We have seen that at , the magnetization vanishes. This happens
continuously with increasing temperature and hence, it is a ``2nd order''
phase transition. In 1st order phase transition the magnetization,
vanishes abruptly. A way to characterize phase transition is though
studying the ``critical behavior'' of the system. First, we have to define
a quantity called ``order parameter'' which vanishes above the critical
temperature, and is finite below it. We clearly see that the magnetization
satisfies this criterion, and is a suitable candidate. The critical
behavior of the system is determined by the functional form of the order
parameter near the phase transition. In this region, physical
quantities show a power law behavior
Another measure of the magnetic fluctuations is the linear dimension of a typical magnetic domain. We expect that this ``correlation
length'' to be the order of the lattice spacing for . Since the
alignment of the spins will become more correlated as approaches
from above, will increase. We can characterize the divergent
behavior of near by a critical exponent
A finite system cannot observe a true phase transition. Nevertheless we expect that if the correlation length is less than the linear dimension of the system, then a finite system will be an accurate representation of the infinite system. In other words, our simulations would yield accurate results comparable to an infinite system is is not too close to .