Statistical Mechanics - Physics at Oregon State University

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