Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
200 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.<br />
approxim<strong>at</strong>ion, the answers in the Bethe approxim<strong>at</strong>ion and in mean-field theory<br />
will be different. If, on the other hand, the correl<strong>at</strong>ion length ξ is much larger<br />
than the size of the cluster, the size of the cluster is not important and the<br />
problem is essentially solved in a mean-field approach. Since ξ diverges near<br />
Tc, the Bethe approxim<strong>at</strong>ion is equivalent to a mean field theory in this limit.<br />
Therefore, the Bethe approxim<strong>at</strong>ion will give the same critical exponents as<br />
mean-field theory. In order to make a real improvement, we need to take into<br />
account correl<strong>at</strong>ions of infinite range.<br />
We can improve the analysis, however. We will use the susceptibility as<br />
an example, but the analysis woks for all critical exponents. Near the critical<br />
temper<strong>at</strong>ure we have χ ≈ A±|T −Tc| γ , plus higher order terms. This means th<strong>at</strong><br />
near the critical temper<strong>at</strong>ure we have log(χ) =≈ log(A±) + γ log |T − Tc|. The<br />
second term will domin<strong>at</strong>e if we are close enough to the critical temper<strong>at</strong>ure.<br />
Hence we have<br />
log(χ)<br />
log |T − Tc|<br />
→ γ (9.57)<br />
near the critical temper<strong>at</strong>ure. Hence if we plot the r<strong>at</strong>io on the left as a function<br />
of temper<strong>at</strong>ure, we should be able to extract the value of γ.<br />
Very simply st<strong>at</strong>ed, so wh<strong>at</strong> can be wrong. First of all, we assume th<strong>at</strong> we<br />
know the value of the critical temper<strong>at</strong>ure. Th<strong>at</strong> might be true or not. Wh<strong>at</strong><br />
happens if we are wrong? If we do not use the exact value of the critical temper<strong>at</strong>ure<br />
a plot of the r<strong>at</strong>io of the logarithms will show two peaks. The r<strong>at</strong>io<br />
will go to zero when the denomin<strong>at</strong>or goes to infinity, or when the temper<strong>at</strong>ure<br />
reaches the erroneous value of the critical temper<strong>at</strong>ure. The r<strong>at</strong>io will go to<br />
infinity when the enumer<strong>at</strong>or diverges, <strong>at</strong> the true value of the critical temper<strong>at</strong>ure.<br />
For example, see figure 9.1 where we have used χ = 1<br />
|t−1| (1 + 0.1|t − 1|)<br />
and moved Tc by a small amount to Tc = 1.01. We can use this behavior to find<br />
the critical temper<strong>at</strong>ure. If we have sufficient d<strong>at</strong>a near the phase transition we<br />
try different values of Tc and see when the peak and zero merge.<br />
But wh<strong>at</strong> is near the critical temper<strong>at</strong>ure? One could guess th<strong>at</strong> the correction<br />
in χ is a factor of the form 1 + B±|T − Tc|. This gives an extra term in<br />
the denomin<strong>at</strong>or equal to the logarithm of this expression, and near the critical<br />
temper<strong>at</strong>ure we now find<br />
log(χ)<br />
log |T − Tc|<br />
|T − Tc|<br />
→ γ + B±<br />
log |T − Tc|<br />
(9.58)<br />
The second term goes to zero, indeed. But how fast? We do not know. See figure<br />
9.2 for a typical case. We used the same form as before, but now with the correct<br />
value of the critical temper<strong>at</strong>ure. In addition, the previous argument assumes<br />
th<strong>at</strong> the behavior near the critical temper<strong>at</strong>ure is given by powers only. There<br />
are theoretical results where near the critical temper<strong>at</strong>ure we have logarithmic<br />
factors, χ ∝ |T − Tc| γ log |T − Tc|. Because the logarithm goes to infinity slower<br />
than any inverse power, such a factor does not change the value of the critical<br />
exponent, but it certainly changes the behavior near the phase transition. We