Standard deviation of anomalous diffusion

In a previous post we have used mean squared displacement to understand diffusion of Brownian and Levy's walks. Alternatively we could use a more familiar statistical measure: variance (or standard deviation).


While mean squared displacement could be seen as more of temporal measurement tool (most of the calculations happen over time), variance (or standard deviation) could be seen as more of ensemble tool (most of the calculations happen over ensemble). I believe that MSD is better suited when the diffusion is homogeneous, while variance (or standard deviation) could be also used when the diffusion is heterogeneous.

Both Brownian walk and Levy's walk considered in the previous post represent homogeneous diffusions. So we have correctly applied MSD, but we can also try using variance (or standard deviation).

Note that I have kept mentioning standard deviation. When dealing with data I prefer standard deviation over variance, because standard deviation has the same units as data, so its interpretation comes a bit more naturally. Therefore I'll be using \( \sigma_t \) parametrized as follows:

\begin{equation} \sigma_t = \sqrt{\frac{1}{N-1} \sum_{i=1}^N ( x^{(i)}_t - \mu_t )^2} . \end{equation}

In the above \( x^{(i)}_t \) is a value of \( i \)-th trajectory at time \( t \) and \( \mu_t \) ensemble average at time \( t \).

Brownian walk

For Brownian walk we will often observe that:

\begin{equation} \sigma_t \sim t^{0.5} . \end{equation}

Note that the exponent is exactly half of what we get using MSD. This is because I am relying on standard deviation and not variance. If I would have used variance the exponents would be exactly the same.

Levy's walk

For Levy's walk we would expect to observe:

\begin{equation} \sigma_t \sim t^{\alpha} . \end{equation}

Usually we will observe super-diffusive regime \( 0.5 < \alpha < 1 \) or ballistic regime \( \alpha = 1 \). Actually [1] suggests that:

\begin{equation} \alpha \approx \frac{3-\gamma}{2} . \end{equation}

The equation above will hold as long as estimate of \( \alpha \) is within allowed value interval.