### Archive

Archive for the ‘Physics’ Category

## On Calculating Entropy

When talking about Pasquali patches that converge and stationary surfaces that they converge to, we discussed that such would stabilize because it wouldn't loop in any shape or form nor would it diverge.  Such Claim we called Entropy.  We didn't provide a mechanism to calculate such. We do that now.

The way we propose we calculate the entropy is by measuring the deviation from the stationary surface.  Thus for the case where we have a Pasquali patch $p_t(x,y)$, $t \in \mathbb{Z}^+$ we could measure entropy by:

that is, by taking the absolute deviation from the stationary surface (notice we use the area element $dA$).  Since in this case the time element is configured at integer times, rather than on continuous time, letting $t \in \mathbb{R}$ implies using the Pasqualian.  Thus we would have

It seems fairly clear that $0 < S(t) < \infty$, but here we must agree that $0$ means that the system is stable and has the most entropy.

We may however, rather than the absolute difference from the stationary surface, choose to emphasize larger changes and (heavily) discount smaller ones.  Thus we could suggest

for $t \in \mathbb{R}$.

## On Infinite Term Functions with Converging Integrals, Part II

This post is a continuation of a previous one. I have developed several preparatory (in preparation of other) claims or theorems:  The first and second claim show that a particular collection of finite polynomial functions have area of 1 in the interval $[0,1]$, and are hence also Pasquali patches.  The third and fourth shows that the finite sum of any such functions actually have converging integrals in the same interval.  The corollaries show that this is not the case if the sum is infinite.  A fun way to summarize this information is soon after developed, and these observations, though simple, lead us to classify all Pasquali patches which are functions of $x$ alone, and therefore all stationary/limiting/stable surfaces, eigenfunctions or wavevectors (from the quantum mechanics point-of-view).

Claim 1. Take $f_i(x) = (i+1) x^i$ with $i = 0 \ldots n$.  Then $\int_0^1 f_i(x) \, dx = 1, \forall i$.

Proof by Definition of Integration (Inducing).  We show that $\int_0^1 f_0(x) \, dx = 1$. The expression equals

We assume that the $k$th element $\int_0^1 f_k(x) \, dx = 1$ although we readily know by the definition of integration that such is true, since

The exact same definition argument applies to the $k+1$th element and

Claim 2. The functions $f_i(x) = (i+1) x^i$ with $i = 0 \ldots n$ are Pasquali patches.

Proof.  A Pasquali patch is a function $p(x,y)$ so that $\int_0^1 p(x,y) dx = 1$.  Let $p(x,y) = f_i(x)$.  Since by Claim 1  $\int_0^1 f_i(x) \, dx = 1, \forall i = 1 \ldots n$, then applying the definition means  $f_i(x) = (i+1) x^i$ are Pasquali patches $\forall i= 0 \ldots n$.

Claim 3.  The finite polynomial $g(x) = \sum_{i=0}^n (i+1) x^{i}$ converges in area from $[0,1]$ to $n+1$.

Proof.  We are looking for

.  The sum is finite so it converges, and there is no issue exchanging the order of the sum and integral. Thus:

Claim 4. Pick $n$ functions from the pool of $f_i(x) = (i+1) x^i$.  For example, pick $f_3(x), f_5(x)$, and $f_7(x)$.  Create the function $h(x) = \sum_i f_i(x)$.  Then $\int_0^1 h(x) \, dx = n$.

Proof by induction.  Since by Claim 2 all $f_i(x)$ are Pasquali patches, it follows their integral is $1$ in the interval (Claim 1).  Picking 1 function from the pool thus gives an integral of 1 in the interval.  Suppose that picking $k$ functions gives $k$ units at the integral in the interval. Now pick $k+1$ functions.  The first $k$ functions give $k$ units at the integral in the interval, and the 1 additional function contributes 1 unit at the integral in the interval.  Thus $k+1$ functions contribute $k+1$ units at the integral in the interval.

Corollary 1. The infinite polynomial $a(x) = \sum_{i=0}^\infty (i+1) x^i$ diverges in area in the interval from $[0,1]$.

Proof.  Take

Here exchanging the order of limit and integral is justified by the fact that, term-wise, the integral converges. Next

Here the second to last step is justified by Claim 3.

Corollary 2.  The infinite polynomial $a(x) - h(x)$ diverges in area in the interval from $[0,1]$.

Proof.  Take the limit

Taking n to infinity applies to $a(x)$ only which we know diverges by Corollary 1.  The same limit  has no effect on $h(x)$ as the sum it is composed of is finite and adds up to an integer constant, say $m$.  We conclude that any infinite collection of terms of $f_i(x)$ diverges, even when a finite number of them may be absent from the sum.

And now sushi.

Corollary 3.  The infinite polynomial  $a(x) - b(x)$ diverges in area in the interval from $[0,1]$ with $a(x), b(x)$ are infinite polynomials constructed by sums of functions picked from the pool $f_i(x) = (i+1) x^i$ and with no repetitions. (Note that the difference of these two infinite polynomials must also be infinite).

Proof. Since the $a(x) - b(x)$ is an infinite polynomial, the integral of such will be an infinite string of ones since the functions it contains are $f_i(x)$ and these are Pasquali patches (Claim 2) and there are no repetitions.  Such infinite sum of ones clearly diverges.

Remark 1.  We can view what we have learned in the claims from a slightly different vantage point.  Create the infinite identity matrix

Next create the following polynomial differential vector

It is clear that

for all rows $i$ of $I$.  We can omit the little $i$ because this definition applies to all rows and:

This of course summarizes Claims 1 and 2.  Next, define the matrix $J$ consisting of rows which are finite sums of rows of $I$ (so that each row of $J$ consists of a finite number of ones at any position, namely $n$ such coming from $n$ picked rows of $I$).  Claims 3 and 4 are summarized in the statement

where $S$ is the vector consisting of the sum of the rows of $J$, which, since it is made up of a finite number of ones at each row, adds up to a constant integer at each row:

Finally, the corollaries can be summarized in the statement in which we create a matrix $K$ consisting of rows with a finite number of zeroes (and an infinite number of ones) or an infinite number of zeroes but an infinite number of ones as well.  It is clear then that

Remark 2. The cool thing about this notation is that it gives us power to conclude several interesting things.  For example, scaling of matrices $I$ and $J$ as by a constant $t$ shows convergence at the integral in the interval $\left[ 0,1 \right]$ of every one of the scaled sums  represented by the rows of such matrices.  Thus:

Corollary 4. Let $I^* = t \cdot I$ and $J^* = t \cdot J$ with $t$ is a scaling factor.  Then the area of each of the infinitely many polynomials represented by the matrices $I^*, J^*$ dot $D$ in the interval from 0 to 1 converge.

Proof.  On the one hand, we have

On the other hand,

Remark 3. Next consider the infinite-matrix formed by convergent sequences (at the sum) at each row,

Depicted is the reciprocals of squares which we know converges at the sum (Basel problem), simply for illustration, but all convergent sequences would be in the $i$th row of $A$.  We have

is convergent by definition.  The cool thing is we can easily prove in one swoop that all sequences that are scaled will also converge at the sum (and the infinite polynomials with coefficients $A \cdot D$ have converging area in the interval from 0 to 1).

Corollary 5. Let $A^* = t \cdot A$ with $t$ is a scaling factor.  Then the area of each of the infinitely many polynomials represented by the matrix entries of $A^* \cdot D$ in the interval from 0 to 1 converge.

Proof.  We have

for all $i$, so this equals

for all $i$.

All of these small and obvious observations lead to this:

Claim 5. The Grand Classification Theorem of Limiting Surfaces (A General and Absolutely Complete Classification of Pasquali patches which are functions of $x$ alone).  All Pasquali patches which are functions of $x$ alone (and therefore possible limiting surfaces) take the form

Proof. We have that, since such $p(x)$ is a Pasquali patch, it must conform to the definition.  Thus

shows this is indeed the case.  To show that "all" Pasquali patches that are functions of $x$ alone are of the form of $p(x)$, we argue by contradiction.  Suppose that there is a Pasquali patch that is a function of $x$ alone which does not take the form of $p(x)$.  It couldn't possibly be one such that is a finite polynomial, since  $A_i$ was defined to be that matrix formed by all convergent sequences at the sum at each row and it can be scaled any which way we like, and this includes sequences with a finite number of nonzero coefficients.  But now it couldn't be any infinite polynomial either, by the same definition of  $A_i$ which includes infinite sequences so that $\sum_j a_{i,j}$ is convergent.  Thus it must be a polynomial formed by dotting divergent sequences (at the sum), but all such have been happily excluded from the definition of $A$.

Remark 4.  Thus, EVERY convergent series has an associated Pasquali patch (which is solely a function of  $x$), and vice versa, covering the totality of the Pasquali patch functions of $x$ universe and the convergent series universe bijectively.

Remark 5.  Notice how the definition takes into account Taylor polynomial coefficients (thus all analytic functions are included) and those that are not (even those that are as yet unclassified), and all sequences which may be scaled by a factor as well.

