Hysteresis and Avalanches

Physicists in the US usually hear about hysteresis first in their sophomore or junior year. The magnetization of the tape lags behind the field from the tape head - so when the field drops to zero, the tape stays magnetized, storing the music.

[hysteresis loop]

In your thermodynamics course, you derive that the area inside the hysteresis loop equals the work done in the cycle. You also may have heard that supercooled water was another example of hysteresis: hysteresis is characteristic of first-order phase transitions.

You likely won't hear about hysteresis again in your courses. It was an unpopular subject for decades. Experimentalists generally tried to get rid of it, so they could get publishable equilibrium data. Theorists cringed from thinking about non-equilibrium, dirty materials with long-range elastic or magnetic forces. But styles change: dirt and non-equilibrium are now a major focus of research in physics.

What's gotten us excited is the noise found in hysteresis loops. Even though they look smooth, hysteresis loops often consist of many small jumps. These jumps can be thought of as the jerky motion of a domain boundary, or as an avalanche of many local spins or domains.

[Irregular Staircase Curve]

Tiny jumps produce popping sounds during the hysteresis loop!

[Picture of Coil, Magnet, and Speaker] Feynman has a nice discussion of this in his ``Lectures on Physics'' (section II.37-3):

``It is not hard to show that the magnetization process in the middle part of the magnetization curve is jerky - that the domain walls jerk and snap as they shift. All you need is a coil of wire [...]

As you move the magnet nearer to the iron you will hear a whole rush of clicks that sound something like the noise of sand grains falling over each other as a can of sand is tilted. The domain walls are jumping, snapping, and jiggling as the field is increased. This phenomenon is called the Barkhausen effect.'' Jeff Urbach has a web page describing the Texas group's experiments.

[Magnification of region where big jump develops]

We've been trying to understand why the avalanches come in such a wide range of sizes. Like earthquakes and 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


Details, Docs, Source.

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


Details, Docs, Source.

Things are a little less confusing in two dimensions. This is a simulation somewhat above the critical point, where none of the avalanches are very large.

Two Dimensions: Closer to Critical Randomness

[Big yellow blotch in islands]

Details, Docs, Source.

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

[Tumbling Avalanche]

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.)

[Critical Exponents]

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:

More information:

This research was paid for by THE US GOVERNMENT through the Department of Energy (DOE #DE-FG02-88-ER45364), the National Science Foundation (NSF #DMR-9419506) and through the Cornell Theory Center.

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Last modified: May 27, 1996

James P. Sethna, sethna@lassp.cornell.edu

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).