Vibrational residence-time sampling bias

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 isIt counts an odd number of switches, not every switch. The expected observed-change count per observation time is thereforewhich 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 $k=0.25\mathrm{p}{\mathrm{s}}^{-1}$ and $\Delta =2\mathrm{ps}$, $p_{\mathrm{flip}}\simeq 0.3161$ and $k_{\mathrm{count}}\simeq 0.1580\mathrm{p}{\mathrm{s}}^{-1}$. Treating same-state occupancy as no-exit survival gives a different naive estimate, $-\ln (1-p_{\mathrm{flip}})/\Delta \simeq 0.190\mathrm{p}{\mathrm{s}}^{-1}$. Both differ from the stipulated true rate.

Under this specific symmetric model, inversion givesUsing 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.^[ref] Relating a spectral memory statistic to residence times requires a physical observation model as well as a transition model.