= Spectral-diffusion frequency-increment variance
The \b[spectral-diffusion frequency-increment variance] quantifies how much a transition frequency changes between two observation times. For a real <stationary process> with mean $\overline\omega$ and <autocovariance function> $C(T)$,
$$
D(T)=\mathbb E[(\omega(T)-\omega(0))^2]=2[C(0)-C(T)].
$$
The mean increment is zero under these assumptions, so its second moment is also its <variance>. Expanding the square proves the identity: the two squared-frequency deviations each contribute $C(0)$ and their cross term contributes $-2C(T)$.
If $C(T)=\Delta^2e^{-|T|/\tau}$, then
$$
D(T)=2\Delta^2(1-e^{-|T|/\tau}).
$$
For an original example with $\Delta=2\,\mathrm{rad\,ps^{-1}}$ and $T=\tau$, $D=8(1-e^{-1})\simeq5.057\,\mathrm{rad^2\,ps^{-2}}$ and the root-mean-square increment is about $2.249\,\mathrm{rad\,ps^{-1}}$. The single-time frequency spread remains $\Delta$; temporal decorrelation does not require the stationary distribution to broaden.
The long-time value $2C(0)$ assumes that the <autocovariance function> tends to zero. It is not a universal upper bound. For a random-phase sinusoid $\delta\omega(t)=A\cos(\Omega t+\Phi)$ with phase uniform over a full cycle,
$$
C(T)=\frac{A^2}{2}\cos(\Omega T),\qquad D(T)=A^2[1-\cos(\Omega T)].
$$
At half a cycle, the frequencies are anticorrelated and $D=4C(0)$; at a complete cycle the increment vanishes. Thus a nonmonotone increment statistic can reflect recurring frequency memory.
For nonstationary observations $X,Y$, the general identity instead is
$$
\mathbb E[(Y-X)^2]=\operatorname{Var}(X)+\operatorname{Var}(Y)-2\operatorname{Cov}(X,Y)+(\mathbb EY-\mathbb EX)^2.
$$
Drift of the mean must not be silently assigned to stationary <spectral diffusion>. Likewise, under an explicit model of independent zero-mean endpoint measurement errors with variance $s^2$, the observed second moment gains $2s^2$. This <spectral frequency-increment measurement noise> term changes when the endpoint errors are correlated.
Primary <two-dimensional infrared spectroscopy> work uses controlled frequency labeling and later readout to constrain frequency memory.<Frequency-frequency correlation functions and apodization in two-dimensional infrared vibrational echo spectroscopy: A new approach by Kyungwon Kwak and others>[Introduction, pulse delays and frequency-frequency correlation function]{ref} The identity above is a model relation; a measured <2D IR center-line slope> needs the stated response assumptions before it supplies $C(T)$.
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