Hysteresis and Avalanches

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.

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.

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!

Watch the sizes of the avalanches during one cycle: they get large in the middle of the run (near the incipient infinite avalanche).

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.

This is a simulation closer to the transition. Notice the big avalanche which goes from one end of the system to the other!

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:

• Download and Run the Program
• Authors
• What is Hysteresis?
• Noise in Hysteresis
• Return Point Memory
• The Critical Point
• The Epsilon Expansion
• Our Papers
• The Texas group's experiments.

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