Brownian motion was first observed soon

after the invention of the microscope.

When everything was being examined under

the microscope.

In 1785, Ingenhousz observed stochastic

motions of pollen and water. Before we knew

the existence of molecules, fantastical

mechanisms were involved to explain this

behaviour.

Now we know better, the stochastic motion

of the pollen is the result of collisions

with water molecules.

This type of stochastic motions abound in

nature.

And in the next slide, I want to show you

a few basic examples.

First of all the stock market

fluctuations, stock market rises and falls

and sometimes crashes as in 2009.

And the underlying motion, S&P index

appears to follow a type of random walk

motion.

Another example is the diffusion of food

colour in water.

Take a ink-dropper and drop a drop of food colour

in a beaker of water; and you will see it

slowly spread out. The way it spread out

is governed by diffusion.

Another example is the cultural diffusion

of practises across different civilizations

And finally, diffusion of fluid through

some porous medium, such as water flowing through

a porous rock

If we want to understand all these phenomena

we need to have a good model.

The many body problem of pollen being

buffeted by water, is much to complicated

to treat analytically.

To make progress, therefore we need to

simplify

This leads to us to random walk model,

which follows Einstein’s very famous

dictum that "A model should be as simple as

possible, but no simpler."

In the random walk model, we disregard all

the collisions between the particle and the

external world. And in instead pose it that

the random motions of the particle arises

intrinsically from its own internal degrees

of freedom. Pictorially, the particle may

be replaced by a drunker, who's successive

steps are in random directions because of

his inebriation.

This is the random walk model.