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So I was trying to motivate the fundamental postulate of statistical mechanics (i.e. all microstates are assumed to be equally probable $-$ even if we can't practically measure them, but only their macroscopic properties) using simple examples.

enter image description here

I came up with the simple analog of a dice, where the six sides $1-6$ are the random microstates and a yes/no-answer machine can only compute some rough "macroscopic" property of the result.

When I tried to incorporate a metaphor for the dynamics, which changes one microstate to another, a mathematical problem emerged:

The idea is to create a two-dimensional plane,

enter image description here $\ \ \ $ (example of a random plane)

with $n$ (e.g. six) differently colored fields and the following property: If you stand on one field of some color and you take a random step to a neightboring field you can end up on any of the other colors. So the problem is to find a colored plane, finite or not, such that every other color can be reached from every colored field. The number of colors $n$ is fixed, but the number of fields on the plane is $n$ or more (i.e. repitition of equally colored fields is allowed).

Up to four colors, I figured that a solution would just be projections of a tetrahedron. For these solution you need only exactly $n$ fields and if you want them to they are even finite or "compact" as board:

enter image description here

But from there on it gets tricky.

For which $n$ is it possible to solve the problem described above and why?

If there are some ways around it, how can the constructions be formalized?

Sadly, I guess there are some topological restrictions above $n=4$. This leads to a more difficult secondary follow up question. Let's say I come up with a plane, which doesn't fulfill the criteria of equal propability for each color, like for example in

enter image description here

where there are some red fields, which don't connect to blue ones. Then if I start somewhere and take $100$ random steps, I will collect fewer blue than yellow steps.

Is there a canonical way to classify the dependency of the descriptive statistics of the random walk on the structure and coloring of the plane?

This gets difficult very fast, but I can imagine inductive investigations on geometries with high symmetry.

I also have a little second question for the dice metaphor, which has no definitive answer and is too trivial for a seperate post:

To define a "macroscopic property" of a dice roll, what are some good/interesting ways to distinguish two out of the six numbers $\{1,2,3,4,5,6\}$? The best thing I could come up with is "these numbers are divisible by 3", but that's kind of lame.

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On the last question: odd primes, powers of $3$, powers of $4$, powers of $5$, powers of $6$, positive powers of $2$, perfect squares, ... –  joriki Apr 16 '12 at 10:01
@joriki: "odd primes" Haha. Yes, I like that one. –  NikolajK Apr 16 '12 at 10:03

1 Answer 1

For $n=7$ you can color a hexagonal tiling of the plane, see here for a picture. (This refers to the "chromatic number of the plane".) There are tilings of the hyperbolic plane with regular $n$-gons of a certain size for every $n\geq 7$. Looking at such figures might give you an idea for finding some "systematic" coloring of such a tiling.

Apart from that it might be worthwile to consider the dual version of your problem. It then becomes a problem about planar graphs.

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