Crumpling paper

Why physicists like Power laws

Power laws are relationships of the form,

y=x A
where A is some constant. Power laws are important because they don't contain any explicit references to scale. For instance, if we were to look at the formula for the volume of a cube,
we can see that there isn't any mention in the formula about how big the cube is. However, if we have an exponential relationship, say that for the quantity of a radioactive isotope over time,
this formula contains an explict reference to the time t0. If we were interested in the decay of the isotope for a time much less than t0, say, if we watched a lump of uranium for one second, we wouldn't notice the mass of uranium diminish much at all. On the other hand, if we were to wait one trillion years, a time much greater than t0 for uranium, it would be practically all gone.

Now, people have known about power laws where the exponent A is some integer like -2,1,3, or something familliar for a very long time. More recently we've found that many systems have power laws where the exponent A is something really different... where it might be a fraction or an irrational number. One of the first appearances of this was in studying the behavior of a fluid around it's critical point (where the gas and liquid phases disappear; this is a very hard experiment to do, it turns out, and recently people at NASA have been doing critical point experiments in zero gravity.

I did my own critical-point experiment using carbon tet for the Physics 510 lab, and the graph above is a log-log plot of the difference in density between the liquid and gas phases vs the temperature below the critical point. We use a log-log graph, because power laws have the conspicuous property that they plot as a straight line on a log-log graph. Suppose we were to take the log of both sides of the first equation,

log y = log xA
log y = A log x
which is a plain straight line where the A is the exponent.

Now, we had a very simple and pretty mean-field theory to understand the behavior of fluids which was very nice except for the fact that it was completely wrong about what happens at this critical point; and it proved to be impossible to make any simple change to the theory that would make it work right. Ken Wilson won a Nobel Prize for his discovery of the renormalization group which turns out to be very useful in all kinds of physics, from high energy theory to turbulence, phase transitions, and chaos. The renormalization group is really a systematic way of thinking about self-similarity...

a property of fractals, and a pretty power law shows up in the definition of fractal dimension.

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