I am following the proof of the Einstein diffusion equation, that draws a connection between the mean-square displacement and velocity autocorrelation. I am confused about a logical leap they make in the derivation, given here: https://nanohub.org/resources/7581/download/Martini_L9_DynamicProperties.pdf, slide 15.
This is how it goes:
I don't understand how the last step takes place. How does one compress this double integral into a single integral? I would really appreciate it if someone could go into the technical derivation of such a step.
I wrote this elsewhere in Latex and took an image because for some reason the Latex wasnt compiling right in the SE input window. I would appreciate any advice you have for me!
I believe what they're doing here is just changing the order of integration: the original double integral is integrating over the region where
$$0 \leq \tau \leq t' \leq t$$
so if we rewrite to switch the order of integration (assuming that the assumptions required for Fubini's theorem hold) we get
$$2\int_0^t d\tau \int_\tau^{t} dt' \langle \textbf{v}(\tau) \cdot \textbf{v}(0)\rangle$$
and because the integrand doesn't depend on $t'$ anymore, we're essentially integrating a constant there, so we get
$$2\int_0^t d\tau(t - \tau)\langle \textbf{v}(\tau) \cdot \textbf{v}(0)\rangle$$
as desired.