9.5. Phase Change¶
Now let’s test whether a random array contains a percolating cluster:
def test_perc(perc): num_wet = perc.num_wet() while True: perc.step() if perc.bottom_row_wet(): return True new_num_wet = perc.num_wet() if new_num_wet == num_wet: return False num_wet = new_num_wet
test_perc takes a
Percolation object as a parameter. Each time through the loop, it advances the CA one time step. It checks the bottom row to see if any cells are wet; if so, it returns
True, to indicate that there is a percolating cluster.
During each time step, it also computes the number of wet cells and checks whether the number increased since the last step. If not, we have reached a fixed point without finding a percolating cluster, so
To estimate the probability of a percolating cluster, we generate many random arrays and test them:
def estimate_prob_percolating(n=100, q=0.5, iters=100): t = [test_perc(Percolation(n, q)) for i in range(iters)] return np.mean(t)
estimate_prob_percolating makes 100 Percolation objects with the given values of
q and calls
test_perc to see how many of them have a percolating cluster. The return value is the fraction of those that have a percolating cluster.
p=0.55, the probability of a percolating cluster is near 0. At
p=0.60, it is about 70%, and at
p=0.65 it is near 1. This rapid transition suggests that there is a critical value of
p near 0.6.
We can estimate the critical value more precisely using a random walk. Starting from an initial value of
q, we construct a Percolation object and check whether it has a percolating cluster. If so,
q is probably too high, so we decrease it. If not,
q is probably too low, so we increase it.
Here’s the code:
def find_critical(n=100, q=0.6, iters=100): qs = [q] for i in range(iters): perc = Percolation(n, q) if test_perc(perc): q -= 0.005 else: q += 0.005 qs.append(q) return qs
The result is a list of values for
q. We can estimate the critical value,
q_crit, by computing the mean of this list. With
n=100 the mean of
qs is about
0.59; this value does not seem to depend on
The rapid change in behavior near the critical value is called a phase change by analogy with phase changes in physical systems, like the way water changes from liquid to solid at its freezing point.
A wide variety of systems display a common set of behaviors and characteristics when they are at or near a critical point. These behaviors are known collectively as critical phenomena. In the next section, we explore one of them: fractal geometry.