Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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172 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.<br />
but now restrict ourselves to density m<strong>at</strong>rices ρ = (ρ) N , where the power means<br />
a direct product and where we use the three parameters a and m as defined<br />
before. This leads to<br />
G(h, T ) min<br />
m,a Tr (ρ) N H − (ρ) N hM + kBT (ρ) N log((ρ) N ) <br />
(8.70)<br />
and this procedure gives an upper bound to the energy. Note th<strong>at</strong> this does not<br />
give any bounds on deriv<strong>at</strong>ives. But we know th<strong>at</strong> deriv<strong>at</strong>ives are monotonous,<br />
and hence we cannot have large oscill<strong>at</strong>ions in them. Therefore, deriv<strong>at</strong>ives have<br />
errors similar to the errors in the free energy!<br />
The energy U − Mh is calcul<strong>at</strong>ed from Tr (H − hM)ρ, which becomes<br />
U − Mh = <br />
· · · <br />
⎡<br />
ρ(σ1, σ1) · · · ρ(σN , σN) ⎣−J <br />
σiσj − h <br />
σ1<br />
σN<br />
<br />
i<br />
σi<br />
⎤<br />
⎦ (8.71)<br />
A given variable σj occurs only in a linear fashion in this summ<strong>at</strong>ion, and<br />
hence the sums over<br />
<br />
the spin-variables either give a factor one or m, because<br />
ρ(σ, σ) = 1 and σ σρ(σ, σ) = m. This leads to<br />
<br />
σ<br />
U − Mh = − 1<br />
2 JNqm2 − hNm (8.72)<br />
as expected. This expression is independent of the values of the complex number<br />
a.<br />
The harder task is to evalu<strong>at</strong>e the entropy, because it has the logarithm of<br />
a m<strong>at</strong>rix in the expression.<br />
The entropy can be obtained from<br />
S = −kB<br />
<br />
<br />
σ1,···,σN σ ′ 1 ,···,σ′ N<br />
< σ1, · · · , σN|ρ|σ ′ 1, · · · , σ ′ N >< σ ′ 1, · · · , σ ′ N| log ρ|σ1, · · · , σN ><br />
(8.73)<br />
So how does one calcul<strong>at</strong>e the logarithm of a m<strong>at</strong>rix? The final result is not<br />
too surprising, it looks like log(xy) = log(x) + log(y), but here we work with<br />
m<strong>at</strong>rices in stead of numbers, so we have to be a little careful. In our case we<br />
define N additional m<strong>at</strong>rices rel<strong>at</strong>ed to ρ by<br />
< σ1, · · · , σN|Rn|σ ′ 1, · · · , σ ′ N >= δσ1,σ ′ 1 δσ2,σ ′ 2 · · · ρ(σn, σn) · · · (8.74)<br />
which is diagonal except for the n-th spin variable. Hence ρ = R1R2 · · · RN<br />
where all the m<strong>at</strong>rices are of rank 2N . These m<strong>at</strong>rices commute, since the off<br />
diagonal elements occur on different blocks th<strong>at</strong> are not connected. So we have<br />
[Ri, Rj] = 0.<br />
The exponent of a m<strong>at</strong>rix is defined via a power-series as usual, eA <br />
=<br />
n 1<br />
n! An , and the logarithm is the inverse of the exponential, eA = B ⇒