# Question about the autocorrelation function of the fluctuating force in the Langevin model for Brownian motion

According to the Langevin model, we have, for the motion of Brownian particles, $$\frac{dv}{dt} = -M\gamma v + \zeta(t)$$ with $$\zeta(t)$$ the random force acting on the particle due to fluctuations.

Then I was told that the autocorrelation time $$\tau_{c}$$ of this fluctuating force is typically of the order of the time interval between two collisions of the ﬂuid particles on the Brownian particle.

This statement puzzles me. I know these two terms $$-M\gamma v$$ and $$\zeta(t)$$ come from the collision (scattering) between the Brownian particles and the fluid particles But I wonder does the statement mean that the autocorrelation function of the fluctuating force $$\langle\zeta(t)\zeta(t+\Delta t)\rangle$$ have the mathematical relation that $$\langle\zeta(t)\zeta(t+\Delta t)\rangle \sim e^{-\gamma t}$$ and $$\tau_{c} \sim 1/\gamma$$?

If there is such a kind of relation, why? And if not, how to show

that the autocorrelation time $$\tau_{c}$$ of this fluctuating force is typically of the order of the time interval between two collisions of the ﬂuid particles on the Brownian particle.

It is possible to write down a generalized Langevin equation, in which the random force term has a memory: $$\frac{dv(t)}{dt} = -\int_0^t dt' \Gamma(t-t') v(t') + \zeta(t)$$ where I have set $$M=1$$ for simplicity. In this case we have $$\Gamma(t) = \frac{\langle \zeta(0)\zeta(t)\rangle}{\langle v^2\rangle} .$$ So, if you want to have an exponentially decaying correlation function for $$\zeta(t)$$, $$\Gamma(t) = \frac{\gamma}{\tau_c} \exp(-t/\tau_c)$$ you must also have an integral of the frictional term involving a memory kernel based on that correlation function. In the limit that $$\Gamma(t)=\gamma\delta(t)$$ (which comes from the above formula by allowing $$\tau_c\rightarrow 0$$) you recover the Langevin equation $$\frac{dv}{dt} = -\gamma v(t) + \zeta(t) .$$ You can find some discussion of this in various books on statistical mechanics, but also in this online account.