## Solution A Percolation Theory Approach

All things flow; nothing abides.

### Heraclitus

The colonization models described previously address the Fermi paradox in terms of the time it might take one or more ETCs to spread throughout the Galaxy. The most recent colonization model, proposed by Geoffrey Landis, presents a more interesting solution to the paradox. Landis bases his model on three key assumptions.94

First, he assumes that interstellar travel is possible but difficult. No dilithium crystals; no warp engines; no USS Enterprise boldly going; just a long, slow haul to the closest stars. As we have seen, this is a reasonable assumption: to the best of our certain knowledge, the laws of physics do not forbid interstellar travel, but they make it time-consuming and costly. Landis thus argues that there is a maximum distance over which an ETC can establish a colony directly. Mankind, for example, may one day establish a colony directly around Tau Ceti (just under 12 light years distant from Earth) but find it impossible to directly colonize any of the stars in the Hyades cluster (150 light years distant from Earth). Any given ETC will find there is only a small number of stars both suitable for colonization and within the maximum travel distance from its home planet. Therefore any given ETC will establish only a small number of direct colonies. More distant outposts can be settled only as secondary colonies.

Second, since interstellar travel is so difficult, Landis assumes a parent civilization will possess only weak (and possibly non-existent) control of its colonies. If the timescale over which a colony develops its own colonization capability is long, then every colony will possess its own culture — a culture independent of the colonizing civilization.

Third, he assumes a civilization will be unable to establish a colony on an already colonized world. (This is tantamount to saying that invasion is unlikely over interstellar distances, which seems reasonable. If interstellar travel is both difficult and costly, then invasion must be even more difficult and more costly. There goes the plot of several Hollywood blockbusters.)

Finally, he proposes a rule. A culture either has a drive to colonization or it does not. An ETC possessing such a drive will definitely establish colonies around all suitable stars within reach. An ETC having no uncol-onized stars within reach will, of necessity, develop a culture lacking the colonization drive. Therefore any given colony will have some probability p of developing into a colonizing civilization, and a probability 1 — p of developing into a non-colonizing civilization.95

These three assumptions, plus the rule, generate a percolation problem. The key task in a percolation problem is to calculate, for a specific system, the probability that there is a continuous path from one end of the system to the other. The word "percolation" comes from the Latin phrase meaning "to flow through," and those who developed percolation theory perhaps had in mind coffee percolation when they named it: to make a drink, water must find a path through the ground coffee and into the pot. Coffee-making is a particular example of the general problem of the diffusion of liquid through a porous solid; but percolation models have also been used to study phenomena as diverse as the propagation of forest fires, the spread of contagious disease in a population, the formation of stars in spiral galaxies, and the behavior of quarks in nuclear matter.96

In essence, percolation is merely a way of filling a large array of empty spaces with objects. (Strictly, percolation theory is valid only for arrays that are infinitely large, so the systems of interest must be large for percolation theory to apply.) The array need not be rectangular, nor need it be two-dimensional: some phenomena are best modeled with a one-dimensional array, others with a three-dimensional array, and still others with higher-dimensional arrays. To fix ideas, though, it is easiest to imagine a large two-dimensional array of N cells, rather like an extended chessboard.

figure 26 The cells in each of these four arrays have been shaded (occupied) at random. In (a), each cell has a 30% chance of being occupied. In (d), each cell has a 60% chance of being occupied. Even in (a) there are "clusters" — cases where two or more nearest-neighbor cells are occupied. (The nearest neighbor of a cell is one that is directly above, below, left or right of the cell.) In (d) there is a "spanning cluster": a path through nearest neighbors from one end of the array to the other.

### Percolation Theory

Suppose each cell of an array has a probability p of being populated. Each cell is independent of the others — just because a particular cell happens to be populated does not mean that its neighboring cells are more or less likely to be populated. Clearly, p x N of the cells will be populated and (1 — p) x N will be empty. If the probability p is large, then the array will contain lots of filled cells; if p is small, then the array will be sparsely populated. Figure 26 shows four computer-generated 8 x 8 arrays. In (a) the probability of occupancy for a cell is 30%; in (b) it is 40%; in (c) it is 50% and in (d) it is 60%. (Physicists deal with much larger simulations than this, of course, but an 8 x 8 grid is fine for the purposes of illustration.) Two occupied cells that are next to each other are called neighbors, and groups of neighbors are called clusters. For the two-dimensional array shown in the illustration, each cell, except those on the edges, can have four neighbors: the cells directly above and below, and to the left and right. Percolation theory deals mainly with how these neighbors and clusters interact with each other, and how their density affects the particular phenomenon being studied. A cluster that spans the length or width (or both) of an array is particularly important in percolation theory. It is called the spanning cluster, or percolation cluster. For an infinite lattice, a spanning cluster occurs only when the probability p is above a critical value pc.97

What has this to do with the Fermi paradox? Well, if Landis is right, we can use the well-honed techniques of percolation theory to simulate the flow of ETCs through the Galaxy. Although percolation problems are difficult to study analytically, they can be easily simulated on computer.

