# 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 `test_perc` returns `False`.

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 `n` and `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.

When `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 `n`.

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.

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