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Next: Exercise 13.6: Qualitative behavior Up: Monte Carlo Simulation Previous: Exercise 13.5: Comparison with

The Ising phase transition

We have seen that at $T=T_c$, 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

\begin{displaymath}
m(T) \sim (T-T_c)^{\beta},
\end{displaymath}

where $\beta$ is the ``critical exponent''. Although $M$ vanishes with $T$, thermodynamic derivatives such as the heat capacity and sucseptibility diverge at $T_c$:

\begin{displaymath}
\chi \sim \vert T-T_c\vert^{-\gamma}
\end{displaymath}

and

\begin{displaymath}
C \sim \vert T-T_c\vert^{-\alpha}.
\end{displaymath}

We have assumed that the exponent is the same on both sides of the transition.

Another measure of the magnetic fluctuations is the linear dimension $\xi
(T)$ of a typical magnetic domain. We expect that this ``correlation length'' to be the order of the lattice spacing for $T \gg T_c$. Since the alignment of the spins will become more correlated as $T$ approaches $T_c$ from above, $\xi$ will increase. We can characterize the divergent behavior of $\xi
(T)$ near $T_c$ by a critical exponent $\eta$

\begin{displaymath}
\xi (T) \sim \vert T-T_c\vert^{-\eta}
\end{displaymath}

A finite system cannot observe a true phase transition. Nevertheless we expect that if the correlation length is less than the linear dimension $L$ 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 $T$ is not too close to $T_c$.



Subsections
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Next: Exercise 13.6: Qualitative behavior Up: Monte Carlo Simulation Previous: Exercise 13.5: Comparison with
Adrian E. Feiguin 2009-11-04