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Brownian Motions When a small particle is suspended in a fluid, it subjected to the impact gas or liquid molecules. For ultra fine particles (colloids), the instantaneous momentum imparted to the particle varies random which causes the particle to move on an erratic path now known as Brownian motion. Figure 1 illustrates the Brownian motion process.
Figure 1. Schematics of a Brownian motion process. The Brownian motion of a small particle in a stationary fluid in x-direction is governed by the following Langevin equation,
where u is the velocity of the particle, |
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| (2) |
and n(t) is a white noise excitation due to the impact of fluid molecules on the particle. The intensity of noise is specified by its spectral intensity given as
| (3) |
where 
is the Boltzmann constant and T is the temperature. It should be emphasized
that the Brown motion occurs in three dimensions and Equation (1) applies only
to the x-component of the motion.
For the stochastic equation given by (1), using the standard linear system analysis, it follows that
| (4) |
where 
is
the power spectrum of the velocity of the Brownian particle, and
is the system function given by
| (5) |
Hence,
| (6) |