The random walk model provides a simple way
to represent to complex random movements
of many everyday objects such as
the wiggly motion of air molecules in a room,
the dispersion of a drop of food coloring in water.
or the ups and downs of stock prices over time.
To be concrete, suppose you could sit on
an oxygen molecule in the air and you could
actually see the world at the molecular scale.
What would you see?
You’d be zooming at roughly three hundred meters at a second.
However before moving very far,
you bashed into another air molecule at roughly the same speed.
Each such collision happens roughly everyone by into a second.
So your trajectory would consist of very short periods
of free motion interrupted by collision
to drastically change your direction.
Think of bumper cars
but speed up by factor of a hundred
and shrunk by factor of ten billion.
The motion of your oxygen molecule
will be dominated by all these collisions.
The motion of all microscopic particles,
for example, molecules, cells and
pollens can be described in this way.
However keeping track of all these
collisions between microscopic particles
and all the other particles in environment
is hopelessly complicated.
So instead we use a simplifying mathematical model
to account for this complex phenomenon
in the case of an air molecule, it is much simpler
to represent its movement as a random walk.
In the random walk model, we nearly say
that a particle changes its direction
and its speed at random times.
These changes and perspectives from collision
between many particles to a single
randomly moving particle is enormously simplified.
You can use the random walk model to ask
and answer many important questions,
like, how far does a particle travel in a given amount of time?
What is the distribution of distances the particle travels?
Or how long does it take for a particle to move a given distance?
In addition to understanding the motion of a particle
in a microscopic world, the random walk model
has many other important applications of larger scales.
How much would you expect to win or lose over time to gambling?
How does a stock market fluctuate up and down?
When do random fluctuations in voltage cause
a neuron in your brain to fire?
Or how do cultural ideas get pass from
one person to another in societies?
Applying the mathematics of random walk’s
to these kinds of questions, represents some of the research
that we do here at the Santa Fe Institute.