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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222 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.<br />
if the series converges.<br />
At this point we have established a general procedure rel<strong>at</strong>ing the partition<br />
function of a system to a partition function containing the combined effects of<br />
the spins in a block of two. The coupling constants ˜ J and ˜ h changed values,<br />
however. They had to be renormalized in order for the expressions to be valid.<br />
Hence the new system is different from the old one, since it is not described<br />
by the same Hamiltonian. It is possible th<strong>at</strong> in certain cases both ˜ J ′ = ˜ J and<br />
˜h ′ = ˜ h. In th<strong>at</strong> case the old and new system do represent the same physics, no<br />
m<strong>at</strong>ter how many times we apply the demagnific<strong>at</strong>ion oper<strong>at</strong>ion. Critical points<br />
therefore correspond to fixed points of the renormaliz<strong>at</strong>ion formulas. One has<br />
to keep in mind th<strong>at</strong> a fixed point does not necessarily correspond to a critical<br />
point; there are more fixed points than critical points.<br />
The one-dimensional Ising model depends only on two coupling constants.<br />
These represent the interaction energy between the spins and the energy of a<br />
spin in an external field. Both constants are scaled with respect to kBT . It is in<br />
general always possible to scale the constants with respect to the temper<strong>at</strong>ure,<br />
since the partition function always combines β and H in a product. In a general<br />
model, one has a number of coupling constants, and a search for critical points<br />
corresponds to a search for fixed points in a many-dimensional space. The<br />
easiest example of such a search is again for the one-dimensional Ising model,<br />
this time without an external field. Hence ˜ h = 0 and 9.161 shows th<strong>at</strong> ˜ h ′ = 0<br />
too. In every step of the renormaliz<strong>at</strong>ion procedure the coupling constant for<br />
the external field remains zero. The renormaliz<strong>at</strong>ion equ<strong>at</strong>ion 9.160 is now very<br />
simple:<br />
˜J ′ ( ˜ J, 0) = 1<br />
2 log cosh(2 ˜ J) (9.167)<br />
Since cosh(x) ex we see th<strong>at</strong> log cosh(2 ˜ J) log e2 ˜ J = 2J˜ and hence<br />
0 ˜ J ′ ˜ J. This also implies g( ˜ J ′ , 0) g( ˜ J, 0), and hence the series 9.166 for<br />
the free energy converges and is bounded by <br />
1 k<br />
k 2 g( J, ˜ 0) = 2g( J, ˜ 0).<br />
Suppose we start with a coupling constant ˜ J = ˜ J (0) . Each iter<strong>at</strong>ion adds<br />
a prime to the value according to the equ<strong>at</strong>ion 9.160 and after k iter<strong>at</strong>ions we<br />
have the value ˜ J = ˜ J (k) . If we keep going, we arrive <strong>at</strong> limk→∞ ˜ J (k) = ˜ J (∞) .<br />
Because of the limiting conditions we have 0 ˜ J (∞) ˜ J.<br />
Since the function g is also decreasing, we can get a lower bound on the free<br />
energy by using the infinite value, and we find<br />
2Ng( ˜ J (∞) , 0) −βG 2Ng( ˜ J (0) , 0) (9.168)<br />
It is clear th<strong>at</strong> the value of ˜ J (∞) depends on the initial value of ˜ J and hence<br />
on the temper<strong>at</strong>ure T . The possible values of ˜ J (∞) can be found by taking the<br />
limit in 9.160 on both sides. Th<strong>at</strong> leads to<br />
˜J (∞) = 1<br />
2 log<br />
<br />
cosh(2 ˜ J (∞) <br />
)<br />
(9.169)