Brownian Motion

Brownian motion was first observed by Dr Robert Brown in 1827, using pollen grains. It was not explained correctly until much later. Albert Einstein set about explaining Brownian motion in 1905.

Dr Robert Brown was a biologist and was studying pollen grains under a microscope. He noticed that the tiny grains were constantly moving and "jigging" around. He explained this by saying that, since pollen grains are from plants and plants are alive, then these grains must also be alive and moving of their own accord.

Years later...

Albert Einstein, in 1905, set about proving that Brownian motion took place due to moving particles, not because pollen grains were alive.

One of Einstein\'s papers gave an explanation of Brownian motion using the kinetic theory of matter. Einstein suggested that the pollen grains moved because they were being bombarded by thousands of tiny water molecules, which were moving very quickly. He even went as far as to work out the speed at which these water molecules were moving.

A random movement of pollen grains suspended in water was observed by botanist Robert Brown in 1827. He thought that the movement must be due to the live pollen, but on observing non-living dye particles in water, he saw the same random dance. This motion was explained a generation later by Maxwell and Einstein. It\'s due to the invisible water molecules hitting the visible particles and moving them a bit. Since the particles are hit continuously from all sides, they moves in a random fashion.

"Brownian motion, isn\'t that the Physics theory I studied in high school?" This is the very instinctive reaction for most of us when encountering the term "Brownian motion". But how much do we really know about it? This report is aimed at unravelling the theory of Brownian motion to give readers a better perception of Brownian motion and its applications in this world of ours.

1. Introduction
Human lives are full of uncertainties, as with many natural phenomena. No one can precisely foresee what will happen in the next second, minute, ... and so on. Rather than accepting the fact that the future is always uncertain, many models and algorithms have been continuously formulated for the prediction of matters involving uncertain elements. One of them is the Brownian model.

2. Historical Background
Brownian motion is a sophisticated random number generator, based on a process in plants discovered by Robert Brown in 1827(Figure 1). He found that small particles suspended in a fluid were in continuous movement and thus, described it as Brownian motion. His discovery did not receive much attention for a long time, until before the turn of the 20th century when Guoy\'s conviction and research (that Brownian motion constituted a clear demonstration of the existence of molecules in continuous motion) brought it to the attention of the Physics world. However, all nineteenth-century research remained at the qualitative level.

It was only in 1905 when a quantitative analysis was brought about, where Einstein[1] succeeded in stating the mathematical laws governing the movements of particles on the basis of the principles of the kinetic-molecular theory of heat. According to this theory, bodies of microscopically visible size suspended in a liquid will perform irregular thermal movements called Brownian molecular motion, which can be easily observed in a microscope. Brownian motion was then more generally accepted because it could now be treated as a practical mathematical model. As a result, many scientific theories and applications related to it have been developed and they subsequently play major roles in the world of Physics.

Fig. 1. Brownian motion of a microscopic particle

3. The Elementary Theory of Brownian Motion
The \'Elementary theory of Brownian motion\' is one of the major investigations by Einstein on the Brownian movement theory in 1908. In this paper, the term "Brownian motion" is referred to as the irregular and unceasing movement of solid microscopic particles when suspended in a fluid medium.

In an undissociated dilute solution, there is a process of diffusion(*), which is caused by the Brownian motion of the suspended thermal molecules. On the other hand, another process proceeding in the opposite direction of that of the diffusion one also occurs. This movement of suspended substances is brought about by osmotic pressure forces(+).

The first step