Physics 170 Projects

Mathematica has a presentation mode, which you may find handy for your presentation. Go to the file menu and select New Slide Show. You can copy into your show the relevant material from your more detailed notebook. I’m just exploring it for the first time, but it seems to have the flexibility you expect for a presentation mode. I have the start of an example here. Be careful about demonstrating your code in realtime; most interesting results take time to generate and your presentation should have those results already computed.

You can include necessary initialization and other code in cells that are hidden, marking them as initialization so they will be run without our having to watch that process. Using the Cell Properties menu, you can uncheck the Open option, which effectively hides the code, but make sure to have it marked Initialization or you will need to unhide it to execute it. To run the presentation, you can use the Palette menu to reveal the Slide Show palette. Then use Evaluate Initialization Cells from the Evaluate menu before starting the slide show.

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. An experiment by Troy Shinbrot, Fernando Muzzio, and Albert Alexander at Rutgers 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.)

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.

The Nobel Prize was awarded to Pierre-Gilles de Gennes in 1991 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 actual 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?

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, Gould, Tobochnik, and Christian.

One approach to simulating fluid flow is with a lattice gas model. Rather than applying “honest” physical laws to the motion of fluid particles, one attempts to capture the inter-particle interactions with a set of rules for collisions between fluid parcels that travel along the directions of a grid. There is a short discussion in Problem 15.21 of Gould, Tobochnik, and Christian, but other sources are readily available.

We have analyzed spin systems assuming that the spins interact only with an external applied magnetic field and with nearest neighbors in a linear chain (the one-dimensional Ising model). Such systems are paramagnetic. In some systems, however, the interaction energy of neighboring spins is large and tends either to align neighboring spins with one another (ferromagnetism) or anti-align them (antiferromagnetism). As the temperature is lowered, these materials undergo a phase 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 Potts model is a generalization of the Ising model of ferromagnetism. In the vector version of the Potts model, the spin direction of each atom takes on one of \(q\) possible values defined by \(\theta_n = 2 \pi n/q\). Nearest neighbors then interact via the Hamiltonian \(H_c = J_c \sum_{\text{n.n.}} \cos(\theta_i - \theta_j)\). When \(q = 2\) the vector Potts model is simply the Ising model. In the limit as \(q \to \infty\) the vector Potts model becomes the XY model.

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.

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.

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 A First Course in Computational Physics by Paul L. DeVries, or in a variety of other sources.

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.

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.

J. S. Boleman discusses the solution of the radial Schroedinger 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.

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.

We have taken a quick glance at percolation by looking at square lattices that may be occupied by one of two values. Other percolation problems might be taken off-lattice (such as dropping blotches of cookie dough at random on a pan), drilling holes through a plate at random positions, etc.

Other problems may be of greater physical interest. 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?

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.

The Sandpile Model, also known as the Bak-Tang-Wiesenfeld model, is a cellular automaton that aims to model “spatially extended dynamical systems.” The model displays self-organizing criticality, meaning that it always reaches a critical state without the fine tuning of any external parameter. The behavior of the critical state is characterized by spatial and temporal power laws and \(1/f\) noise. This model has been extended to many systems beyond sandpiles, including earthquakes, the human brain, and how bills are passed in Congress.

In the one-dimensional version, each cell in a row that represents half a sandpile has height \(z_i\) and slope \(m_i\), where \(m_i = z_i - z_{i+j}\). Adding a grain at site \(i\) causes \(z_i\) and \(m_i\) to increase by 1 and \(m_{i-1}\) to decrease by 1. If \(m_i > m_{\rm c}\) for a critical value of slope \(m_{\rm c}\), cell \(i\) topples by sending one grain of sand to the next cell to the right.