nd real avalanches, the noise pulses in hysteresis span a huge range: one domain, thousands, or even millions. We think it has to do with a change in the shape of the hysteresis loop. If the system is very clean, the first domain to be pushed over by the external field can push over its neighbors, leading to an infinite avalanche which turns most of the domains. If the system is very dirty, the domains will probably flip only a few at a time: the avalanches get stopped by the dirt. Our model has a transition where the infinite avalanche disappears:
This shows how the shape of the hysteresis loop changes as the randomness is changed from large to small. Notice the big jump at small randomness. Selecting the picture will show an animation.
The broad range of avalanche sizes now makes sense: near the place where an infinite avalanche is about to occur, there will be lots of rather large avalanches! Here is an animation (thanks to Chris Pelkie, Cornell Visualization Service.) of the avalanches in a 3D model magnet, near the disorder and field where the infinite avalanche first appears.
We've written an applet that allows you to see the hysteresis loop formed along with the avalanches, and allows you to change the disorder away from the critical point. On the left, the horizontal axis is the external field; the vertical axis is the magnetization. On the right, you see the avalanches as they occur during the simulation.
Three Dimensions: Cascade
Watch the sizes of the avalanches during one cycle: they get large in the middle of the run (near the incipient infinite avalanche). Try lowering the disorder to two and "Generate Loop". Is there a big jump in the hysteresis loop? Is it associated with a single huge avalanche? Try raising the disorder to three, and generate a loop. Do the avalanches all stay small? Is there a big jump in the hysteresis loop? Try clicking inside the hysteresis loop on the left to raise and lower the external field (horizontal axis); you should be able to generate subloops inside the hysteresis loop.
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:
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- What is Hysteresis?
- Noise in Hysteresis
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- The Critical Point
- The Epsilon Expansion
- Our Papers
- The Texas group's experiments.