Velocity Correlation Function

Question:
Consider the Langevin equation

with < f (t) >= 0.

a) Calculate the velocity correlation function < v(0)v(t) > at all times. This function shows how velocity is correlated with its initial value. What are the values of this function at t=0 and t → ∞?

b) Calculate

Solution:
a) By multiplying Langevin equation by exp(−t ξ/m) , we get,

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