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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216 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.<br />
which gives up to second order<br />
<br />
j(j − 1)<br />
1 + +<br />
4<br />
(N − j)(N − j − 1)<br />
−<br />
4<br />
or<br />
or<br />
<br />
N(N − 1)<br />
x<br />
4<br />
2<br />
1 + 1<br />
(j(j − 1) − jN − N(j + 1) + j(j + 1) + N) x2<br />
4<br />
1 + 1 2 2<br />
2j − 2jN x<br />
4<br />
We now have for the correl<strong>at</strong>ion length<br />
and after approxim<strong>at</strong>ing the logarithm<br />
<br />
ξ ≈ −j log 1 + 1<br />
−1 j(j − N)x2<br />
2<br />
ξ ≈<br />
2<br />
(N − j)x 2<br />
But now we are able to approxim<strong>at</strong>e the value of x by −2e −2βJ and obtain<br />
ξ ≈ e4βJ<br />
2(N − j)<br />
(9.132)<br />
(9.133)<br />
(9.134)<br />
(9.135)<br />
(9.136)<br />
(9.137)<br />
This is a different result, indeed. In the limit of zero temper<strong>at</strong>ure the hyperbolic<br />
tangent is very close to one, and all terms in the pair correl<strong>at</strong>ion function play<br />
a role. But the divergence is still exponential in stead of a power law, so th<strong>at</strong><br />
qualit<strong>at</strong>ive conclusion did not change. This is another consequence of the fact<br />
th<strong>at</strong> <strong>at</strong> zero temper<strong>at</strong>ure there is not really a phase transition.<br />
The previous results have some interesting consequences. First, suppose<br />
th<strong>at</strong> the correl<strong>at</strong>ion length is equal to the length of the chain. The whole chain<br />
will act in a correl<strong>at</strong>ed fashion. At a given time it is very likely th<strong>at</strong> all the<br />
spins point in the same direction and one would be tempted to call the chain<br />
magnetic. There will be times, however, where part of the chain is magnetized in<br />
one direction, and part in the other. Such a disturbance has a higher energy and<br />
will disappear. Nevertheless, this disturbance might switch the magnetiz<strong>at</strong>ion of<br />
the whole chain. An average over an infinite time therefore gives a non-magnetic<br />
st<strong>at</strong>e! If we make the chain longer, we have to go to a lower temper<strong>at</strong>ure for the<br />
correl<strong>at</strong>ion length to be equal to the chain-length. Fluctu<strong>at</strong>ions become more<br />
unlikely, and it will take longer for the chain to switch its magnetiz<strong>at</strong>ion. From<br />
a fundamental point of view, the chain is still non-magnetic. From a practical<br />
point of view, the chain is magnetic in a meta-stable st<strong>at</strong>e. Therefore, we could<br />
define a transition temper<strong>at</strong>ure T ∗ by ξ(T ∗ ) = N. But phase transitions are<br />
only defined in the thermodynamic limit, and we have limN→∞ T ∗ = 0. But it<br />
does tell us when finite size effects play a role. For a given N we can find T ∗