Other Courses • Physics 170 • Schedule • Project • Code Examples • Notes

Projects

The project allows you to apply some of the skills you have developed in the course to explore a problem of your own choosing. The scale of a project should be roughly akin to one of the problems in the course, or perhaps slightly more expansive. It should be grounded in a problem of physical interest and should aim to solve the problem in a numerically insightful way.

As shown in the Schedule, your project results will be presented to the class during the final three weeks of the course. The project should also be documented in a fashion similar to the problems in the course, with a clear verbal description of the physics, the relevant equations, and a summary of the results, in addition to the code itself.

The text by Gould and Tobochnik has many wonderful ideas for projects. Some of these, and others as well, are listed below. See The American Journal of Physics, The Journal of Computational Physics, the list of references for the course, and the usual physics journals for further ideas on projects.

A description of the topic and scope of your project is due March 6.

Some Possible Topics

The following list has some suggestions. Feel free to come up with your own.

Chaotic Granular Mixing has been observed for the first time. Studies of chaos in the mixing of fluids is common, but it was thought that grains mixed by a combination of steady motion and diffusion. Now an experiment by Troy Shinbrot, Fernando Muzzio, and Albert Alexander at Rutgers (shinbrot@sol.rutgers.edu, 732-445-6710), using identical (initially segregated) red and green particles in a cylindrical drum being gently tumbled, shows that grains can spontaneously interpenetrate chaotically, and the green-red interface was fractal in nature. Even more unexpected was the speed at which the interface grew in complexity---many orders of magnitude greater than expected. These results should have an impact on the mixing industry, which worries about how long and how hard to mix commodities such as pharmaceuticals, explosives, makeup, and powdered foods. (Troy Shinbrot et al., Nature, 25 February 1999; see also www.aip.org/physnews/graphics/.)

Random Walks in Odd Geometries In free space, the average distance covered in a random walk increases as the square root of the number of steps. Surprising behavior can arise when obstacles are placed in the way of the diffusing particles, or when the particles are animate and propel themselves. Random walks of microorganisms, proteins, and other chemicals play an important role in the life of living organisms, in medical technology, and basic research in chemistry, biology, environmental science, and many other disciplines. See, for example, the marvelous book by Howard C. Berg, Random Walks in Biology, which has been placed on reserve in Sprague.

Polymers The Nobel Prize was recently awarded to Pierre de Gennes for advances in understanding of the physics of polymers and liquid crystals. A polymer consists of N linearly linked units (monomers), where N can be in the thousands. Because N is so large, polymers are ideal candidates for statistical and random-walk analyses.
When a high polymer is dissolved in a good solvent, it is free to assume a great many configurations. One simple property of a polymer "macromolecule" is its length from end to end, R_N. It can be shown that in two dimensions the end-to-end length depends on the 3/4 power of R. In three dimensions, the exponent is close to 3/5.
What is the exponent for a long polymer chain in three dimensions? How many monomers does the polymer need before asymptotic behavior is observed? How many monomers does the polymer need before the asymptotic behavior is observed in two dimensions?

Molecular Dynamics A common approach to solving continuous problems in a discrete way is to impose a grid. One might study vacancies in a crystalline lattice, for example, but allowing atoms to exist only at the "proper" lattice sites and let them explore the possibility of a switch in a stochastic fashion. Another approach is to allow the atoms to move continuously under the influence of forces from all the other atoms. Such an approach is called molecular dynamics and provides a way to look for equilibrium configurations, to study diffusion phenomena, and to study transient temporal behavior. Many books have nice introductions. See, for example, Harrison for a brief section, Gould and Tobochnik, Giordano, or Heermann, among others.

Paramagnetic to Ferromagnetic Transition We have analyzed spin systems assuming that the spins interact only with an external applied magnetic field. Such systems are paramagnetic. In some systems, however, the interaction energy of neighboring spins is large and tends either to align neighboring spins (ferromagnetism) or anti-align them (antiferromagnetism). As the temperature is lowered, these materials undergo a transition from paramagnetic to (anti)ferromagnetic behavior.
Simulate this behavior in a two- or three-dimensional spin system using the Ising model for the interaction energy. What is the transition temperature below which long-range ordering of spins is observed? How does the magnetic moment in the ferromagnetic case depend on temperature near this critical temperature?

