Calculating diffusion coefficient from jump statistics
If we have a random process, such as when looking at Brownian Motion, we can look at the individual displacements $\Delta x$, and we'll get[@catipovic2013Improving the quantification of Brownian motion]
$$\textrm{Var}(\Delta x) = \left< \Delta x^2\right> - \left<\Delta x\right>^2$$
For standard Brownian processes, we also have $\left<\Delta x\right> = 0$
And from here, we have that the variance is equal to the mean-squared displacement, which can be re-written as the known equation:
$$\left<\Delta x^2\right> = 2Dt$$
In this case $t$ is the time between frames in the measurement (which is given as constant).
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