Hi there. I'm stuck with this thing. I was studying Brownian motion from the book a first course on statistical physics, by Linda E. Reichl. In page 251 from her book, she treats Brownian motion from the Langevin equations.
I have a doubt on the derivation of the correlation function for the velocity and position in Brownian motion from the Langevin equations.
The Langevin equations are:
I have then that for a brownian particle:
is a Gaussian white noise with zero mean, such that .
The assumption that the noise is Gaussian means that the noise is delta-correlated:
Now, the book makes use of the fact that , and gives for the correlation function:
I think that this must be used:
But I'm not sure of it, and I don't know what are the intermediate steps between (1) and (2).
The thing is that the integrals I get confuses me. For a simpler case I was considering the average of velocity.
The book gives (subject to the condition that ):
So I think that this should mean:
I can put all that in terms of an integral of time using the equations for brownian motion:
And using (1):
is clearly a Gaussian, but I don't know what "shape" it has (I thought of using the moments to get that shape, but the deviation give something with a dirac delta in zero, which makes no sense to me). Anyway, I think the integrals can be computed just using the given facts: and , and thats the point that concerns me. How to use those facts to compute the integrals. I don't know if I'm giving the proper interpretation to the notation neither.
Any help will be appreciated.
PD: For a discussion on Brownian motion and Langevin equations you can check this: http://web.phys.ntnu.no/~ingves/Teac...loads/kap6.pdf
I'm interested on the formal derivation, step by step of the correlation functions, and the average velocity, because some things are not that clear to me.