Statistical Mechanics - Physics at Oregon State University

166 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.
−4
−2
m
2.0
1.6
1.2
0.8
0.4
0.0
0
−0.4
−0.8
y
−1.2
−1.6
−2.0
Figure 8.1: β ∗ = 1
2 , h∗ = 2
and right hand sides of this equation, it is easy to see that one either has one
or three solutions. In figure 8.2 we show the results for β ∗ = 2 and h ∗ = 0.1,
and now we see three solutions.
−4
2.0
1.6
1.2
0.8
0.4
0.0
−2 0
−0.4
m
−0.8
y
−1.2
−1.6
−2.0
Figure 8.2: β ∗ = 2 , h ∗ = 0.1
When there is only one solution, we have found the function m(β ∗ , h ∗ ) and
are able to calculate the internal energy and the partition function from. In case
there are three solutions, we need to compare the magnetic Gibbs energy for
the three solutions and find the lowest energy solution. Note that when |h ∗ | > 1
there is only one solution, the model system is always magnetized. In a strong
magnetic field the spins are always aligned in a ferromagnetic manner.
Spontaneous magnetic order.
2
2
4
4