Consider a 50 × 50 square array of spins. Initially assign all sites the spin value of +1 and select one spin randomly. The simulation involves flipping a random lattice point and determining if the net energy decreases. If the energy decreases because of the flip, then the new state is allowed, and the simulation continues. If the new lattice configuration causes the overall energy to increase, then a random number between 0 and 1 is generated.

If the exponential of the temperature and change in energy is less than this randomly generated number, then the state is allowed, and the simulation continues. If this exponential is less than the randomly generated value, the flip is not allowed, the flipped lattice point is returned to its previous state, a different random lattice point is chosen, and the simulation continues.

Thus, we summarize the algorithm in the following step form:

a. Pick a site x on the lattice at random and flip its spin, changing its state from x to x"

.

b. Compute the trial change in energy U = U(x"

) − E(x), where U(x) = − 

k∈ℵ(x)

sxsk sk is the spin of site k and ℵ(x) is the set of neighbors of site x.

c. If U < 0, accept the flip.

d. If U > 0, generate a random number r, where r ∈ (0, 1). If r <

exp(−βU), accept the change in sign, where we assume that β = 0.5;

otherwise, reject the flip and return the spin to the previous value.

e. Repeat this process 2499 more times (for the 50 × 50 sites), choosing a new site at random. This constitutes one sweep of the lattice.

450 Markov Processes for Stochastic Modeling After several sweeps to flush out the initial spin assignments, this procedure generates spin configurations that naturally follow the real probability function.