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Two Dimensions: Large Randomness
Two Dimensions: Closer to Critical Randomness
This is a simulation closer to the transition. Notice the big avalanche which goes from one end of the system to the other!
The Infinite Avalanche: Three Dimensions
Near this transition, the system looks self-similar: not only are there avalanches of all sizes, but the individual avalanches have holes of all sizes. The avalanche at left started in the blue region and ended in the pink region. Notice that the surface is rugged on many scales; also, if you look carefully you can see a small tunnel through it. (Click on the picture for a larger view.)
This self-similar behavior is governed by certain universal critical exponents: universal here means that different systems (e.g., theory and experiment) will have the same exponents. For example, the probability of having an avalanche of size s at the critical point varies as s to the power tau. We've been running lots of systems on the Cornell Theory Center supercomputer, in order to extract these critical exponents in 2, 3, 4, and 5 dimensions. Karin Dahmen has been using the renormalization group to predict these exponents as a function of dimension: these theoretical methods converge best near the ``upper critical dimension'', which for our problem is six. After an amazing amount of hard work, we've found great agreement between theory and experiment:
- Download and Run the Program
- What is Hysteresis?
- Noise in Hysteresis
- Return Point Memory
- The Critical Point
- The Epsilon Expansion
- Our Papers
- The Texas group's experiments.