Goals of this course

In this course, you will learn some about both computer programming and physical science (and maybe something about biology and finance.) We will start by discussing some ideas about the deep links between computers (physical objects) and physical science (which we understand using computation.) For most of the course, you will learn how to integrate the two disciplines of computer programing and physical science in order to model processes in the universe. We will use the computer for exploration. Applications will include chaos (unpredictability in areas such as weather and the spinning of moons), fractals (geometric objects with details at all sizes, as in a tree or coastline), formation of patterns (as in tiger spots), and waves. We will end with a discussion of advanced technologies for computers.

Methods and Tools

We will study how to set up problems and some techniques and software that are available to solve these problems. As part of learning how to run simulations, you will create virtual "demonstrations" of physical phenomena. Given the complexity of the universe and the simulations that can be carried out today, visualization and presentation of the results is important to working with computers. We will discuss how to draw conclusions from simulations: what intuition and insight one can gain and how to extract numerical answers.

The computational tools that will be used include Python, a high level object oriented language which can be used to quickly write code. We will apply the Visual library (VPython), which allows for visualization on a variety of platforms, including Linux, Windows, and Macintosh operating systems. This is free software that you can also install on your own computer, if you have one and wish to. We will use MATLAB, also available on the public clusters and for student purchase, for manipulating and visualizing data.

Topics

This course will undergo some development as we go along, so the topics may shift some. Currently, the plan is to cover the following:

Chaos and Fractals

In the study of motion, the characterization of "chaos" is used for systems which have very unpredictable behavior. This type of behavior is quite different from the motion of a single planet orbiting a star or a falling object. The unpredictabliity does not come from ignorance, but from the "sensitivity" of the motion to small effects. Often, the behavior of chaos is reflected in the fractal nature of the path the system follows (in "phase space"). A fractal shape is one that does not have a simple dimension like 1 or 2, but is a non-integral number, say the square root of 3. The geometry of fractals, which often resemble clouds or coastlines, can be stunning.

General principles in computational science

What is a computer? How does one set up and solve a problem on the computer? Does one take a "Monte Carlo" approach (based on random sampling) or a deterministic approach? What is a useful "solution" and how does one assess and present the results of a simulation? What types of visualization are useful? What types of approximations can be used? How accurate is the answer? What resources are needed?

Random walks

A recurring theme in modelling is that of "random walks". We will use the idea of randomly moving object to study the motion of small particles, the behavior of light in clouds, the motion of bacteria, and the price of stock options.

Patterns and waves

Besides treating simple objects as points, we will look at the dynamics of objects with extension, such as surfaces and three-dimensional objects. We will look at models that create patterns that have more or less disorder, similar to the spots or stripes of animals, and the simulation and visualization of waves.

Advanced computing

How will evolving technology and concepts, including DNA computing, quantum computing, and smaller silicon devices affect science? What is the relationship between the physical world and computing?