Brownian Motions

Computer Simulation Procedure

As noted before, the Brownian force n(t) may be modeled as a white noise stochastic process. White noise is a zero mean Gaussian random process with a constant power spectrum given Equation (3). Thus,

,

(25)

The following procedure was used by Ounis and Ahmadi (1992) and Li and Ahmadi (1993). Choose a time step (The time step should much smaller than the particle relaxation time). Generate a sequence of uniform random numbers, &;(between 0 and 1). Transform pairs of uniform random numbers to pairs of unit variance zero mean Gaussian random numbers. The can be done using the following transformations:

(26)

(27)

The amplitude of the Brownian force then is given by

(28)

The entire generated sample of Brownian force need to be shifted by , where U is a uniform random number between zero and one.

Figure 2. Numerically simulated Brownian force.