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Quantum physics, general relativity, *and* fractals!
http://www.newscientist.com/article/mg20127011.600-can-fractals-make-sense-of-the-quantum-world.html?full=true
--
Darren New, San Diego CA, USA (PST)
There's no CD like OCD, there's no
CD I knoooow!
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Darren New wrote:
> Quantum physics, general relativity, *and* fractals!
Hmm. Well, I can't claim relativity, but...
Take one metallic lattice. Each atom is a tiny magnet. At low
temperatures, the majority of the atoms line up, yielding a macroscopic
magnetic field. At high temperatures, most atoms do not line up,
yielding no macroscopic field. So, at low temperatures, the lattice is
ferromagnetic, while at high temperatures it is paramagnetic. But at
what temperature does it change from one to the other?
Ah, well, that depends. On a great many things. But one trick is to use
"renormalisation". That is, take your lattice, step back from it, and
"average out" the alignment of the individual atoms. At a low
temperature, you have an essentially stable lattice with small
variations. Averaging blurs out the small variations, leaving only
overall uniformity - in other words, the lattice appears to be at a
lower temperature after renormalisation. Similarly, a warm lattice
appears hotter after renormalisation.
In short, repeatedly renormalising a given lattice will make its
apparent temperature tend upwards or downwards. If you can write a
formula for this, then you can take any starting temperature, repeatedly
apply this formula to it, and eventually end up with either a very low
or a very high number, indicating that the lattice is ferromagnetic or
paramagnetic at the temperature you started with.
The formula for doing a renormalisation depends on the nature of the
lattice and the number of possible atom alignments. Some scientists were
looking at this, and found that the phase transition (the temperature at
which ferromagnetic becomes paramagnetic) was surprisingly difficult to
predict. So they decided to use complex-valued temperatures (which,
obviously, don't really exist) to try to make the maths clearer.
Oh, they made the problem clearer alright. The phase transition is a
fractal. (!) But hey, what do you expect, iterating a nonlinear function
like that?
Even weirder, sometimes the model predicts additional magnetic phases
which only occur at complex-valued temperatures.
Personally, I just use this stuff to make pretty pictures...
http://www.icd.com/tsd/fractals/beginner4.htm
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