# Brownian movement

The irregular motion of small particles suspended in a liquid or a gas,caused by the bombardment of the particles by molecules of themedium: first observed by Robert Brown in 1827. The movement of particle (random) suspended in a fluid is called Brownian motion. Brownian motion's mathematical model has several applications in real world.

1. Colloidal solution consist of dispersion medium and dispersed phase. The dispersion medium may be liquid, solid or gas and the dispersed phase may also be solid, liquid or gas.

2. Some example of the colloidal solutions are Milk which is the emulsion of liquid in liquid, which means the dispersed phase is liquid and dispersion medium is also liquid. Fog is an Aerosol, which liquid particles dispersed in the gas. Smoke is also an colloidal solution of solid in gas.

3. The Brownian motion is the zigzag movement of the dispersed phase in the continuous dispersion medium.

A one dimensional model which describes a particle that usually undergoes Brownian motion was published in 1906 by Smoluchowski. The collisions are assumed with M>>m (M is actually the mass of the test particle, whereas m is the mass of any of the singleton particles that composes the fluid).

After collision the test particle’s velocity will increase by ΔΔV (mΔΔM) v if V is test particles velocity and fluid particle velocity is v. If the number of collisions from the right is NR and the number of collisions from the left is NL, then the changed particle velocity after N collisions is given by ΔΔV (2NR –N). The multiplicity is then given by,

(N!)/(NR)!(N−NR)!(N!)(NR)!(N−NR)!

And total possible states are 2N. Thus the probability that the particle will hit from the right NR times is

PN(NR)PN(NR) = N!2NNR!(N−NR)!