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I am reading the Zwanzig's book and I have a confusion about the average of the fluctuating force and its distribution.

As it says $F(t)$ is a random variable that means it has a probability distribution means the probability of getting the different amplitude of Fluctuating force at the same time $t=\epsilon$(say) is according to the distribution curve, that is the meaning of distribution curve.

Then it says the time average of the $F(t)$ over small time gap is zero,

$\frac{\int_{0}^{\tau}F(t)dt}{\tau}=0$;

From this relation, it is clear to me that the curve of $F(t)$ vs $t$ cross the $t=0$ axis many time over the small interval of time. But then He said that the distribution curve of $F(t)$ is Gaussian, How he came to this conclusion.

What is the relation between $F(t)$ vs t and the probability distribution curve of $F(t)$?

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  • $\begingroup$ doi.org/10.1016/0301-4622(91)87208-M $\endgroup$ – user191954 Nov 17 '18 at 15:16
  • $\begingroup$ When many random forces add together the resulting net force has Guassian distribution by virtue of central limit theorem. $\endgroup$ – Deep Nov 21 '18 at 6:15

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