“Blog” less; write journal papers!

“‘Blog’ less; write journal papers.”

That’s my NYR for 2018.

Allow me to explain.


My research is not experimental, neither is it connected with, say, design of a new machine or development of a new manufacturing process. The most concrete aspect my work involves only computational modeling. But that too is not of the kind which engineering researchers typically undertake. I don’t do FEM of this multi-physics problem or that. What I work on are some very fundamental issues of physics and engineering.

My research thus is decidedly theoretical, often bordering on being “speculative.” It tends to concentrate on fundamental aspects. For decades by now, I have been trying to tackle some of the trickiest, deepest or very abstract problems (e.g. foundations of QM). At other times, I have been busy just isolating something new as a problem in its right (e.g., instantaneous action-at-a-distance in diffusion, or non-uniqueness of solution to the diffusion equation, or the fundamentality of stress vis-a-vis strain, or mode transitions in ideal vibrations and their relation to vibrations in the real mechanical system, or the physical meaning of the delta of calculus of variations….).

OK, there are some simple experiments here and there I might do. But they are not a very significant aspect of my work. The experiments are more in the nature of illustrations (e.g. melting snowman). They are not even fully in the nature of quantitative validations, let alone the prime vehicles to discovery. So, they are just “potatoes” of my research. The meat is: deep theoretical issues themselves. That’s what it’s like when you say “fundamental.”

The only way in which you can formulate or tackle such problems—fundamental or foundational—is by being a bit “relaxed” about both the specifics of your topic and the way you go about tackling it.

If you believed too much in the existing theory, you wouldn’t be able to spot unidentified problems with it or find new solutions to the known ones. If you try to do theoretical research and if you still try to stick to a schedule like what they do in experimental research (say in designing and fabricating a gadget, complete with bill of materials, or in developing a process, complete with prototype 1, prototype 2, etc.), you wouldn’t able to even get off to a decent start. After all, a schedule can be made from only those ingredients that are already known to you, not of never seen possibilities or unknown ideas. And, while in experimental research, reality has a wonderful way to throw up new possibilities, you have no such luxury in theoretical research. Every “never seen” possibility has to be forged by your own mind. If you don’t think in a relaxed manner, you are never going to believe that the issue is easy enough for you to tackle it.

But one unintended consequence of it all is that, in theoretical research like mine, it’s easy (far too easy in fact) to get a bit too relaxed. It is easy to pursue too many diverse theoretical threads, and in examining them, to run around in circles and so keep on getting back to the same points again and again.

But now I have come to realize that perhaps time has come to stop pursuing new threads in my research and to consolidate what has already been learnt.

The best way I can think of for doing the latter is: writing papers.

In particular, I have to kick aside this one habit: writing things down only when and as “inspiration” strikes.

Writing thoughts down (maintaining pocket diaries) has done a world of good to me. But this long-pursued activity seems to have by now come, in my case, to the point of diminishing marginal utility.

In place of this habit (of keeping on idly brain-storming and noting down possibilities it throws up) I would now like to put in place another habit: writing things (papers, actually) down in a structured, routine, regular, day-to-day, and time-bound manner. Allow me to explain this part too.

Given the way I have pursued my research (and in fact, given even the very nature of problems I ended up tackling), it would have been impossible for me to say something like this:

“OK! January, diffusion paper! February, stress-strain paper! March and April, QM position paper!”

“… What, in February, I don’t write something on QM? neither on diffusion? How ridiculous?”

That is how I would have reacted. But not any more.

Instead, I am now going to be a bit “bureaucratic” about my research. (UGC and AICTE folks ought to be happy in discovering a new soul-mate in me!)

What I am going to do is what I indicated just minutes ago. I am going to make some kind of a “time-table”: this period, work (i.e. actually write papers about) only this particular problem. Leave aside all other issues. Just finish that particular paper. Only then move to those other, more interesting (even alluring) issues in a next delimited period specifically allocated for that. I will have to pursue this policy. And I had better.

After all, while “passively” letting myself jump from issues to issues has yielded a lot of new insights, there are any number of issues where I have “hit the plateau” by now—and I mean those words in a positive sense. By “hitting the plateau,” I mean not a loss of creativity or originality, but a sense, even a firm realization (based on logic) that a certain stage of completeness is already achieved.

And that’s why, I am going to concentrate on “professionally” writing papers, in the next year. Following some kind of a time-bound schedule. As if I were writing a report, or delivering a software product on its schedule. So, it’s high time I became a bit less “creative” and more “professional,” to put it vaguely.

Since I will not be pursuing this bit of this idea or that bit of that idea a lot, I will be blogging less. And since a lot of my research seems to have actually “hit the plateau” in the above-mentioned, positive sense, I would instead be writing papers.

Hence the “slogan”: “`Blog’ less, write journal papers!”

That’s my NYR for 2018…. though I wouldn’t wait for 2018 to arrive before getting going on it. After all, a new year is just an excuse to make resolutions. The digits in the date aren’t important. A definite, demarcated change (“quantum jump” if you will! [LOL!]) is. But a change of the last digit in the YYYY, since it comes only after as long a period as one complete year, is a good time to making the required definite change.

So, there. I will keep you posted, with very brief notes here and there, as to how this paper-writing “business” is actually progressing in my case. My immediate plan is to get going writing the diffusion papers, and to finish writing them, right in January 2018.

Let’s see how things actually progress.


A Song I Like:

This is that Marathi song which I said I had liked a lot during my childhood vacation (see my last 2–3 posts). I still like it. It is the one which has a decidedly Western touch, but without spoiling or compromising on the Indian sense of melody. …

(Marathi) “raajaa saarangaa, maajyaa saarangaa”
Music: Hridaynath Mangeshkar
Singer: Lata Mangeshkar
Lyrics: Shanta Shelke


Bye for now, make a time-table you can stick to, and also take care to execute on it. … Best wishes for a happy and prosperous new year!

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Yo—5: Giving thanks to the Fourier transform

Every year, at the time of thanksgiving, the CalTech physicist (and author of popular science books) Sean Carroll picks up a technique, principle, or theory of physics (or mathematics), for giving his thanks. Following this tradition (of some 8 years, I gather), Carroll has, for this year, picked up the Fourier transform as the recipient of his thanks. [^]

That way, it’s quite a good choice, if you ask me. …

…Though, of course, as soon as I began reading Carroll’s post, a certain thing to immediately cross my mind was what someone had said concerning Fourier’s theory.

Fourier’s is the most widely used theory in the entire history of physics, he had said, as well as the most abused one . … The words may not be exact, but that was the sense of what had been said. Someone respectable had said it, though I can’t any longer recall exactly who. (Perhaps, an engineer, not a physicist.)

The Fourier theory has fascinated me for long; I have published not just a paper on it but also quite a few blog posts.

To cut a long story short, I would pick out (i) the Lagrangian program (including what is known as the Lagrangian mechanics as well as the calculus of variations, the stationarity/minimum/maximum/action etc. principles, the Hamiltonian mechanics, etc.) and (ii) the Fourier theory, as the two basic “pillars” over which every modern quantum-mechanical riddle rests.

Yes, including wave-particle duality, quantum entanglement, EPR, Bell’s inequalities,  whatnot….

As I have been pointing out, the biggest good point that both these theories have in common is that they allow us to at all perform at least some kind of a mathematical calculation of the analytical kind—even if, often times, only in a physically approximate sense—in situations where none would otherwise be possible.

The bad point goes with the good point.

The biggest bad point common to both of them is that they both take some physics that actually occurs only locally (say the classical Newtonian mechanics) and smear it onto a supposedly equivalent “world”—an imaginary non-entity serving as a substitute for the actually existing physical world. And, this non-entity, in both theories (Lagrangian and Fourier’s) is global in nature.

The substitution of the global mathematics in place of the local physics is the sin common to the abuse of both the theories.

Think of the brachistochrone problem, for instance [^]. The original Newtonian approach of working with the local forces using \vec{F} = d\vec{p}/dt (including their reactions), is in principle applicable also in this situation. The trouble is, both the gravitational potential field and the constraints are continuous in nature, not discrete. As the bead descends on the curve, it undergoes an infinity of collisions, and so, as far as performing calculations go, the vector approach can’t be put to use in a direct manner here: you can’t possibly calculate an infinity of forces, or reactions to them, or use them to incrementally calculate the changes in velocities that these come to enforce. Thus, it is the complexity of the constraints (or the “boundary conditions”)—though not the inapplicability of the basic governing physical laws—that make Newton’s original approach impracticable in situations like the brachistochrone. The Lagrangian approach allows us to approach the same problem in a mathematically far simpler manner. [Newton himself was one of the very first to solve this problem using this alternative approach which, later on, to be formalized by Lagrange. (Look up the “lion’s paws” story.)]

Something similar happens also with the Fourier analysis. Even if a phenomenon is decidedly local, like diffusion of the physically distinct material particles (or parcels) from one place to another, the Fourier theory takes these distinct (spatially definite) particles, and then replaces them by positing a global non-entity that is spread everywhere in the universe, but with some peak coinciding with where the actual particles physically are. The so-smeared non-entity is the place-holder [!] for the spatially delimited particles, in Fourier’s theory. The globally spread-out entity is not just an abstraction, but, really speaking, also an approximation—a mathematical approximation. And as far as the inaccuracies in the calculations go, it turns out, this approximation does work out very well in practice. (The reason is not mystical. It is simply that the diffusing particles (atoms/molecules) are so small and so numerous in the physically existing universe.) But if you therefore commit the error of substituting this approximate mathematical abstraction in place of the exact physical reality, you directly end up having the riddles of QM.

If you are interested in pursuing this matter further, you should see my conference paper, first. (Drop me a line if you haven’t already downloaded it when it was available off my Web site, or can’t locate it any other way.) … Though I have also written quite a few posts on the topic, they don’t make for the best material—they are far too informally written (meaning: written completely on the fly and without any previously thought out structure at all). They also too lengthy, and often dwell on technical aspects that are too detailed.

And, that way, they don’t have much mathematical depth, anyway.

But since I seem to be the only person in the entire world who has ever thought along these lines (and one who continues to care), you may want to have a look at myQ detailed musings, too: [^] [^] [^][^].

(… And, no, as far as this issue goes, by no means am I done. I would continue exploring this topic further in my research, also in the future. Though, let me wind it all up for now… This was supposed to be a short and sweet post—a “Yo” post!)

* * * * *   * * * * *   * * * * *

A Song I Like:

(Marathi) “ekaTyaane ekaTe gardeet chaalaave”
Singer: Avadhoot Gupte
Lyrics: Mangesh Padgaonkar
Music: Shreedhar Phadake

[May be I should post a translation of this song some time later. … Also, of that another Marathi song which I have run just a few posts ago, viz., “man pisaaT majhe…” As to that song (“man pisaaT”) I know for a fact that a lot of Marathi-“knowing” people have never bothered to carefully go through the actual words, they have never tried to put them in some kind of a context, and thus, paying only a fragmentary attention here and there, they have come to associate something of a too abstract and weird (or “artsy”) kind of a sense to it. Their appreciation of that song rests mostly on the musical tune and the singer’s rendition, but their sense of the lyrics seems to be quite off the mark. The actual song isn’t of a meaningless “artsy” kind, and I hope to bring out what I think is the original sense of that song, too. And, as far as the present song goes, there isn’t just an  innovative sort of tune and a wonderful rendering by the singer. There also is a very beautiful piece of poetry lying underneath. … It’s a young new song (it came out only in 2010), but with an obvious touch of class to it. The original CD is just Rs. 100. … Enjoy…. More, later]

[E&OE]

 

There’s something wrong about the diffusion equation—but what exactly is it?

As promised last time, let me try to give you a “layman’s” version of the trouble about the diffusion equation.

1. Physical Situations Involving Diffusion

First of all, we need some good physical situations that illustrate the phenomenon of diffusion, in particular, the simplest linear 1D diffusion equation:

\alpha \dfrac{\partial^2 u}{\partial x^2} = \dfrac{\partial u}{\partial t}

Here is a list of such models:

  • Think of a long, metal railing, which has got cold on a winter morning. [I said winter, and not December. No special treatment for Aussies and others from the southern hemisphere.]  Heat the mid-point of the railing using a candle or a soldering iron. The heat propagates in the rod, increasing temperatures at various points, which can be measured using thermocouples. Ignoring higher-order (wave/shocks) effects, the conduction of heat can be taken to follow the abovementioned simple diffusion equation.
  • Think of a container having two compartments separated by a wall which carries a small hole. The entire container is filled with air (say, 1 atm pressure at 25 degree Celsius), and then, an electromechanical shutter closes down the hole in the internal wall. Then, place an opened bottle of scent in one of the compartments, say, that on the left hand side. Allow for some time to elapse so that the scent spreads practically evenly everywhere in that compartment. (If you imagine having a fan in that compartment, you must also imagine it being switched off and the air-flow becoming stand-still on the macro-scale). Now, open the internal hole, and sense the strength of the scent at various points in the right-hand side compartment, at regular time-intervals. [I was being extra careful in writing this model, because the diffusion here can be directly modelled using the kinetic theory of gases.]
  • Take a kitchen sponge of fine porosity, and dip it into a bucket of water, thus letting it fully soak-in the water. Now, keep the sponge on a table. Take a flat piece of transparent glass, and place it vertically next to the sponge, touching it gently. Then, place a drop of ink at a point on the top surface of the sponge, right next to the glass. Observe the flow of ink through the sponge.

Even if this post is meant for “layman” engineers/physicists who have already studied this topic, I deliberately started with concrete physical examples. It helps freshen up the physical thinking better, and thereby, helps ground the mathematical thinking better. (I always believe that by way of logical hierarchy, the physical thought comes before the mathematical thought does. Before you can measure something, you have to know what it is that you are measuring; the what precedes the how.)

2. Mathematical Techniques Available to Solve the Diffusion Equation

Now, on to the mathematical techniques available to solve the above-mentioned diffusion equation. Here is a fairly comprehensive (even if perhaps not exhaustive) list of the usual techniques:

  • Spectral:
    • Analytical: The classical Fourier theory. Expand the initial condition in terms of a Fourier series (or, for an infinitely extended domain, a Fourier integral), and find the time evolution using separation of variables
    • Numerical: Discretize the domain and the initial condition, and also the time dimension. Use FFT to numerically compute the Fourier evolution. (If you are smart: chuck out the FFT implementation you wrote by yourself, and start using FFTW.)
  • Usual Numerical Methods:
    • FEM: Weak formulation.
    • FVM: Flux-conservation formulation
    • FDM: Based on the Taylor series expansion. For a 1D structured grid, it produces the same system as FEM.
  • The “Unusual” Numerical Method—the Local Finite Differences: Discretize the time-axis using the Taylor series expansion (as in FDM). On the space side, it’s slightly different from FDM. Check out p. 15 of Ref [1]. Practically speaking, almost none models the diffusion equation this way. However, we include it at this place to provide a neat progression in the nature of the techniques. If it helps, note that this technique essentially works as a CA (cellular automaton).
  • The Stochastic Methods
    • Brownian movement: By which, I mean, Einstein’s analysis of it; Ref. [3]. BTW, the original paper is surprisingly easy to understand. In fact, even the best textbook expositions of it (e.g. Huang’s Statistical Physics book) tend to drop a crucial noting made in the original paper. (In fact, even Einstein himself didn’t pay any further attention to it, right in the same paper. It was easier to spot it in the original paper. More on this, below, or later.)
    • The random walk (RW)/Monte Carlo (MC)/Numerical Brownian movement. For our limited purposes (focusing on the simple and the basic things), the three amount to one and the same thing.

The Solution Techniques and the Issue of the Instantaneous Action at a Distance

Now go over the list again, this time figuring out on which side each technique falls, what basic premise it (implicitly or explicitly) assumes: does it fall on the side of a compact support for the solution, or not. Here is the quick run-down:

All the spectral methods and all the usual numerical methods involve solution support extended over the entire domain (finite or infinite). The unusual numerical method of local finite differences involves a compact support. The traditional analysis of Brownian movement is confused about the issue. In contrast, what the numerical techniques of random walk/MC implement is a compact support.

Go over the list again, and make sure you are comfortable with the characterizations. You should be. Except for my assertion that the traditional analysis of Brownian movement is confused.

To explain the confusion, we have to go to Ref. [1] again.

In Ref. [1], on p. 16, the author states that:

“… A simple argument shows that if h^2/\tau \rightarrow 0 or +\infty, x may approach +\infty in finite time, which is physically untenable.”

However, in the same Ref [1], on p. 2 in fact, the author has already stated that:

“It is easily verified that u(t,x) = \dfrac{1}{(2 \pi t)^{d/2}} \exp(-\dfrac{|x|^2}{2 t}) satisfies [the above-mentioned diffusion equation.]”

Here, the author does not provide commentary on the nature of the solution, as far as the issue of IAD is concerned.

For a commentary on the nature of solution, we here make reference to [2], which, on p. 46, simply declares (without a prior or later discussion of the logical antecedents or context, let alone a proof for the declaration in question) that the function \dfrac{1}{(4\pi t)^{n/2}} e^{- \dfrac{|x|^2}{4 t}} (where n is the dimensionality of space) is the fundamental solution to the diffusion equation; and then, on p. 56, goes on to invoke the strong maximum principle to assert infinite speed of propagation—which is contradictory to the above-quoted passage in Ref [1], of course, but notice that the solutions being quoted is the same.

BTW, the strong maximum principle suspiciously looks as if its native place is the harmonic analysis (which is just another [mathematicians’] name for the Fourier theory). And, this turns out to be true. [^]

So, back to square one. Nice circularity: You first begin with spectral decomposition that first posits domain-wide support for each eigenfunction; you then multiply each eigenfunction by its time-decay term and add the products together so as to get the time evolution predicted by the separation of variables in the diffusion process; and then, somewhere down the line, you allow yourself to be wonder-struck; you declare: wow! There is action at a distance in the diffusion equation, after-all!

Ok, that’s not a confusion, you might say. It’s just a feature of the Fourier theory. But where is the confusion concerning the Brownian movement which you promised us, you might want to ask at this point.

The Traditional Analysis of the Brownian Movement as Confused w.r.t. IAD

Well, the confusion concerning the Brownian movement is this:

Refer to Einstein’s 1905 paper. In section 4 (“On the irregular movement…”) he says this much:

“Suppose there are altogether n particles suspended in a liquid. In an interval of time \tau the x-Co-ordinates of the single particles will increase by \Delta, where \Delta has a different value (positive or negative) for each particle. For the value of \Delta a certain probability-law will hold; the number dn of the particles which experience in the time interval \tau a displacement which lies between \Delta and \Delta + d\Delta, will be expressed by an equation of the form

dn = n \phi(\Delta) d\Delta

where

\int_{-\infty}^{+\infty} \phi(\Delta) d\Delta = 1

and \phi only differs from zero for very small values of \Delta and fulfils the condition

\phi(\Delta) = \phi( - \Delta) .

We will investigate now how the coefficient of diffusion depends on \phi, confining ourselves again to the case when the number \nu of the particles per unit volume is dependent only on x and t.

Putting for the particles per unit volume \nu = f(x, t), we will calculate the distribution of the particles at a time t + \tau from the distribution at the time t. From the definition of the function \phi(\Delta), there is easily obtained the number of the particles which are located at the time t + \tau between two planes perpendicular to the x-axis, with abscissae x and x + dx. We get

f(x, t + \tau) dx = dx \cdot \int_{\Delta = -\infty}^{\Delta = +\infty} f(x + \Delta) \phi(\Delta) d\Delta

[…

… we get …]

\dfrac{\partial f}{\partial t} = D \dfrac{\partial^2 f}{\partial x^2} (I)

This is the well known differential equation for diffusion…”

[Bold emphasis mine.]

In the same paper, Einstein then goes on to say the following:

“Another important consideration can be related to this method of development. We have assumed that the single particles are all referred to the same Co-ordinate system. But this is unnecessary, since the movements of the single particles are mutually independent. We will now refer the motion of each particle to a Co-ordinate system whose origin coincides at the time t = 0 with the position of the centre of gravity of the particles in question; with this difference, that f(x, t)dx now gives the number of the particles whose x Co-ordinate has increased between the time t = 0 and the time t = t, by a quantity which lies between x and x + dx. In this case also the function f must satisfy, in its changes, the equation (I). Further, we must evidently have for $x > or < 0$ and t = 0,

f(x,t) = 0 and \int_{-\infty}^{+\infty} f(x,t) dx = n .

The problem, which accords with the problem of the diffusion outwards from a point (ignoring possibilities of exchange between the diffusing particles) is now mathematically completely defined [his Ref 9]; the solution is:

f(x,t) = \dfrac{n}{4 \pi D} \dfrac{e^{-\frac{x^2}{4Dt}}}{\sqrt{t}}

The probable distribution of the resulting displacements in a given time t is therefore the same as that of fortuitous error, which was to be expected.”

[Bold emphasis mine]

Contrast the bold portions in the above two passages from Einstein’s paper. Both the passages come from the same section within the paper! The first passage assumes a probability distribution function (PDF) that has compact support, and proceeds, correctly, to derive the diffusion equation. The second passage reiterates that the PDF must obey the same diffusion equation, but proceeds to quote a “known” solution that has x spread all over an infinite domain, thereby simply repeating the error. … To come so close to the truth, and then to lose it all!

Well, you can say: “Wait a minute! He changed the meaning of f somewhere along, didn’t he?”

You are right. He did. In the first passage, f referred to the PDF of particles density at various locations x; in the second passage, it refers to the PDF of particles undergoing various amounts of displacements from their current positions. The difference hardly matters. In either case, if you do not qualify x variable in any way, and in fact quote the earlier, infinite-domain result for the diffusion, you implicitly adopt the position that the PDF is extended to \infty. You thereby end up getting IAD (instantaneous action at a distance) back into the game.

This back-and-forth jumping of positions concerning compactness of support (or IAD) is exactly what Ref [1] also engages in, as we saw above. The difference is that, once in the stochastic context, the Ref [1] is at least explicit in identifying the infinite speed of propagation and denying it a physical tenability. Even though, by admitting the classical solution, it must make an inadvertent jump back to the IAD game!

In contrast, the issue is very clear to see in case of the numerical methods—even if no one discusses IAD in their contexts! The most spectacular failure of the successive authors, IMO, is their failure to distinguish between the local finite differences and the usual FDM. If you grasp this part, the rest everything becomes much more easy to follow. After all, you get the random walk simply out of randomizing the same local-propagational process which is finitely discretized in the local finite differences technique. The difference between RW/MC on the one hand and FDM/FEM/FVM on the other, is not just the existence or otherwise of  randomness; it also is: the compactness of the solution support. … I wish I had the time (or at least the inclination) to implement both these techniques and illustrate the time evolution via some nice graphics. For the time being at least, the matter is left to the reader’s imagination and/or implementation. Here, let me touch on one last point.

Why This Kolaveri Confusion, Di?

Mathematicians are not idiots. [LOL!] If so, what could possibly the reason as to why a matter this “simple” has not been “caught” or “got” or highlighted by any single mathematician so far—or a mathematical physicist, for that matter? Why do people adopt one mind-set, capable of denying IAD, when in the stochastic realm, and immediately later on, adopt another mind-set, that explicitly admits IAD? Why? Any clue? Do you have any clue regarding this above question? Can you figure out the reason why? Give it your honest try. As to me, I think I know the answer—at least, I do have a clue which looks pretty decent to me. … And, as indicated above, the answer is not in the nature of the change of mind-set when people approach the problems they regard as “deterministic” vs. the problems they regard as “probabilistic” or “stochastic.” It’s not that…. It’s something different.

Do give it a try, but I also think that it will probably be hard for you to get to the same answer as mine. (Even though, I also think that you will accept my answer as a valid one, when you get to know it.) And the reason why it will be hard for you—or at least it will be so for most people—is that most people don’t think that physics precedes mathematics, but I do. If only you can change that hierarchy, the path to the answer will become much much easier. A whole lot easier.

That precisely is the reason why I included the very first section in this post. It doesn’t just sit there without any purpose. It’s there to help give you a context. Mathematics requires physics for its context. Anyway, I don’t want to overstretch this point. It’s not very important.

Knowing for a fact that two classes of theories, speaking about the same mathematical equation which has been studied for a couple of centuries, but have completely different things to say when it comes to an important issue like IAD—that is important.

Important, as from the quantum mechanics viewpoint. After all, check out p. 4 of Ref [2]. It lists the Schrödinger equation right after the diffusion equation. And while at that page, notice also the similarity and differences between the two equations, stripped down (i.e. suitably scaled and specialized) to their bare essences.

Resolving the riddles of the quantum entanglement is as close as the Schrödinger equation is to the heat equation—and then as close as resolving the confusions concerning IAD is, in the context of the diffusion equation.

… We must know why physicists and mathematicians have noted the two faces of the diffusion equation, but have remained confused about it. … Think about it.

May be another post on this entire topic, some time later, probably sooner than later, giving you my answer to the above question. In the meanwhile, remember to let me know if you can give any additional information/answer to my Maths StackExchange question on this topic [^]. BTW, thanks are due to “Pavel M” from at that forum, for pointing out Evans’ book to me. I didn’t about it, and it seems a good reference to quote.

References:

[1] Varadhan, S. R. S. (1989) “Lectures on Diffusion Problems and Partial Differential Equations,” Notes taken by Pl. Muthuramalingam and Tara R. Nanda, TIFR, Springer-Verlag.
[2] Evans, Lawrence C. (2010) “Partial Differential Equations, 2/e,” Graduate Studies in Mathematics, v. 19, American Mathematical Society
[3] Einstein, A. (1905) “On the movement of small particles suspended in stationary liquids required by the molecular-kinetic theory of heat,” Annalen der Physik, v. 17, pp. 549–560. [English translation (1956) by A. D. Cowper, in “Investigations on the Theory of Brownian Movement,” Dover]

* * * * *   * * * * *   * * * * *

[This post sure would do with a couple of edits in the near future, though the essential ideas are all already there. TBD: I have to think whether to add the “Song I Like” section or not. Sometime later. It already is almost 2700 words, with many latex equations. … May be tomorrow or the day after…]

[E&OE]

Transient diffusion with compact support throughout—not just initially

[An update made on 2 January 2013 appears near the end]

The following is the question I raised today at the Mathematics Stack Exchange [^]. (It is only today that I became a member there.)

The Question:

Assume the simplest linear diffusion equation: \alpha \dfrac{\partial^2u}{\partial x^2} = \dfrac{\partial u}{\partial t}, where u is the temperature and \alpha is the thermal diffusivity.

The domain is finite, say, [-100, 100]. (If the assumption of an infinite domain makes it possible (or more convenient) to answer this question, then please assume so. However, the question of interest primarily pertains to a finite domain.)

Assume that the initial temperature profile has a compact support, say over [-1, 1].

After the passage of an arbitrarily small but finite duration of time:

(i) would the temperature profile necessarily have support everywhere over the entire domain?

(ii) or, is it possible that a solution may still have some compact support over some finite interval that is smaller than the whole domain?

Can it be proved either way? Given the sum totality of today’s mathematics (i.e. all its known principles put together), is it possible to pick between the above two alternatives in general?

A subsidiary question only if the alternative (ii) is possible: please supply an example, better so, it is of a kind wherein the initial profile is infinitely differentiable, e.g. the bump function e^\frac1{x^2-1}.

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I was expecting answers to affirm instantaneous action at a distance. The first full answer to arrive [^] confirms this anticipation. However, it’s too late in the night to go and point out the Brownian movement-related objection to it. I will do it some time tomorrow. (It’s a mathematics forum. I am not too comfortable writing maths-related comments directly. I have to first translate my thoughts from physics to mathematics.) Also, tomorrow, I will come back here and decide on the fly whether to add the usual final section “A Song I Like,” or not. Also, I will add the tags to this post, tomorrow.

In the meanwhile, think about it and see if you wish to answer or have an interaction on this topic, at Math[s] StackExchange.

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Update on 2 January 2013: I have now begun interaction at the Maths StackExchange forum. Also, now am adding the following section (even if I still go jobless), and the post tags. Guess I will have to be back very soon with another post on this same topic. A sort of “layman’s” (at least an engineer’s) version of the problem, and my position about it. I often find that by loosening the demands of the mathematical rigor a bit (though not the rigor of the basic logic or thought), a layman’s version helps bring in more context more easily, and thereby is actually helpful in delineating issues—both problems and solutions—better. So I will attempt doing that in a next post.

A Song I Like:
(Hindi) “swapn jhade phool se, mit chubhe shool se…” (“karvan guzar gaya, gubaar dekhate rahe”)
Lyrics: Neeraj
Singer: Mohammed Rafi
Music: Roshan

[E&OE]