# Statistical Mechanics Mean Field Model

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1. Feb 22, 2015

### Chris B

I don't think I've fully grasped the underlying ideas of this class, so at the moment I'm just sort of flailing for equations to plug stuff into...

1. The problem statement, all variables and given/known data

Show that in the mean field model, M is proportional to H1/3 at T=Tc and that at H=0, M is proportional to (Tc - T)1/2

2. Relevant equations

It'll take me forever to write it out here, but M equals some coefficients times tanh(x) where x depends on H, M, and T

3. The attempt at a solution

In class we approximated M at T=0 by taking the first term in the Taylor series expansion of tanh(x) which turns out to just be x. We were given the hint that we need to take the next term in the series (why?) to do this problem. The next term in the series is -x3/3 so the whole thing is
M=N/V gμbS(x-x3/3)
where x = gμbS(H+λM)/kT
I also know that Tc = λN(gμbS)2/VK
I plugged Tc in for T and by rearranging got something that was proportional to H1/3 minus H/λ
For the next part I'm not sure what to do. If I plug in zero for H the whole thing is zero. I don't see how I'm going to get (Tc - T)1/2 if I'm substituting Tc for T.

2. Feb 25, 2015

### RUber

These are two different scenarios. In part A, you assume T=Tc. In part B, the only assumption is that H=0.