calculate_relaxation_time

mdcraft.analysis.polymer.calculate_relaxation_time(times: ndarray[float], acf: ndarray[float]) → float

Calculates the orientational relaxation time \(\tau_\mathrm{r}\) of polymers using end-to-end vector autocorrelation function (ACF) time series \(C_\mathrm{ee}(t)\).

A stretched exponential function with \(\tau\) and \(\beta\) as coefficients,

\[C_\mathrm{ee}(t)=\exp{\left[-(t/\tau)^\beta\right]}\]

is fitted to the ACF time series, and the relaxation time is estimated using the first moment of \(C_\mathrm{ee}\),

\[\tau_\mathrm{r}=\int_0^\infty C_\mathrm{ee}(t)\,dt =\frac{\tau}{\beta}\Gamma\left(\frac{1}{\beta}\right)\]

where \(\Gamma\) is the gamma function.

Parameters:

timesnumpy.ndarray

Changes in time \(t-t_0\).

Shape: \((N_\mathrm{frames},)\).

Reference unit: \(\mathrm{ps}\).

acfnumpy.ndarray

End-to-end vector ACF \(C_\mathrm{ee}(t)\).

Shape: \((N_\mathrm{groups},\,N_\mathrm{bins},\,N_\mathrm{frames})\).

Returns:

relaxation_timefloat

Average orientational relaxation time \(\tau_\mathrm{r}\).

Reference unit: \(\mathrm{ps}\).