Solids, Fluids, and Dynamics

from 'Life At the Edge of Chaos', Chris Langton, 1991

With cellular automata, and other spatially extended dynamical systems, we are experimenting with "artificial matter" (or "programmable matter" in Toffoli's words ). That is, we are experimenting with "material" for which we have precise control over the behaviors of the individual "atoms" or "molecules" of which it is constituted.

It is somewhat surprising that, despite the many different varieties of atoms and molecules that constitute "real" materials, almost every known substance comes in one of three flavors: solid, liquid, or gas. As it is possible to continuously transform liquids into gases and vice versa without passing through a phase transition, they are taken to constitute a single, more general phase of matter: fluids. Therefore, there are really just two fundamental phases of matter - solids and fluids - and so we should not be surprised to find two similar fundamental phases in our "artificial" materials, despite the large number of ways that we can put them together. The important point here is that solids and fluids are dynamical rather than merely material, categories.

We know solids and fluids primarily as states of matter, rather than as universal classes of dynamical behavior, because up until quite recently, everything that exhibited dynamical behavior was fundamentally material in constitution. Now, however, with the availability of computers, we are able to experiment with dy- namical behaviors per se, abstracted from any particular material substrate. What we find is that, despite having abandonded the material basis of solids and fluids, we are nonetheless left in possession of solid and fluid dynamics! Thus, we are safe in assuming that these fundamental classes of dynamical behavior do not inhere in material per se, but rather in the way in which the material is organised.

The most important point, however, is that these two universality classes of dynamical behavior are separated by a phase transition. As we have seen the dynamics of systems operating near this phase transition provides the basis of support for embedded computation. Thus, a third catagorary of dynamical behaviors exists in which materials - or more broadly, dynamical systems in general - can make use of an intrinsic computational capacity to avoid either of the two primary categories of dynamical behavior by maintaining themselves on indefinitely extended transient.

Since computers and computations are specific instances of material and dynamical systems respectively, they are also ultimately bound by these same universality classes. Therefore, computer "hardware" can behave like a solid, like a fluid, or, like something in between.

An interesting open question to pursue here is whether one can resolve the fluid phase of cellular automata (and other dynamical systems) even further into liquid and vapor phases. The careful reader may have noticed that I have consistently used the phrase "in the vicinity of a second-order phase transition" rather than the phrase " at a second-order phase transition." This is in part due to the fact that it is hard to determine whether or not one is precisely "at" a critical point when working with finite systems. But it is also due to the fact that in most of the experiments on cellular automata leading to these results, the regime exhibiting the most complex dynamics appears to be just slightly below (to the ordered side) of the critical transition itself. Most physical systems exhibit a liquid phase or the "ordered side" of their critical point, and liquids exhibit extremely complex dynamics.

It is somewhat surprising that liquids are very poorly understood. Although the structures of both solids and gases are well studied, it has proven difficult in practice to characterize the structure of liquids. As might be expected from their location between the solid and the gas phases, liquids exhibit much more complicated behaviours than either of the other two phases. Recent work by Stanley suggests that the hydrogen-bond network of liquid water exhibits very complex dynamics, and super-cooled water can apparently exhibit critical dynamics.

It may be appropriate to view liquids as constituting a broadened phase transition in the phase portrait of a material. Liquids occupy just the right spot "at the edge of chaos" in the phase portrait of many materials to be candidates for fur ther investigation along the lines of the results reported in this paper. The complex transition regime might be co-extensive with the liquid regime for many real and "artificial" materials.

Four dynamical phases of cellular automata