Instructor: Nigel Goldenfeld

Time: 2.00-3.20pm Mondays and Wednesdays

Place:
158 LLP

OFFICE HOUR: 4-5 Mondays ESB 3-113

**Please sign-up for the email list of the class, to receive homework assignments, last
minute announcements, class cancellations etc. You will not be able to receive the homework assignments without doing this. IMPORTANT: use only your @illinois.edu email address. Everyone attending the class needs to do this, even if you are auditing or not taking the class for credit. [Please note: Doing this does not change your registration status for the class. The email address is just used by the teacher to contact all the students in the class, including those auditing.]**

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. A list of typos or mistakes is here. Please update this open document if you find more errors.

**Contact and office hour information for the grader:
Weizhen Jia (weizhen2@illinois.edu): office hour 430A Loomis, 5-6 pm on Tuesdays**

If you wish to make an appointment to see the graders outside their office hour, please make appointment by email only, not phone. Please note that they will not respond to any email that is not an @illinois.edu email address, so do not use gmail etc for course matters.

**Handouts and
Homework**

**Further reading**

**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.

**Fermi-Pasta-Ulam-Tsingou**

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.

**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).

**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:

- An Ising simulation which displays on your web browser using Java.
- A variety of other computer simulations from the Center for Polymer Studies at Boston University.
- Growth of fractal objects -- Diffusion limited aggregation (DLA). A nice Java simulation.
- Gas dynamics and the approach to the Maxwell-Boltzmann distribution.
- Physics of traffic flow.
- Maxwell's Demon game
- Molecular dynamics from Stark Design, an educational software company. MS Windows can download a nice molecular dynamics simulation program. See also the article in Scientific American.

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*

Telephone: (217)-333-8027

Office: 3113 ESB

Nigel Goldenfeld's Home Page

Department of Physics home page

University of Illinois home page

Updated January 2020