7.4: Phase change

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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 that do.

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.