Nature is the deepest source of inspiration when it comes to knowledge and creativity. In 1827, the biologist Robert Brown was looking at pollen grains under a microscope when he noticed something about their movement. They were floating around in the water, but their motion was quite peculiar. Rather than moving in a graceful manner, like a log floating on a calm river, the particles exhibited a jerking, random-looking motion. The observation was repeated by other scientists, but all efforts to come up with an explanation were in vain. In 1905, Albert Einstein came along and presented a wonderful explanation for this natural phenomena.
To get a better idea of Brownian Motion, imagine a tightly packed crowd at a concert or a sporting event, where all the people are waving their hands in the air. Now, imagine throwing a beach ball into the crowd, where the waving hands of the crowd will bat it around. Each time the ball touches a hand, it gets flicked off in a new direction, so the ball seems to be undergoing random motion. If you imagine a vast crowd of people and watch the balls’ motion for an extended period, you will notice the Brownian motion in effect. This kind of motion is a strange thing: a continuously wriggling, twitching jiggle that looks like nothing we see in the everyday world. Fundamental processes tend to be smooth, like the curving path of a falling leaf blowing in the wind, but those particles of pollen seem to have an angular motion to them. The angular motion makes them jump in one direction and then another without ever settling down.
Not surprisingly, then, some rather strange mathematics are required to make a good model of it. Ordinary Calculus, the bread and butter of physical science, has a hard time nailing down Brownian Motion because of its spiky changes in direction; in fact, a whole new system of calculus had to be invented to cope with Brownian Motion. The mathematical model of Brownian Motion took some time to develop, but when it did, it represented a new kind of object. As a result, variations of it have been used to attack problems that other techniques could not quite handle. Biologists have used Brownian models to make improved models of the behavior of birds, fish, and insects moving in large groups; it’s been used to enhance noisy digital signals, such as improving medical ultrasounds. It is also very commonly used in financial asset prices and informs major decisions.
In-depth Brownian Motion
Now, let’s take a deeper dive into understanding Brownian Motion. In the above equation, the function stands for the position of the particle at that instant of time, where n is the scale factor and z is the small random displacement. However, don’t worry about digesting some mathematical monstrosity, as we will be taking a more intuitive approach in understanding this equation. Let’s consider a man named John and assume that he had a bit too much to drink. In his attempt to get back home, he tries to walk in a straight path through a field. However, each step he takes involves a random stagger to the left or right; from a bird’s-eye view, his path looks like a jagged line. This is what is known as a random walk. John moves steadily forward, but he also waves sideways through the field. When John gets to the other side of the field, he might get lucky and reach the gate or end up getting stuck in a hedge. Thus, a question we may ask is… what are his chances of reaching the gate? Although it’s not a straightforward answer, we can get this answer from probability theory.
We should expect John to make it to the gate (in a technical sense) because his leftward and rightward staggers should cancel one another out. Naturally, any given walk could land him in a hedge-related predicament, but if we had to make a bet on the final position, we should bet on the gate, as that’s where he will end up most often. We may also ask the question of how far John might be from the gate, whether he lands on the other side of the field 10 meters or 30 meters from the gate. John’s path has what mathematicians call a Markovian Property, which says that at every moment of the walk across the field, the next step only depends on where he previously happened to be. In real life, John might start veering off to the left, making more steps to the left likely, but this would not be a Markovian property anymore. This property is important, as it greatly simplifies things.
So far, so good, but we still don’t have a model for Brownian Motion. Suppose John takes n steps forward (where n is the number of steps taken by John), each accompanied by a stagger either to the left or the right. The total in effect motion is one that is diagonal. To turn this in to Brownian motion, we start increasing the value of n; more specifically, we make John start taking smaller steps. If he takes steps half the length, then n doubles. Now, let’s keep doubling n until it becomes very large. What’s surprising is that the answers to questions about this process—such as the chances that John will end up in a certain part of the far boundary—settle down to stable values as n gets larger. This encourages us to look at John’s walk as n goes off towards infinity (Zeno’s Dichotomy), which results in the mathematical process known as the Wiener Process.
Now, let’s shift away from the crass example of a drunk John back to the pollen grain. When a pollen grain floats, it is being buffeted by millions of water molecules that are whizzing around and bouncing off each other in such a complex way that it is effectively random. Now, each second, a single grain of pollen is bombarded thousands of times, each time gaining a minuscule nudge. The overall effect of these nudges is the wandering motion that Robert Brown noted.
Now, it has to be said that there is a big difference between millions of tiny particles and the Weiner Process, with the primary difference being the millions of minuscule nudges. Even though these infinities can be wrestled into mathematical order with a bit of extra work, they only approximate a physical situation. Still, they approximate it very well, and the Weiner Process itself turns out to be highly useful in other areas and subjects, such as high finance. In 1900, there was a PhD thesis titled the Theory of Speculation by Louis Bachelier. In it was an analysis of the movements of prices on the Paris stock exchange using the then-new theory of Brownian Motion. Bachelier’s ideas did not gain recognition until the 1960s, but as the computer age dawned, Brownian Motion became a popular way to simulate share price movements in the future. Random walk techniques model unpredictable behavior that evolves, in which each change is independent of the previous ones; Brownian motion takes this to the ultimate conclusion.