Complexicon: Random Walk

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.