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equally one could use the solution of the Langevin equation: $$m\dot{v} = -\zeta v +\delta F(t)$$ witch is: $$v(t)=v(0)e^{-\zeta t \over m }+ \int \limits_0^t dt'e^{-\zeta (t-t') \over m } \delta F(t)$$ doing the ensamble average gives: v(t)=\overbrace{\langle 1 \rangle}^{=1} v(0)e^{-\zeta t \over m }+ \int \limits_0^t dt'e^{-\zeta (t-t') \over m } ...