Brownian Motions

The autocorrelation of the particle velocity field defined as (with a bar standing for the expected value) is the inverse Fourier transform of the power spectrum function. i.e.,

(7)

Hence

(8)

From (6) and (7) it follows that

(9)

The mass diffusivity is defined as

for large t,

(10)

where x(t) is the position of particle given by

(11)

Using (11), one finds

(12)

Changing variables, after some algebra it follows that

(13)

Thus,

(14)

Using (6) or (9) in (14), we find

(15)