= Vibrational residence-time sampling bias
\b[Vibrational residence-time sampling bias] occurs when finite-time observations miss exits and returns between frequency states, or when imperfect state assignment creates apparent transitions. The resulting apparent dwell behavior need not equal the underlying <vibrational frequency-state residence time>.
For an original symmetric two-state <continuous-time Markov chain> with exit rate $k$ in either state, the probability that two samples separated by $\Delta$ differ is
$$
p_{\rm flip}(\Delta)=\frac{1-e^{-2k\Delta}}2.
$$
It counts an odd number of switches, not every switch. The expected observed-change count per observation time is therefore
$$
k_{\rm count}=\frac{p_{\rm flip}(\Delta)}\Delta,
$$
which approaches $k$ only as the interval becomes small. At long intervals it approaches $1/(2\Delta)$ even if the true process switches much faster.
With $k=0.25\,\mathrm{ps^{-1}}$ and $\Delta=2\,\mathrm{ps}$, $p_{\rm flip}\simeq0.3161$ and $k_{\rm count}\simeq0.1580\,\mathrm{ps^{-1}}$. Treating same-state occupancy as no-exit survival gives a different naive estimate, $-\ln(1-p_{\rm flip})/\Delta\simeq0.190\,\mathrm{ps^{-1}}$. Both differ from the stipulated true rate.
Under this specific symmetric model, inversion gives
$$
k=-\frac{\ln(1-2p_{\rm flip})}{2\Delta},\qquad 0\le p_{\rm flip}<\frac12.
$$
Using the exact model probability recovers the chosen rate. An estimated probability near $1/2$ makes the inverse very sensitive to uncertainty, and the nonlinear inverse is not automatically an unbiased finite-sample estimator.
Misclassification can bias in the opposite direction. If a physically unchanged two-state label is independently reversed with probability $q$ at each observation, the apparent transition probability is $2q(1-q)$. At $q=0.1$ it is $0.18$ despite no physical transition. Faster sampling alone does not remove this labeling error.
A useful measurement comparison varies sampling interval and checks state assignment against suitable references. In <two-dimensional infrared spectroscopy>, a <2D IR waiting time> separates frequency labeling and later response, but spectra are not automatically direct individual-state trajectories.<Frequency-frequency correlation functions and apodization in two-dimensional infrared vibrational echo spectroscopy: A new approach by Kyungwon Kwak and others>[Introduction and Section II, pulse timing and response model]{ref} Relating a spectral memory statistic to residence times requires a physical observation model as well as a transition model.
Back to article page
