AnnouncementsInstructor: Nigel Goldenfeld
Time: 2.30 -- 3.50pm Tuesday and Thursday
Place: Loomis Lab 136
Please register for the email list for the class, so that last minute announcements, class cancellations etc. can be sent to you. This applies even if you are not taking the class for credit. Thanks.
The Gradebook is available online.
The lecture notes for the class are online to people at UIUC. You may download these for your own use only. Please do not redistribute them to anyone else. Thank you.
Handouts and Homework
Further reading
Fluctuations in Temperature in Statistical Mechanics
This is a somewhat confusing and controversial subject, because the discussion relates to the basic assumptions of statistical mechanics. Here are a few interesting articles on the topic, but please read them critically.
Computationally efficient models of fluid dynamics
Here is a simulation (due to Thomas Pohl) that you can run in your web browser, showing the solution of a fluid dynamics problem. In this simulation, the fluid is contained in a box, and the lid of the box is sliding to the right, causing the fluid to move. You can visualise the flow, put obstacles in and play around with the simulation. It does not use a conventional numerical discretization of the Navier-Stokes equations, but instead uses some of the cellular automata methods described in class.You can read more about the lattice gas approach to fluid dynamics and an extension of the idea known as lattice Boltzmann hydrodynamics in this reasonably pedagogical article:
- Lattice Boltzmann methods (a thesis with lots of pedagogical material) by J.M. Buick. See especially a brief overview of numerical methods for solving fluid dynamics problems, detailed introduction to lattice-gas methods (with a discussion of its drawbacks), and a detailed introduction to the lattice Boltzmann model.
- 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 (most prominently by Dirk Helbing, but going back many decades to work of Lighthill, Prigogine and others) 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.
Fermi-Pasta-Ulam
The Fermi-Pasta-Ulam problem is still a topic of interest, and a recent review can be found in this article.
Glass
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).
Computer Simulation
An important area of statistical mechanics is using computers to evaluate thermal averages. Although we do not have time in this course to discuss this, let me point out some useful resources:
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.
Software
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.
Nigel Goldenfeld
Department of Physics home page
University of Illinois home page