# Why the distribution of Fluctuationg force in brownian motion has gaussian distribution?

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 $$F(t)$$?

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