Question

Why is the correlation function of fluctuation force in Brownian motion related to a delta function?

Answer 1

★★For the first step to derive fluctuation-dissipation theorem, I find

★★⟨F(t)F(t′)⟩=2Bδ(t−t′)⟨F(t)F(t′)⟩=2Bδ(t−t′)

★★where BB is a constant, and F(t)F(t) is a random fluctuating force with Gaussian distribution, which is being called white noise

★★The delta is indicating that if F(t)=XF(t)=X, then F(t+δt)F(t+δt) = a random variable with a Gaussian distribution

Answer 2

The delta is indicating that if F(t)=X, then F(t+δt) = a random variable with a Gaussian distribution; no matter how small δt

is.

In more technical terms, the delta function is the temporal autocorrelation function corresponding to a physical process that has no memory, ie. one "time frame" is completely independent to the next.

And the average is across the ensemble of all possible F(t)s.