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[Phys-L] Re: Critical Phenomena: Feynman Vol. 2: Fig. 36-15



Steven Ratliff wrote:

Would you please help me with
a question. I looked at Feynman's figure 36-15 ("spontaneous
magnetization as a function of temperature for Nickel"). However, I did
not see a systematic variation. The first and seventh data points are on
the theoretical curve. The second through sixth data points are below the
curve, and the eighth through eleventh data points are above the curve.
Please explain.

Well, I agree it's not super-clear from Feynman's figure. It's easy
for me to see, because I've looked at a lot of other data of this
kind *and* I know the modern theory, using renormalization group
(RG) ideas ... so I know exactly what to look for.

The thing that jumps out at me concerns the four points closest to
Tc. They fall systematically above the curve. The premodern theory
(which assumes there is a Taylor series around T=Tc) makes a quite
firm prediction about the shape of the curve, namely that M should
go like sqrt(t), where t is the distance from Tc, in the appropriate
scaled units, i.e. t := (Tc-T)/Tc. It had been known for decades that
the data did not follow this rule. Rather than t^.5, the data always
seemed to follow something closer to t^.3 ... which is a function that
drops _more steeply_ to M=0 as t goes to zero, i.e. as T goes to Tc.
This _steep drop_ is exactly what we see in Feynman's figure.
Note the exponent is closer to .3 than to .33333, and in any case
there is just no chance of writing t as a power series in M (let
alone vice versa).

Additional data (and additional theoretical insight) comes from studying
other observables, for instance the heat capacity, which does funny
non-Taylor-expandable things near T=Tc.

I spent a few minutes trying to find an accessible tutorial on
renormalization group -- without success. I'll get back to you if
I find anything.
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