# 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,
V=x^{3}
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,
Q=e^{t/t0}
this formula contains an explict reference to the time t_{0}.
If we were interested in the decay of the isotope for a time much less
than t_{0}, 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
t_{0} 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 x^{A}
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.

[ INDEX |
states |
* power laws * |
other systems |
hand crumpling |
cylindrical crumpling |
audio |
results |
triangular grids ]

Fractal generated with
`FRACTINT`

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