I was watching The Planets on the telly with Tim and it set me to thinking.

Describing the early solar system

We're starting with a cloud of gas and dust. Particles of dust can clump together to form pebbles and so on up to planets (if the dynamics go that way) and may break one another in collisions. Gas can condense to liquid - and will typically do so on a seed such as a particle of dust. Collisions between wet dust (and, indeed, droplets) are apt to stick the pieces together. Entropy ensures that collisions generate heat: taken with the dynamics of the gas, this gives fairly complex thermodynamics. Add to all this the gravitational field of all this matter, taking into account any clumping in which the matter engages, and you've got your work cut out.

Still, gravity should properly be described by general relativity, which means discussion of the process as happening on a smooth manifold whose geometry encodes the gravitational field. A smooth manifold should be described in terms of charts - `small patches' of space-time, stitched together by equivalence on their overlaps. So, in setting out to describe the evolution of the solar system, describe events using charts.

We have plenty of scope for choice in how we chose charts: for any `suitably small' region of space-time, we can chose a chart which describes what's going on in that region in some fairly straight-forward way. I want to look at building up an atlas of space-time out of such charts.

Each chart gets to be described as a finite state machine (we may as well imagine I'm building a simulator here) which can evolve with time. We can compute its internal dynamics exactly, or use some heuristic description in terms of a suitable statistical model and a few parameters (such as the amounts of dust of various sizes and chemical compositions, the vapour pressures of the gasses present and the momentum flux densities of dust and gas) expressed locally to the chart. (We can adapt our computational strategy in response to the dynamics within the chart.) In the overlaps of charts, we will need computational strategies for the charts to `tell one another' their respective contributions to what's going on. One model would have charts disjoint, abutting one another at boundaries: each neighbour makes its contribution to the rates of change of properties of the boundary, while responding to the consequent time-evolution of the boundary.

Given a history (of which we'll mainly remember the recent past) for each chart up to some moment, we can vary the chart's spatial extent (even as measured in its own terms) in response to dynamics (this must happen when charts abutting one another correspond to locally inertial frames moving with respect to one another). We can take two (or several) neighbouring charts and group them together - which will involve evaluating the equivalences between the charts on their overlaps (e.g. boundaries) so as to express the old descriptions in new terms. We can equally break up a chart in a similar way (e.g. when one region in an almost-homogeneous region begins exhibiting enough local abnormality that a change of description will make life simpler).

We can, indeed, analyse the history up to some given moment, construct a chart overlapping the ones previously used and begin extrapolating that chart's internal dynamics forwards: it then engages in the usual interaction with charts it overlaps. If it's dynamics are easier to compute than those it overlaps, it may be worth reducing the ranges of these - pulling back from its turf. This may, in turn, leave us free to simplify our descriptions of the old regions.

In short, the entire system is going to be an evolving population of interacting finite state machines, each computing the dynamics of a region of space-time. Each knows its neighbours and converses with them to discover what's coming at it from the outside of its boundary and to let them know what effect it's having in response.

Populate a suitably uniformish patch of space with stuff at a suitably early time, using a non-spinning chart with respect to which the stuff has no total momentum but does have some angular momentum. Break up that chart into little bubbles moving (locally) with the flow, giving reasonable thermodynamic variation in properties of the bubbles. Time-evolve the resulting mess. It'll probably be worth keeping track of how charts relate to the original global chart, using this to build up a description of what we see overall (but not trying to use it as a chart in which to solve more of the dynamics than controlling drift in the bulk properties). The global chart might, for instance, serve as the agreed standard with respect to which each of the other charts communicates information to its neighbours.

Describing the modern solar system

Much the same setup should work fine - just there'll be fewer charts in use. After all, we expect this to be the long-term solution to the equations ...

Written by Eddy.
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