Page 368 - Numerical Methods for Chemical Engineering

Markov chains and processes; Monte Carlo methods 357

If two sites (i, j) and (m, n) are neighbors (to the “north,” “south,” “east,” or “west”), they
interact with a coupling strength J to favor parallel alignment (either both up or both down),
with an energy contribution

−J, if spins are parallel
[ν] [ν]
E ij,mn =−JS S mn = (7.241)
ij
J, if spins are anti-parallel
In addition, each site interacts with an external magnetic field H according to the state of
its spin, which has a magnetic moment µ. The total energy of the lattice in state ν is then
N
N N J
N

E [ν] = Hµ S [ν] − S [ν] S [ν] + S [ν] + S [ν] + S [ν] (7.242)
ij ij i−1, j i+1, j i, j−1 i, j+1
i=1 j=1 2 i=1 j=1
We divide by 2 in the second sum in the energy expression to avoid overcounting each
interacting pair. To mimic the behavior of an infinite lattice, we use periodic boundary
conditions, in which for neighboring points outside the N × N simulation cell we assume

[ν] [ν] [ν] [ν] [ν] [ν] [ν] [ν]
S = S S = S S = S S = S (7.243)
−1, j N, j N+1, j 1, j i,−1 i,N i,N+1 i,1
For each state ν, we define the net magnetization and the order parameter

N
N N 1
N

m [ν] = µ S [ν] σ [ν] = S [ν] (7.244)
ij 2 ij
i=1 j=1 N i=1 j=1
If all spins are “down,” σ [ν] =−1, and if all spins are “up”, σ [ν] = 1. If σ [ν] = 0 there is
no net order of spins in the lattice. Ising lattice MC.m simulates a 2-D Ising lattice and is
called by the following code:

MCOpts.N = 50; MCOpts.mu = 1; MCOpts.H = 0;
MCOpts.J = 1; MCOpts.kb T=5;
*
MCOpts.Nequil = 50 (MCOpts.Nˆ2); MCOpts.Nsamples = 50000;
MCOpts.freq sample = MCOpts.N; MCOpts.make plots = 1;
Ising lattice MC(MCOpts);

make Ising lattice MC movie.m uses the results of Ising lattice MC.m to make a movie
showing the spin fluctuations.
An infinite 2-D Ising lattice has a critical point at the Curie temperature T c , k b T c ≈ 2.269 J,
above which there is no net order in the absence of an external field, and below which the
system has a net surplus of either “up” or “down” spins. A strong external field can induce
spin order even above T c , but higher fields are required at higher temperatures. Figure 7.14
shows two sample states from Monte Carlo simulations. Figure 7.14(a) shows the positions
of spin-up sites in a disordered state for the base case simulated by the code above, while
Figure 7.14(b) shows that imposing an external field H < 0 results in mostly spin-up sites.
For further discussion of Ising lattices, see Chandler (1987).