Diffusion in Disordered Media Random walks on a periodic lattice can be used to model diffusive properties in crystalline solids, but different behavior may arise when the medium is disordered. Many materials of interest are indeed amorphous and disordered, and they may exhibit diffusion behavior very different from similar materials that are crystalline. Diffusion of electrons is simply related to the electrical conductivity of the medium by the Einstein relation (p. 406 of Kittel and Kroemer), so if you can calculate the diffusion constant, you can also get the conductivity.

Diffusion Limited Aggregation Snow flakes, lightning, cracks along geological faults, and many other phenomena develop through the random addition of subunits. A simple model, called Diffusion Limited Aggregation (DLA), begins with a seed particle at the origin. A second particle is added at some distance from the seed, and allowed to conduct a random walk (on a square or triangular lattice, for example) until it bumps into the seed, at which point it sticks. Additional particles are introduced one at a time and a aggregation is built whose properties you can investigate both visually and analytically.

Tomography Computer-assisted tomography is one of several modern imaging techniques that have revolutioninzed the practice of internal medicine. CAT x-ray scans work by sending a collimated x-ray beam through a "target" slice at several different angles in the plane of the slice. From the dependence of the transmitted intensity on position across the beam and on incident angle, Fourier transform analysis can produce an image of the two-dimensional slice. A brief introduction is given at the end of Chapter 6 in DeVries.

Optics The analysis of all but the simplest optical systems is done by computer simulations, including ray tracing. Diffraction calculations can be done analytically only for very simple geometries, but can be accomplished numerically for much more general situations. The dynamics of excited-state populations in lasers, both cw and pulsed, can also be investigated numerically.

Vapor Deposition on Crystal Surfaces Chapter 4 of An Introduction to Computer Applications in Applied Science, by Farid F. Abraham and William A. Tiller, gives a readable description of a Monte Carlo simulation of the process of depositing a thin film on a crystalline surface.

Many-Electron Atoms J. S. Boleman discusses the solution of the radial Schrödinger equation using a self-consistent potential for potassium. (Am. J. Phys., 40, 1511) The self-consistent field method uses an approximate wave function to calculate a new potential which is then used to calculate a better wave function, using an iterative approach. The results of this calculation are in good agreement with the experimental electron energy levels. Griffin and McGhie (Am. J. Phys. 41 1149) give a similar treatment, using the Thomas-Fermi approximation for the potential. You might try to implement the Hartree-Fock method.

Band Structure Calculations This is a challenging topic for someone studying solid state physics, but very interesting and important in understanding the behavior of crystals (metals, semiconductors, and insulators). Possible approaches include the orthogonalized plane wave method (OPW), discussed in a chapter by Herman et al. in Methods in Computational Physics, Vol. 8. A tight-binding method (Linear Combination of Atomic Orbitals) is another approach.

Percolation Imagine throwing blotches of paint at a square canvas. For definiteness, imagine that the blotches are of roughly uniform size and much smaller than the side of the canvas. Assuming that you have terrible aim, about how many blotches must hit the canvas before a connected path runs from one side to the other?
This is an example of a percolation problem. There are many others that are more physically interesting. For example, how does the conductivity of a mixture of conducting and insulating objects depend on the composition of the mixture? How do nonmagnetic impurities affect the magnetic properties of a material? How do certain kinds of diseases spread through a population?

The Peddler (or Traveling Salesman) Problem In this basic operations research problem, a peddler aims to visit N cities following a route such that no city is visited twice and the total distance traveled is a minimum. All known exact solutions require computation that increases exponentially with the number N. In practice, therefore, when the number of cities to visit is large, one must use a (much) more efficient strategy to find a route that is near the optimal one. Such a technique is called simulated annealing, because of its analogy to the physical process of removing defects from a crystal by raising the temperature to near the melting temperature. Prof. Lyzenga uses this technique to look for a multidimensional parameter fit to a model describing the flow of crustal zones between earthquakes.