Readers with some programming expertise can set up the Landis model and study for themselves the distribution of ETCs under different model parameters. Figure 27 shows a typical result.

figure 27 A slice from a typical percolation simulation on a simple cubic lattice in three dimensions. For this array pc = 0.311, while the simulation is for p = 0.333. The black circles denote "colonizing" sites, and the gray circles denote "non-colonizing" sites. The absence of circles denotes sites that have not been visited. Note the irregular shape of the boundary and the large voids. Does Earth perhaps lie in one of the voids?

figure 27 A slice from a typical percolation simulation on a simple cubic lattice in three dimensions. For this array pc = 0.311, while the simulation is for p = 0.333. The black circles denote "colonizing" sites, and the gray circles denote "non-colonizing" sites. The absence of circles denotes sites that have not been visited. Note the irregular shape of the boundary and the large voids. Does Earth perhaps lie in one of the voids?

As in any percolation problem, the final lattice depends upon the relative values of p and pc. In the Landis model, if p < pc, then colonization will always end after a finite number of colonies. Growth will occur in clusters, and the boundary of each cluster will consist of non-colonizing civilizations. If p = pc, then the clusters will show a fractal structure, with both empty and filled volumes of space existing at all scales. If p > pc, then clusters of colonization will grow indefinitely, but small voids will exist — volumes of space that are bounded by non-colonizing civilizations. We produce a Swiss-cheese model of colonization: civilizations span the Galaxy, but there are holes.

The percolation approach thus suggests that colonizing extraterrestrials have not reached Earth for one of three reasons. First, p < pc, and any colonization that has taken place stopped before it reached us. Second, p = pc, and Earth happens to be in one of the large uncolonized volumes of space that inevitably occur. Third, p > pc, and Earth is in one of the many small unoccupied voids. Which explanation is most probable? To answer this we need to know the value of the colonizing probability p and also the typical number of stars available for colonization. Of course, we have absolutely no idea of what a reasonable value for p might be; Landis takes p = 3, which is as good as any other estimate. As for colonization sites, Landis argues that suitable candidates exist only around stars sufficiently similar to the Sun (in other words, single main-sequence stars within a restricted spectral range). Within a distance of 30 light years of Earth there are only five candidate stars, so a reasonable guess for this number is 5. These values produce a model that is close to critical: there are large colonized volumes of space and equally large empty volumes of space. According to the Lan-dis model, then, we have not been visited by the many ETCs that exist in the Galaxy because we inhabit one of the voids.

The percolation approach addresses the Fermi paradox in an attractive way. Rather than attributing a uniformity of motive or circumstance to ETCs, it assumes civilizations will have a variety of drives, abilities and situations. The resolution of the paradox arises naturally as one possible consequence of the model. Of course, it is possible to argue about the details of the model; Landis himself does so in his paper. For example, the model ignores the peculiar motion of stars. Stars are not fixed, like the squares on a chessboard, but instead move relative to each other. Although the relative movement of stars is slow, it might affect the percolation model. It is also possible to suggest ways to improve the analysis. For example, we could develop more complex models, taking into account Galactic boundaries, habitable zones and the actual distribution of stars. One can also challenge the basic assumptions of the percolation approach. For example, is it realistic to assume the existence of a distance horizon, beyond which no civilization will ever colonize? After all, if a civilization can travel 50 light years, would a trip of 100 light years really be so much more difficult? And what of the assumption that only a few suitable stars will lie within the horizon? A suitably advanced civilization may well find it possible — indeed preferable — to construct habitats around a variety of stellar types. Furthermore, the Landis model assumes colonization will take place directly by members of an ETC. We will see in the next section that colonization might instead take place by probe — a process that is decidedly not described by a percolation model. If just one civilization successfully deployed probes to colonize the Galaxy, then the percolation model of Landis would fail.

Finally, even if this approach explains why we have not been visited, can it explain why we have not heard from an ETC? This question is par ticularly significant if one of the p > pc cases is true, and we inhabit a void surrounded on all sides by advanced civilizations. Even if daughter civilizations become independent of their parents, surely they would want to communicate with each other? Keeping in contact using radio or optical channels would be trivial compared to the problem of physically traveling between stars. It is hard to believe all these civilizations would travel, and then adopt and maintain a policy of silence. So why have we not overheard just one of these conversations? Why have we not seen a single "we are here" beacon? (In the Landis model, ETCs should have nothing to fear in revealing their position: one of the inputs to the model is that colonization of an inhabited system is so difficult it never takes place.) Why have we not seen one example of a massive engineering project, of the kind an advanced ETC might undertake? The answer to all these questions, of course, may simply be that we have not looked hard enough nor listened long enough. Nevertheless, although a percolation model provides an elegant explanation of why we have not been visited, I personally find it ultimately unconvincing.

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