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.