Physics 117 Project

Overview

Groups of three or four students will be formed within the first month of the course. Each group will model a system that displays statistical behavior. The findings of the group will be summarized in a web-based report, which will be linked to this page.

Some Java source code and examples

Contents

Topic	Team
Percolation	Andrew Batley, Jamie Hadden, Wilson Mui
Separation of Particles of Unequal Mass and Size	Libby Schoene, Jason Brudvik, Virginia Stoll
Growth of Snowflakes	Valerie Arndt, Jonathan Erickson, Seth Foreman
Neural Networks	Matt Gay, Jeff Hartline, Ben Schmidel
Modeling the Ideal Hash	Brett Jestrow, Tyson MacDonald, John Walseth
Magnetron Toy	Jimmy Corno, Tom Driscoll, David Mann
Party Dynamics	Guillaume Mauger, Anand Patil, Drew Rollins
Paramagnetic to Ferromagnetic Phase Transition	Ryan Quadri, Ian Wiener, Nigel Wright
Traffic	Mike Rust, Martin Smith-Martinez, Paul SanGiorgio

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Schedule

Date	What's Due
October 12	Groups formed and project topics identified
October 15	Project page stating project topic and including names of participants created; URL sent to me.
Monday, November 1	Code should be planned and begun, and needed resources in place; a summary of the status of your code must be posted on the project page.
Tuesday, November 2	A draft of the page should be in place. It will almost certainly not contain data, but it should have a clear exposition of the problem and a list of references.
Monday, November 15	Demonstration of working code; hand in excerpted output or a summary of the code's capabilities.
Tuesday, November 16	All parts of the project web page except those requiring data are due: these include the relevant background for the project, theoretical analysis, description of your approach, references, etc.
Monday, December 6	Final version of page due

Guidelines

• Groups   The project should be a collaborative effort involving three or four students. Each student should contribute to each phase of the project and to the final report. Each group will be asked to describe how each member contributed to the project and to apportion the credit for the entire project among the group members.
• Topics   Please decide with whom you will work and on what topic by Monday, October 11. Identify a source of information on simulations for your system of choice (see references). Once you have identified a problem to work on, I can provide you with sources that discuss algorithms and programs.

  Prepare a web page with this information and the members of your group and e-mail me the URL (uniform resource locator, which is the address of your page). This URL will be transformed over the duration of the project into your final report.

• Content   Your investigation should not be purely descriptive, but should include some computations. A computer simulation of a dynamical system would make an excellent project. See the suggested topics on chaos and orbital dynamics below for topics conducive to simulation.

• Report   The URL for each group's project will be linked to this page so everyone can watch your project evolve.

Topics

The following list of topics is meant only as a suggestion and should not be taken as restrictive in any way.

• 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?

• 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?

• 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?

• 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. Those of you who attended Wendy Panero's talk may recall that she and Prof. Lyzenga use this technique to look for a multidimensional parameter fit to a model describing the flow of crustal zones between earthquakes.

• 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.

References

• H. Gould and J. Tobochnik, An Introduction to Computer Simulation Methods, Second Edition (Addison-Wesley, Reading, Massachusetts, 1996),

• D. W. Heermann, Computer Simulation Methods in Theoretical Physics, Second Edition (Springer-Verlag, Berlin, 1990).

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Updated 12/3/99 .