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One account I have heard many times about spontaneous symmetry breaking is a pencil standing on its tip; it can fall in any direction, and thus has symmetry. When it falls, the symmetry is broken because the physical system is no longer symmetrical, though the laws describing it did not themselves break the symmetry. Can spontaneous symmetry breaking be predicted, or is this example an oversimplification?


It cannot be predicted because it is based on the small random motions that formulas and models do not include. All of science is composed of simplified models that do not fit reality perfectly but are much easier to work with than reality. It is much easier to call a baseball a single object than billions of atoms vibrating randomly within a ball-shaped pattern. Heat and turbulence are two factors that often contribute break these symmetries. Various models of heat and turbulence can predict average effects over time and position, but not specific details. Doing so would require tracking all of the individual molecules in the air and all the molecules that these air molecules crash into.

Dr. Ken Mellendorf

In general we cannot predict which way a system will go (i.e. which direction the idealized pencil will fall), but we can often predict "when" such symmetry breaking will occur. Indeed if we could predict the direction beforehand, that would imply that the system is not truly symmetric.

The case of the pencil is a bit of a simplification, but an "idealized fully symmetric pencil" serves as a useful illustration. However, a better and physically more correct example has to do with magnets. In that case, given some assumptions, we can predict at what temperature the magnetic will be symmetric and at what temperature it will break that symmetry. We cannot tell which direction the symmetry will break at, but we can tell at what temperature it will happen. I will try to describe this below though I fear my explanation may not be entirely transparent.

One standard example of predicting when the transition occurs has to do with magnets. Suppose you have physically symmetric magnetized ferromagnet (yes, you can have a demagnetized ferromagnet!) at room temperature. The magnetization comes from a net over-all preferred direction of alignment of the magnetic domains within the magnet. Not all the domains point in that direction, indeed most do not. But a majority will and that is the magnetic field we observe outside from the magnet. As you heat up the magnet, the magnetic field will decrease as the magnetic domains within it begin to decrease their magnetization. The increase in temperature allows more variation in the individual magnetic moments. At high enough temperature, there is enough thermal energy in the system to overcome the magnetic energy and the previous magnetic alignment is destroyed. At this high temperature the ferromagnet has in fact become a paramagnet with it's magnetic moments all pointed in random directions and changing direction rapidly. At this point the system is symmetric with regards to magnetization. Any direction is equal and there is no preferred alignment!

If the magnet is now cooled sufficiently, the magnetic energetics will become significant again and it will become favorable for the moments and domains to align. In the absence of any outside influence or intrinsic property, there is no preferred direction. The material "spontaneously" chooses a particular direction and the earlier symmetry is broken. This direction will also be random relative to the original direction before you heated the magnet. In fact, if you heat and cool it repeatedly, each time will produce a new direction.

Michael S. Pierce

You have a proposition that is not true. Symmetries can be, and are, to use your words broken. The same pencil standing on its eraser could, in principle, stay there in indefinitely. Consider the simple molecule CO2. When it stretches symmetrically: O<----C----> its symmetry does not change. But if it stretches asymmetrically O<---C------->O its symmetry is reduced. That this happens is observed experimentally because the asymmetric stretch absorbs infrared radiation, but the symmetric stretch does not. A similar change in geometric symmetry changes when the molecule bends -- that motion is active in the infrared spectrum.

Vince Calder

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