= Vibrational frequency-state residence time
A \b[vibrational frequency-state residence time] is the duration a probe remains in a defined frequency environment before leaving it. It differs from the time over which the probe's frequency retains statistical memory, and from a <vibrational relaxation> lifetime describing excitation loss.
For a state with constant exit rate $k$, let $S(t)$ be the probability of no exit by time $t$. The small-interval exit model gives
$$
\frac{dS}{dt}=-kS,\qquad S(0)=1,\qquad S(t)=e^{-kt}.
$$
The residence-time density is $f(t)=ke^{-kt}$, with mean $1/k$ and <variance> $1/k^2$. The <exponential distribution> is memoryless: $S(t+s)/S(t)=e^{-ks}$. These conclusions require a constant-rate model, rather than following from the name of a spectral state alone.
For a two-state <continuous-time Markov chain> with A-to-B rate $a$ and B-to-A rate $b$, the mean residence times are $1/a$ and $1/b$. The normalized <two-state vibrational frequency memory> instead decays as $e^{-(a+b)T}$. Returning after an exit contributes to occupancy and correlation but not to survival without an exit.
An original symmetric example uses $a=b=k=0.25\,\mathrm{ps^{-1}}$. Mean residence time is $4\,\mathrm{ps}$, while frequency-memory time is $2\,\mathrm{ps}$. Starting in A, the probability of being in A after $T$ is
$$
P_{AA}(T)=\frac{1+e^{-2kT}}2,
$$
whereas no-exit probability is $e^{-kT}$. At $T=2\,\mathrm{ps}$ these are about $0.684$ and $0.607$.
This produces <vibrational residence-time sampling bias> if a same-state observation is treated as proof that no exit occurred. Using $-\ln P_{AA}(2)/2$ as an exit-rate estimate gives about $0.190\,\mathrm{ps^{-1}}$, below the stipulated $0.25\,\mathrm{ps^{-1}}$. Faster time sampling can resolve more excursions, but the estimator must still account for missed transitions and state-assignment uncertainty.
In <two-dimensional infrared spectroscopy>, the <2D IR waiting time> separates initial frequency labeling from later response readout. Primary response theory relates such measurements to frequency memory under explicit approximations.<Frequency-frequency correlation functions and apodization in two-dimensional infrared vibrational echo spectroscopy: A new approach by Kyungwon Kwak and others>[Introduction, pulse timing and frequency-frequency correlation function]{ref} A fitted memory decay is therefore not automatically a directly measured single-state residence time. Nonexponential survival can require additional states, time-dependent rates or a different physical model.
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