I thought it might be nice to have a thread to discuss the "thermodynamics" of Life. What can we say about how random life patterns behave on average?

I'll start us off. I've been thinking about how the initial density of a soup influences the final contents after it has stabilized. Let's work on tori to avoid bothersome edge effects. Experimental results show that the final population of the universe is maximised when the initial density is somewhere around 0.37. As the initial density tends towards 0 or 1 the final density tends towards 0, because the cells die from having too few or too many neighbours.

This means that for any p < 0.37 there's a q > 0.37 such that the two universes evolve to the same average final densities. What would be interesting to know is if despite having similar densities these two universes can nevertheless be distinguished by looking at the frequencies with which particular objects appear in them.

I don't know what the answer is in general, but I can say something about the case where p is very small and q is very close to 1.

The result of soups with very low density is known as Sparse Life. Lets fix a size of torus and let p tend to 0. Eventually most of our soups will be totally empty, and only a very few will contain a single object. Because p is very small, p^2 is very small compared to p, p^3 is very small compared to p^2, and so on. So almost all objects that appear will be those that have predecessors with the minimal possible number of cells. In Life all patterns with only 1 or 2 cells die out, so we'll look at patterns with 3-cell predecessors. These turn out to be just the blinker and the preblock. Each blinker has two possible predecessors (itself and its successor). Each block can form from 4 different preblocks. So the nonempty universes will contain either blocks or blinkers, with blocks being twice as likely. The average final density will be (4*4)p^3 + (2*3)p^3 = 22p^3.

One can then do the same analysis as q tends to 1 (let's call it Dense Life). In a universe of all live cells, the next generation is all dead. So again all universes will end up empty unless enough dead cells happen to appear close enough to each other in the initial configuration that some cells aren't overpopulated. A cell needs at least 5 dead neighbours to be alive in the next generation. If we want 3 live cells in the next generation then at least two of them must not be orthogonally adjacent. But cells that aren't orthogonally adjacent can share at most 3 neighbours. So at least these 3 neighbours and two other cells on either side must be dead, bringing us to seven dead cells in all. By my count the blinker has 24 7-cell predecessors, and the preblock doesn't have any. The average final density will be (24*3)(1-q)^7 = 72(1-q)^7.

So we've discovered two things: the relation between p and q is that 22p^3 = 72(1-q)^7, and the two universes can totally be distinguished! One has twice as many blocks as blinkers, and the other has no blocks at all.