Instructor: Nigel Goldenfeld
Time: 2.00-3.20pm Mondays and Wednesdays
Place: 158 LLP
OFFICE HOUR: 4-5 Mondays ESB 3-113
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The Gradebook is available online.
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Contact and office hour information for the grader:
Weizhen Jia (email@example.com): office hour 430A Loomis, 5-6 pm on Tuesdays
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Handouts and Homework
Computationally efficient models of fluid dynamics
You can read more about the lattice gas approach to fluid dynamics and an extension of the idea known as lattice Boltzmann hydrodynamics here:
- An interactive web course on cellular automata methods to simulate everything, from fluid dynamics to chemical reactions is available here (you will need to allow your web browser to do a pop-up).
- In my own research, I use modeling in this spirit. Here is an example that received a lot of attention: using computationally efficient methods to study the evolution of landscapes at geothermal hot springs (be sure to click through to the second page after you have read the introduction). These methods are not at all inferior to conventional numerical techniques: in this paper, we show how the space-time discrete simulation method gives the same results as analytical theory based on conventional modeling using the equations of fluid dynamics, and agrees with experimental data.
- Another fantastic example of the use of such methods is to model the fluid dynamics of crowds of people. In a sense, the people are the particles in the automaton, but their collective behaviour is inevitably that of fluid dynamics. Such methods can even save human lives, by designing public spaces to prevent "crowd turbulence" and other fatal outcomes of statistical mechanics.
A good reference to applications is Lattice Gas Cellular Automata: Simple Models of Complex Hydrodynamics by D. Rothman and S. Zaleski. These same authors have a somewhat different review article entitled Lattice-gas models of phase separation: interfaces, phase transitions, and multiphase flow in Reviews of Modern Physics, 66, 1417 (1994).
Modern research on these and other methods is reported in the condensed matter and comp-gas (cellular automata and lattice gas) preprint archives at Los Alamos.
The Fermi-Pasta-Ulam-Tsingou problem is still a topic of interest, and a recent review can be found in this article. An important historical note is here, and the original report (which was never published) is here. A visualization showing the surprising result is here.
In class we talked briefly about the problems of equilibration and the notion that glass is a solid that flows on a long enough timescale. In fact, it does not seem to be true that cathedrals in Europe have windows that are thicker at the bottom than the top due to this effect. Quantitative calculations show that the timescale for the requisite flow is too long. Read more about this in Science News Online; in Discover magazine; and in the American Journal of Physics 66, 392 (1998).
Exact solutions of the two dimensional Ising model
Here is a recent article that reviews briefly analytic solutions of the two-dimensional Ising model, and in particular the mapping to a system of free fermions which is what enables the analytic solutions. The article has references in it to much earlier work, especially the seminal review article of Schultz, Mattis and Lieb, Rev. Mod. Phys. 36, 856 (1964). Some historical context and a brief discussion of the way in which the Onsager solution has influenced subsequent developments is given in this article, written on the 50th anniversary of Onsager's solution.
Statistical Mechanics on the WWW
Link to recent papers on the statistical mechanics section of the condensed matter preprint archive at Los Alamos. Note that many, many papers relevant to statistical mechanics and phase transition physics are not listed here, but appear in the condensed matter section.
There are several computer simulations of the Ising model and other simple statistical mechanics model available on the WWW. With these you can visualise equilibrium behaviour of correlations, dynamic critical behaviour and even the kinetics of the approach to equilibrium. These include the following links:
Some of the modern applications of phase transition and RG ideas are to complex dynamical systems in economics, sociology, computer networks etc. A good resource is the Santa Fe Institute and the New England Complex Systems Institute.
Office: 3113 ESB
Nigel Goldenfeld's Home Page
Department of Physics home page
University of Illinois home page