Calculating diffusion coefficient from mean squared displacement data

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To extract the diffusion coefficient from a nanoparticle tracking analysis experiment, especially when looking at diffusive brownian motion, we can calculate $\left<\Delta x^2\right>$ from the tracks themselves as:[]

$$\left<\Delta x^2(\tau)\right> = \frac{\sum_{i=1}^{N_i}(x(\tau_i + \tau)-x(\tau_i)^2)}{N_i}$$

In this case, instead of looking at the distribution (see: calculating diffusion coefficient from jump statistics), we can directly fit the temporal correlation, assuming the following relation:

$$\left<\Delta x^2\right>(\tau) = 2D\tau$$