Velocity correlation functions

[thrillhouse86]

Hi, I am going through Non Equilibrium Statistical Mechanics by Zwanzig and I can't follow, the step below:

I have the equation:
$$<x^{2}> = \int^{t}_{0}ds_{1}\int^{t}_{0}ds_{2}<v(s_{1})v(s_{2})>$$

I can't show that:
$$\frac{\partial <x^{2}>}{\partial t} = 2 \int^{t}_{0}ds<v(s)v(t)>$$

I'm sure that the answer lies with that fundamental theorem of calculus, but I can't show it. For one thing, do I apply the product rule to the two integrals above ?

Thanks

 

[DrDu]

call the upper indices (which are now both called t) a(t)=t and b(t)=t. Then use that the total derivative (well, total at least with respect to t) is d/dt=da/dt d/da+db/dt d/db.