[I am not really the best person to answer this,
but since nobody else is answering here is my best shot]
given a starting patch of quasicrystal and a (non-deterministic) rule for adding atoms to it, is the Fourier transform of the full, infinite pattern well defined?
In summary it depends on the rule. [It may also be a tautology since quasi-crystals are required to have well defined discrete spectra by definition,
but I assume that is not important here]
In more detail the Fourier transform or spectrum of a pattern is only a function of the pattern, regardless of how it was produced.
The question is how do the rules constrain the possible final configurations and thus their Fourier transforms.
A non deterministic process can conceivably produce a perfectly regular pattern with a simple spectrum.
If a process is sufficiently nondeterministic it may lead to perfectly regular on some runs and wildly complex patterns on others, in which case the rules of the process would not really tell us anything about the final spectrum.
At another extreme the process may be such that the final pattern is the same regardless of the non deterministic choices made along the way which only affect the order in which the pattern is produced, in which case the spectrum is fixed by the rules of the process.
In between you get systems where the space of final patterns is large but has regularities which are reflected in the spectra.
Real process have probabilities associated with their nondeterministic choices,
which leads to a probability density function on the final patterns and on the spectra.
Probabilistic rule lead statistical regularities in the produced patterns and their spectra.
The classic examples of this are white, pink and brown noise.
These are random processes whose final pattern in the time domain is unpredictable but their spectra are well defined and have a high degree of regularity in that their amplitude-frequency relationships follow specific power laws. I do not know how the processes producing quasi-crystals produce the particular spectra they do, except that they must be biased to producing paterns with approximate symmetries that cannot be realized exactly by regular crystals, @user23660's comment and reference look like promising pointers
Some examples to illustrate the ideas described above, as requested in the comments:
For a very simple example of an eventually/asymptically deterministic system start with a square lattice that has an "atom" at the origin. The rule is to put an atom on an empty spot next to an existing atom with equal probability. In the long run you will end up with a mostly symmetric, mostly convex blob around the origins with some variations in the exact shape of the boundary between runs, but for very long runs the result will be essentially the same.
For types of stochastic patterns where all indidviduals are distinct but have large scale regularities which produce recognizable spectra consider natural patterns sand dunes, tree bark, girraffe spots, finger prints and other kinds of textures. Quasicrystals & Penrose tilings are vaguely similar to this, but that they are made of components that are perfectly regular.
For a system that can produce any kind of pattern, simple or complex, you can take the rule to put an atom anywhere. You can then put the atoms in a regular arrangement or chaotically.
This is a cheat though becauuse if you made the rule probabilistic the probaility of getting a regular pattern is virtually 0. You would almost certainly get a random pattern, but I think it would still have a well defined spectrum the same way white noise does.
Unfortunately I do not have a good simple example of a probabilistic system that is likely to produce both complex and regular patterns with significant probability. Fluid dynamics is a real systems that is somewhat like that: you get laminar flow at low Reynolds number but become increasingly turbulent as Reynolds number increases. It may also help to think about Conway's Life. It is deterministic, of course, but depending on the starting configuration it can produce almost any kind of behaviour, so the rules do not really tell you what kind of pattern, and thus spectrum, you will get in the long run and, since Life is actually Turing complete, you can not predict the eventual patterns even if you have the initial configuration, except by essentially running the rules. This is due to the undecidability the reachability problem for Turing machines.
It may help to look at (stochastic, asynchronous) cellular automata.
Stephen Wolfram's New Kind of Science has many examples of discrete systems on the order chaos boundary. Here is also a talk by Wolfram.
This post imported from StackExchange Physics at 2014-05-25 06:52 (UCT), posted by SE-user Daniel Mahler