msd_non_diffusive

mdtools.dynamics.msd_non_diffusive(t, a, x)

Non-diffusive Mean Square Displacement (MSD) as fuction of time.

\[\langle \Delta\mathbf{r}^2(\tau) \rangle = a \tau^x\]

with \(\Delta\mathbf{r}(\tau) = |\mathbf{r}(t_0+\tau) - \mathbf{r}(t_0)|\) being the displacement vector after a diffusion time \(\tau\). \(a\) and \(x\) are just parameters. If \(x=1\), the equation describes diffusive motion. In this case, \(a\) is equal to \(2dD\), with \(D\) being the diffusion coefficient and \(d\) being the number of spatial dimensions of the diffusion process. The brackets \(\langle ... \rangle\) denote averaging over all particles of the same species and over all possible starting times \(t_0\).

Parameters

• t (scalar or numpy.ndarray) – Array of times for which to calculate the (non-diffusive) MSD.

• a (scalar or numpy.ndarray) – Prefactor.

• x (scalar or numpy.ndarray) – Exponent.

Returns

msd – (Non-diffusive) mean square displacement MSD(t) = a*t^x.

Return type

scalar or numpy.ndarray

See also

mdtools.dynamics.msd()

Calculate the diffusive MSD according to the Einstin relation as function of time

Notes

If more than one input argument is a numpy.ndarray, the arrays must be broadcastable to a common shape.

Examples

>>> mdt.dyn.msd_non_diffusive(t=np.arange(4), a=2, x=3)
array([ 0,  2, 16, 54])