Behavior of a System at a Critical Point

The conquest of Bastille

The conquest of Bastille
[Watercolor from the collections of the National Library of France. Public domain.]

Revolutionary ambiances are characterized by a great sensitivity of the crowd to suggestions and the capacity of common action of people who individually can influence only the few neighboring people. The properties of a critical system are quite similar. Whereas microscopic interactions are local, a system exhibits long-range macroscopic order at criticality. Criticality also manifests itself by a giant response to small external fields.

This site presents student projects related to a series of lectures on the physics of critical points. Each project’s goal was to illustrate by simulation the behavior of physical quantities near a critical point.

    On this site, basic properties illustrated include:

  • domain size distributions;
  • order parameters, magnetic susceptibilities, and specific heats;
  • spin-spin and energy-energy correlations;
  • correlation lengths;
  • correlation times for the Metropolis algorithm;
  • dynamic exponent z for Metropolis dynamics.
    The project was extended by individual team projects to include:

  • Electrical resistance of percolation clusters,
  • Yang-Lee approach to phase transitions, theoretical prediction and numerical verification, and
  • Transparency window for the 2D Ising film on fixed domain substrate.

For the 1D Ising model, simulations were checked against an exact solution. For the 2D Ising model, the known critical behavior of the Onsager solution and exact numerical solutions for small lattices were used to compare theoretical predictions with results of simulations. Both Metropolis and Wolff algorithms were used and compared. In addition, the bootstrap method of error calculation was implemented and tested.

The students wrote simulation algorithms in C++ code. The code and related documentation is accessible in a Google Code repository. A Guide to the Projects describes their structure and gives detailed information about how to run programs of the various student projects. Persons wishing to contribute may add new elements to a project, e.g., to illustrate properties helpful for a deeper understanding of critical behavior. Initially, the student projects were based on the 1D and 2D Ising models, but contributors may add projects for other models. The contributions should be send to: