On Patch by Patch Products, Part II - (RWLA,MCT,GT,AM Part IV)

Last time I talked about a concept I invented, and based on my studies on Markov chains.  They are, essentially, "continuous matrices" (a surface on $[0,1] \times [0,1]$) with the property that they add to 1 if we take the integral with respect to $x$ for any $y$, in analogy to the requirement in the usual Markov matrix treatment.  I dubbed such "patches," and explained a way to construct them.  In my previous post, I began thinking that patches seem to be very special, in the sense that self patch powers can represent the state of a liquid in time, if we allow ourselves to be a little imaginative.  Let's say that we disturb a uniform distributed patch to an initial state, the initial state patch, like this:

$p(x,y) = 1 - cos(2 \pi x) cos(2 \pi y)$

It is easy to see that if we integrate with respect to $x$ our result is 1, so that it is indeed a patch. (I also constructed this function by letting $f_1(x) = cos(2 \pi x), g_1(y) = cos(2 \pi y), f_2(x) = 1$ and calculated that $g_2(y) = 1$ using the technique I talked about here).

Let's say we have depressed the liquid at the four corners and center of the confined space (necessarily a cube of dimensions $1 \times 1 \times h$), essentially giving it energy. Next calculate the patch powers (as described in my previous posts).  Interestingly, if we map the patch powers of such liquid, they will converge to a steady state, just like Markov matrixes would:

$p(x,y) = 1 - cos(2 \pi x) cos(2 \pi y)$

$p_2(x,y) = 1 + \frac{cos(2 \pi x) cos(2 \pi y)}{2}$

$p_3(x,y) = 1 - \frac{cos(2 \pi x) cos(2 \pi y)}{4}$

$p_4(x,y) = 1 + \frac{cos(2 \pi x) cos(2 \pi y)}{8}$

$p_5(x,y) = 1 - \frac{cos(2 \pi x) cos(2 \pi y)}{16}$

$p_6(x,y) = 1 + \frac{cos(2 \pi x) cos(2 \pi y)}{32}$

The evolution in time of this particular patch is easy to guess (although I should, technically, prove this by induction... I do in my next post):

$p_t(x,y) = 1 - \frac{cos(2 \pi x) cos(2 \pi y)}{(-2)^{t-1}}$

for $t \in \mathbb{Z^+}$, and letting this parameter represent both time and the patch power.  Of course, if we integrate with respect to $x$ any of these, the result is 1, and so, they are, indeed, patches.

I would like to state in a different post the conditions in which a steady-state is achievable; my suspicion is that, in analogy to Markov chains, steady-state happens if the patch is non-zero for some power (and above) on $[0,1] \times [0,1]$, a property that is called regularity within that context, and of course, I would like to be able to calculate the steady state as easily as it can be done with discrete Markov chains (I was afraid that, in this particular example, I wouldn't be able to achieve steady state because of the initial patch having zeroes at the corners and center).  It's pinned as one of my to-dos.  At any rate, the fact that there are patches that converge to a state (a 2D surface), and, specifically, that can converge to the uniform distribution surface, suggests that such systems, from the viewpoint of Physics, must dissipate energy and there is linked the concept of entropy.  Of course from a probabilistic point of view, entropy in this sense is non-existent; patches merely describe the probability of movement to another "position" on each $y$ fiber.

There are of course patches that do not converge to the uniform surface distribution, but to other types: in my previous post, the patch I constructed converged to a plane that is tilted in the unit cube.  I wonder if such cannot have a physical interpretation that relates it to gravity: the liquid experiences a uniform acceleration (gravity) normal to the (converging) plane, which of course says, from a physical interpretation, "the cube is tilted."  Again from the probabilistic point of view, the concept of gravity is an explanatory link to Physics, but the end-state arises without its action on the fluid at all!

There are fun topological considerations too: the fact that we can do this on the unit cube does not preclude us from doing it in, for example, a unit cylinder (a cup or mug!), provided we can find the appropriate retract-into-the-square function and vice-versa.  This I think might be very interesting to map movement in all kinds of containers.

I have already talked about a couple potential venues in Group Theory, which I really would also like to go into further at some point.

As in other posts, another possible area of investigation is the evolution of the surface in smaller bits of time. I was able to link, in previous Markov treatments, discrete representations of Markov chains to continuous time differential equations.  Here is where it would be immensely interesting to see if patches, under this light, do not converge to partial differential equation representations.  Which leads me to the last point, regarding Navier-Stokes turbulent flow (which I admit know very little about), and a potential link to its differential equation representation:

Here is why I think that turbulent flow can be explained by generalizing patches a little bit, to "megapatches" (essentially 3d-patches or tensors), since, now we can think not of a 2D surface converging, but a 3D one in time: a water sphere in space (I once saw a cool video on this and was left thoroughly fascinated) or a water balloon being poked could be understood this way, for example, so that the movement of water throughout the flexible container could be similarly traced (by mapping the probability of movement in the container)!  I need to flesh this out a little more, but I think it's also very interesting, potentially.

These studies make me ask myself, again: what is the relationship between stochastic processes and deterministic representations?  They seem to be too intimately linked to be considered separate.

1. October 15th, 2010 at 02:59 | #1

Interesting, if your patch really DOES model the surface of a liquid. How can you confirm this experimentally? If you could, and your patches did model reality, they would be very useful indeed....

Thought: Would they model a cloth surface in a similar manner? If you have a strobe light and some photographic equipment you could actually try to map your patch to the movement of the cloth (or a sheet, etc). Then you could see whether your model fits and whether your powers of the patch really DOES (or not) describe the evolution of the cloth surface in time. I would be very interested to see such results, as they seem to me highly publishable, iff your patches do indeed describe reality.

2. October 15th, 2010 at 03:07 | #2

Re: probabilistic entropy.

A concept similar to this does exist, and it is called, not surprisingly, the "entropy". The definition of it is: H(X) = E[I(X)] or in other words, the entropy of a random process, X, is the expected value of the information content of X (You can also write
H(X) = - sumof [p(xi) * log2(p (xi))], see for instance: http://en.wikipedia.org/wiki/Entropy_(information_theory) ). So entropy and "information" content of a random process is intimately linked. I would recommend you have a look at Cover and Thomas' "Elements of Information Theory" for a great discussion of this fascinating topic.

3. October 15th, 2010 at 08:57 | #3

Yes, I know about probabilistic (information) entropy: I wrote a post about its underuse in everyday life, and it arose in my studies of "surprise" or "surprisal," and money valuations. I also used it in my Beale cipher investigations. I also know there is a continuous version of the equation you posted... Also, I don't think this would work with a carpet: there have to be "molecules" underneath sliding into a particular position (with some probability), is my belief, and that is why incompressible fluids (liquids) seemed good candidates. Incidentally, I first thought of naming patches(ixes) "carpets." Thanks for the comment!