopy magnet generates a field $\mathrm{B_o}$ and that the interactions with the spins in the local environment generates a much smaller field $\mathrm{B_{loc}}$ (not necessary aligned with $\mathrm{B_o}$.
Owing to Brownian collisions between solvent molecules and the protein, the atoms associated with these spins move, hence making $\mathrm{B_{loc}}$ a function of time ($\mathrm{B_o}$ is constant).
We define a time correlation function$$\mathrm{G(t,\tau}) = \overline{\mathrm{B_{loc}(t)B_{loc}(t+\tau)}},$$ where $\mathrm{G(t,\tau})$ is a stationary random function (see Time evolution of correlation functions (specifically Onsager's hypothesis) in time correlation link)
Hence $\mathrm{G(t,\tau})$ only depends on $\tau$, the delay in measuring $\mathrm{B_{loc}}$.
So, for simplicity we set $t = 0$ and note that $$\mathrm{G(t,\tau)} = \overline{\mathrm{B_{loc}(0)}^2} \mathrm{e}^{-\frac{\tau}{\mathrm{\tau_c}}},$$ where $\mathrm{\tau_c}$ 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 prove exponential decay behavior for time correlation functions, in general?
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$\begingroup$ In fact, this problem has very little to do with NMR Spectroscopy. It is much more related to statistical mechanics. $\endgroup$– HighEnergyJul 21, 2016 at 18:00
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$\begingroup$ Does anyone know how to prove the exponential decay of the time correlation function? $\endgroup$– HighEnergyJul 25, 2016 at 18:22
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