Calculating diffusion coefficient from mean squared displacement data

<p>To extract the diffusion coefficient from a <a class="wikilink" href="/nanoparticle_tracking_analysis/">nanoparticle tracking analysis</a> experiment, especially when looking at diffusive <a class="wikilink" href="/brownian_motion/">brownian motion</a>, we can calculate $\left&lt;\Delta x^2\right&gt;$ from the tracks themselves as:[<a class="litnote tooltip" href="/literature/@catipovic2013">@catipovic2013<span class="tooltiptext">Improving the quantification of Brownian motion</span></a>]</p> <p>$$\left&lt;\Delta x^2(\tau)\right&gt; = \frac{\sum_{i=1}^{N_i}(x(\tau_i + \tau)-x(\tau_i)^2)}{N_i}$$</p> <p>In this case, instead of looking at the distribution (see: <a class="wikilink" href="/calculating_diffusion_coefficient_from_jump_statistics/">calculating diffusion coefficient from jump statistics</a>), we can directly fit the temporal correlation, assuming the following relation:</p> <p>$$\left&lt;\Delta x^2\right&gt;(\tau) = 2D\tau$$</p>