Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 269
(a) (b)
Figure 5.17: (a) A typical microstate of the Ising model. (b) The same microstate in the lattice
gas picture with spin up replaced by a particle and spin down replaced by an empty cell.
(a) What is the relation of the q = 2 Potts model to the usual Ising model? In particular, what is
the relation of the interaction energy K defined in the Potts model and the interaction energy
J defined in the Ising model?
(b) ProgramPottsModelimplements the Wang-Landau algorithm for the Potts model on a square
lattice. Run the program for L = 16 and various values of q and verify that the peak in the
heat capacity occurs near the known exact value for the transition temperature given by
Tc = (ln(1+ √ q)) −1 . You will need more than 100,000 Monte Carlo steps to obtain reliable
data.
(c) Choose q = 2 and q = 3 and observe the energy distribution P(E) = Ωe −βE at T = Tc. Do
you see one peak or two?
(d) Choose q = 10 and observe the energy distribution P(E) at T = Tc. You should notice
two peaks in this distribution. Discuss why the occurrence of two peaks is appropriate for a
first-order transition.
5.9 *Lattice Gas
The Ising model is useful not only because it is the simplest model of magnetism, but also because
it can be applied to many other systems. Two common applications are to fluids and to binary
alloys. In the fluid model a spin of +1 represents a particle and a spin of −1 represents a void
(see Figure 5.17). The hard core repulsion between particles at short distances is modeled by
the restriction that there is at most one particle per site. The short-range attractive interaction
between particles is modeled by the nearest-neighbor Ising interaction.
Binary alloys are modeled in a similar way with +1 representing one type of atom and −1
representing a second type. The Ising interaction models the tendency of like atoms to be near
each other because the attraction between like atoms is stronger than that between unlike atoms
in binary alloys. In this section we will focus on the fluid model, which is called the lattice gas.