## On Scaled Patches and Time-Warping

Suppose we have an idealized canal of width $1$, on which a fluid flow has been established in some remote past. Let us focus solely on the dynamics of the surface.  Pick a spot along the canal which we will call $t_0$.  Next pick a spot $y_0 \in [0,1]$ along the width of the canal, which we will monitor.  Pick a second spot $t_1$ down the canal, some distance from the original spot we picked.  Now let us assume that, up the canal at some remote point, a paper boat has been released.  We will only care about the boat if the boat passes through $(t_0, y_0)$ which we have picked, and we will write down the resultant position at $t_1$.  Let us do this a number of times with any number of boats, and obtain a distribution of the position of the boat at $t_1$, saving it. Next let us repeat the experiment, this time focusing on $y_0^\prime$, save the resultant distribution at $t_1$, and so on and so forth, until we are comfortable having mapped the totality of positions at $t_0$.  Let us next put together (stack, respecting the natural order of $y$) all the distributions we obtained at $t_1$.  We now have a discrete surface which we can smooth to obtain a Pasquali patch.

Let us now look at position $t_2$ which is the same distance as $t_1$ is from $t_0$.  Having defined the dynamics of the system (from a single Pasquali patch), the dynamics at $t_2$ can be theoretically described by $P^2$.  We can therefore ascertain the probability that we will find the boat at $t_2$ along the width of the canal.  In fact, at $t_n$, $n$ very large, we can ascertain the probability that the boat will be at any position along the width.  It should be close to $P^\infty$.  More importantly, a great distance from the origin (any distance, not necessarily a distance $n \cdot \Delta t_n$), the position probability is aptly described by $P^\infty$.  See Figure 1 and Figure 2.

We can experimentally create Pasquali patch and use it for prediction. We can perform the measurement at an arbitrary distance.

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We can then use Pasquali patch powers for position prediction down the canal, at position n times delta t down the origin.

This simple thought experiment brings about several questions. What if the dynamics of the surface system are described by the Pasquali patch, but at points which are not a distance $\Delta t_n$ apart? In other words, what if the description is apt but at points that are not linear in distance? This curious situation suggests a time anomaly, and therefore a manner in which we can measure time warps (by measuring the actual time differences between Pasquali patches). See Figure 3.

In this schematic a Pasquali patch and its powers do describe the system, but at non-equidistant points. The arrow of time is warped.

Next, we looked at the surface dynamics of the system. If we add a depth variable to the canal, we can in theory produce a Pasquali cube, which would measure the dynamics of any point on the $[0,1] \times [0,1]$ cross-section a discrete distance down the canal (and any distance very far from our origin).

A third question arises when we consider the same canal, but whose width opens by a scalar (linear) amount a distance from our chosen origin.  There is no reason we cannot “renormalize” the width (set it equal to 1 again) at a point some set distance from our chosen origin, and proceed with our analysis as before.  See Figure 4.

In this schematic the width of the canal grows linearly, but Pasquali patch dynamics are conserved, suitably scaled.

In a subsequent post I'd like to reconsider the slit experiment under this new light, see where it takes us.