Claim 6. Let $f(x)$ is Maclaurin-expandable so that

Then

Proof.

for some $i$ row of $A$.  Such a row would have to be of form

Then the integral

Remark 6. Notice that all Maclaurin-expandable functions converge in area (have stable area) in the interval from 0 to 1, a remarkable fact.

Example 1.  Take

By applying Claim 6, it follows that

Remark 7. Now we have a happy way to construct (any and all) Pasquali patches which are functions of $x$ alone, merely by taking a sequence which is convergent at the sum.

Remark 8. Quantum mechanically, we now know all possible shapes that a stationary (limiting) eigen wavevector can take.

Remark 9. This gives us extraordinary power to calculate convergent sums via integration, as the next examples show.  It also gives us extraordinary power to express any number as an infinite sum, for example.

Part I v17

## A Remark based on False Claim 1

So I was thinking that, the statistical description of a dynamical system (as one described by a generator Pasquali patch) really does give us a lot of power in computing the probable "position" of a particle (photon, electron) moving in space at different (integer) time intervals.  If the quantum mechanical supposition of time having a minimum discreteness (Planck-time) is correct, we can find the "finest" Pasquali patch generator that will give a complete description of the dynamical system.  Any Pasquali patch generator descriptive of the system which is not this "first" will generate an accurate, yet less refined ("coarser") version of the system (this is what we mean by Claim 3 of the previous post, in that such Pasquali patch will be "contained" in the finest description, yet is not the finest), and in fact either system of course converges to the same steady state (this is what is meant by Claim 4).  If we are able to find a continuous description (like the Shrodinger equation, via a "Pasqualian") of such system then we are in luck (this description would be the finest, though non-discrete, description), and I speculate though I cannot be sure yet that either discrete descriptions will be contained in such.

Whatever the description of the dynamical system via a generator Pasquali patch (or a Pasqualian), each Pasquali patch represents the transition (position) probabilities of a particle (photon, electron) moving within that system.  If we suppose that the particle moves with same velocity (take photons in vacuum as an example), then each Pasquali patch power is descriptive of the transition-position probability at equally spaced spacial or distance intervals.  Though I've remarked about this before, if it were the case that, for a particle with a particular (steady) velocity, the Pasquali patch power is not exactly equally spaced in distance intervals, it must mean that the arrow of time is bent (time is moving faster for smaller-spaced intervals, slower for longer-spaced).  We have not yet described accelerating particles but at present that is not of our interest.

However, we can tell if time is passing so long as each Pasquali patch description is different at each (equal or unequal) interval.  If a single Pasquali patch were to describe the system at EACH distance interval, there is no way to know if time is moving at all.  Take for example the False Claim 1 of the previous post.  We had the collection $\mathbb{Q} = \{Q^1, Q^2, \ldots, Q^k, \ldots\}_{k \in \mathbb{Z}^+}$ with $Q = q(x,y)$ being an explicit function of $y$ and converging to $Q^\infty = 1$.  We could track the time-distance interval via the Pasquali patch power, so that $1$ was the first distance interval from start (we take it as given that time is not being bent, so that a fixed distance implies the passage of 1 unit of time), $2$ was the second distance interval (2 units of time), and so on, and we could tell if time were bent if each power were descriptive of different distance intervals.  Furthermore, since each $Q \in \mathbb{Q}$ is different, this implies each position transition probability is different and the system is in movement.

This is definitely not the case with the collection $\mathbb{P} = \{P^1 = 1, P^2 = 1, \ldots, P^j, \ldots \}_{j \in \mathbb{Z}^+}$ which also converges to $P^\infty = 1$.  Since at each distance interval the movement probability is the same (uniform), one cannot be convinced that each power represents a distance interval equal to equally spaced time intervals or different-spaced time intervals.  Where we could with $\mathbb{Q}$ ascertain that time was moving, we cannot with $\mathbb{P}$.  The statistical description cannot tell if the system is frozen.

When a system has reached the steady state (which, is the highest entropy state!), there is no way to tell if time flows, as the statistical description is and forever will be unchanging.  Recall that the steady state for a Pasquali patch is always a function of $x$ alone, say $p(x)$.  Furthermore recall that any power of $p(x)$ is always $p(x)$ itself (see Compendium).  We reach an impasse: is time flowing normally, faster than what is conventional, slower? At such a point it is impossible to say, at least from the statistical point of view.  We would have to track particles individually in order to ascertain if they have deviated their path at all (it could be the case that they shifted to all positions with equal probability or in the shape of $p(x)$, e.g., but we cannot be sure of either situation).

## 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.

//

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.