## Brownian motion and random walksOn this page, you will learn about In the introduction, we said that we wanted to study randomly growing surfaces, but what does that mean, exactly? What does it mean for something to be random and how can a surface grow randomly? To answer these questions, we will start more carefully and talk about random walks of particles. Imagine a gas molecule in the air: it moves around on its own until it hits another gas molecule which makes it change direction. Since there are so many gas molecules in the air, it will constantly bump into other molecules (roughly \(10^{14}\) hits per second - that equals the total number of Google searches performed worldwide during 79 years!) and it will be just as likely to be hit from another particle on the left as it will be to be hit on the right. The bumps therefore cancel each other out, so after a long time interval, it will barely have moved at all, even though it makes really quick jerks all the time. The way the gas molecule moves will turn out to be important to studying randomly growing surfaces, so we will keep going on this track for a while! ## Random walksFirst, we want to try to model how this gas molecule moves in the simplest possible way, and you will explore one of these models in the following exercise.
This is a very simple model of how the gas molecule can move, but it is also close to reality! To see a larger example, the following is a two-dimensional random walk generated in the same way as the exercise.
A (bigger) two-dimensional random walk Even though the motion is quick and jerky, the particle doesn't get very far for large times - just like for gas molecules! Maybe we are on to something! ## Brownian motionReal gas molecules can move in all directions, not just to neighbors on a chessboard. We would therefore like to be able to describe a motion similar to the random walk above, but where the molecule can move in all directions. A realistic description of this is
To get started, the following is a simulation of a gas, and one particle is marked in yellow. Its path describes a Brownian motion \(B_t\) at time \(t\).
Gas molecule (yellow) describing Brownian motion normal distribution with variance \(t_2 - t_1\).
Now, Einstein realized that even though the movements of all the individual gas molecules are random, there are some quantities we can measure that are not random, they are predictable and can be calculated. One such quantity is the density \(\rho\) of the gas molecules. Einstein showed that the density satisfies a differential equation \[ \frac{\partial \rho}{\partial t} = D\frac{\partial^2 \rho}{\partial x^2}, \] called the
To generate a Brownian motion, follow the following steps: we want to generate a brownian motion at times \(0, 0.1, 0.2, … , 1\). \(B_0\) is defined to be \(0\). By definition, \(B_{0.1} - B_0\) is normally distributed with variance \(0.1\), so generate one such number and let that be the value of \(B_{0.1}\). \(B_{0.2} - B_{0.1}\) is again normally distributed with variance \(0.1\), so generate one such number and add that to \(B_{0.1}\) to get the value of \(B_{0.2}\). Write a program that continues this procedure!
## Further reading:J. Matson, ‘‘Crowd Forcing: Random Movement of Bacteria Drives Gears’’, https://www.scientificamerican.com/article/brownian-motion-bacteria/. Feynman, ‘‘Feynman Lectures on Physics’’, http://www.feynmanlectures.caltech.edu/I_41.html. Central Limit Theorem, https://en.wikipedia.org/wiki/Central_limit_theorem.
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