Randomness
A stochastic process is one that has some random element. A common example is Brownian motion: Take a small seed in water. We'd expect water molecules to randomly bump into the seed fairly evenly on all sides. However, since they are moving randomly, sometimes more hit on one side than on the other, causing the seed to jerk to one side. These fluctuations continue in time, giving the appearance of a randomly dancing seed. After a long time, the position you'd expect the seed to be gets wider and wider, but is still centered at the same point. Stochastics deals with examining the characteristics of random behaviors such as this. Other examples include: The concentration of chemicals undergoing a chemical reaction or the populations of species of animals.
I study the way in which descriptions of random processes behave differently for very large numbers of elements. Specifically, I'm currently studying a model of two species of animals that explore a terrain for food which helps them multiply. Using a continuum model or a discrete model can yield different results as the animals travel through low food regions.
Resources
- Stochastic Processes in Physics and Chemistry. N.G. van Kampen. A canonical text for stochastic processes and non-equilibrium statistical mechanics, now in it's 3rd edition. Level: Graduate.
- Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences. Crispin W. Gardiner. Another canonical text for stochastic processes and non-equilibrium stat mech. Few exercises within, but a good reference. Level: Graduate.