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!)

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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]

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[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}.

* * * * *   * * * * *   * * * * *

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]

A little more on my research on the diffusion equation

Alright. Here we go again. … In my last post, I had mentioned a bit about how attending the recent ISTAM conference at Pune, had helped me recall my thoughts on the diffusion equation and all. In particular, I had mentioned in that post how I had discovered a Berkeley professor’s paper only after publishing my own paper (in ISTAM, 2006, held at Vishakhapattanam), and that I would revise my ISTAM paper and send it over to a journal.

The gentleman in question is Prof. T. N. Narasimhan. Unfortunately, I gather, he has passed away in 2011 [^][^]. Here is the group of his relevant papers:

Narasimhan, T. N. (1999) “Fourier’s heat conduction equation: history, influence, and connections,” Reviews of Geophysics, vol. 37, no. 1, pp. 151–172

Narasimhan, T. N. (2009) “The dichotomous history of diffusion,” Physics Today, July 2009, pp. 48–53

Narasimhan, T. N. (2009) “Laplace, Fourier, and stochastic diffusion,” arXiv:0912.2798, 13 Dec 2009

Narasimhan, T. N. (2010) “On physical diffusion and stochastic diffusion,” Current Science, vol. 98, no. 1, 10 January 2010, pp. 23–26

All these papers are available somewhere or the other on the ‘net. (Copy-paste the paper titles in a Google Scholar search, and you should get to the PDF files.)

The first paper is the most comprehensive among them. In this paper, Prof. Narasimhan discusses the historical context of the development of Fourier’s theory and its ramifications. The paper even gives a very neat (and highly comprehensive) table of the chronology of the developments related to the diffusion equation.

In this paper, unlike in so many others on the diffusion equation, he explicitly (even if only passingly) looks into the issue of the action-at-a-distance. However, the way he discusses this issue, it seems to me, he perhaps had entirely missed the crucial objections that are to be made against the idea of IAD—the same basic things which, in yet another context, lead people into believing in quantum entanglement in the sense they do. (And, how!) In fact, in this paper, Narasimhan does not really discuss any of those basic considerations concerning IAD. If so, then what is it that he discusses?

Narasimhan uses the term “action at a distance” (AD), and not the more clarifying instantaneous action at a distance (IAD). However, such a usage hardly matters. It’s true that if it’s just the AD that you take up for discussion, then it’s the issue of the absence of a mediating agency or a medium (or the premise of a contact-less transmission of momenta/forces) that you highlight, and not so much the instantaneity of the transmission, even if the latter has always been implied in any such a discussion. As people had observed right in the time of Newton himself, the assumption of instantaneity, of IAD, was there, built right at the base of his theory of gravitation, even if it was billed only as an AD theory back then. The difference between IAD and AD is more terminological in nature.

But then, Narasimhan is not even very explicit in his positions with respect to AD either. To get to his rather indirect remarks on the AD issue, first see his discussion related to Biot and the particles approach that he was trying (pp. 154, the 1999 paper).

Today, i.e., after the existence and acceptance of the kinetic molecular theory for more than a century, a modern reader would expect the author to say that someone who believes in, or at least is influenced by, a particles-based approach would naturally be following a local approach, and hence should be found on the side of denying the IAD. Instead, though the author does not explicitly take any position, from the way he phrases his lines, he seems to suggest that he thinks that someone who adopts a particles-based approach would have found AD to be natural. This contradiction was what I had found intriguing initially.

But then, soon enough, I figured out a plausible way in which the author’s thought-train might have progressed. He must have taken the gravitational interaction between n number of bodies as the paradigm of every AD theory, and therefore, must have come to associate (I)AD with any particles-based approach—in exact opposition to the local nature of the particles-based theories of the 19th century and the later techniques (e.g. LBM, SPH, etc.) of the 20th century. That can only be the reason why Narasimhan makes the AD-related comments the way he makes them, especially in reference to Biot’s work (pp. 154).

I would try to gather more historical material, and in any case, address this issue in my forthcoming paper. That’s what I meant when I said I would revise my paper and send it to a journal. I didn’t mean to say that I would be revising my position—I would be only clarifying it to a greater detail. My position is that a kinetic theoretical model i.e. a particles-based approach, the default way people interpret it, does not involve IAD.

Anyway, back to Narasimhan’s paper. Further on this issue, on page 155 of the same paper, the author states the following:

“Essentially, Fourier moved away from discontinuous bodies and towards continuous bodies. Instead of starting with the basic equations of action at a distance, Fourier took an empirical, observational approach to idealize how matter behaved macroscopically.”

[Bold emphasis mine]

In this passage, to be historically accurate, in place of: “action at a distance” the author should have said: “a discrete/particles-based approach;” and in place of: “an empirical, observational” he should have said: “a continuum-based” approach. After all, none, to my knowledge, has ever empirically or experimentally observed an infinite speed of heat transmission—none possibly could.

Now, of course, the Fourier theory does not really acquire its IAD nature because it’s a continuum theory. The reasons are different; however, that’s yet completely different point. See my ISTAM paper for more details.

Coming back to Narasimhan’s paper, of course, the above-mentioned flaws present in it are wholly minor. On the other hand, his paper carries excellent and comprehensive commentary on so many other important aspects, including the historical ones. Indeed, he is to be lauded and thanked for at least including the (I)AD issue in a paper on diffusion, despite being at Berkeley. …

These days, given the attitudes of the people at places like Berkeley, Stanford, MIT, etc., they would seem to carry this attitude towards my paper on diffusion: “Uh. But it all was already known; wasn’t it?” No, it was not. That precisely is (and has been) the point. Unless you have read my paper, what goes by being “known” would squarely consist of something like the following:

“There is IAD in Newton’s theory of gravity. And also, in the Fourier theory, though its effects are quantitatively negligible, and so, we can always neglect it in analysis and interpretation. So, there is this IAD in the partial differential diffusion equation, and this fact has always been known. And, lately, some attempts have been made to rectify this situation, e.g. the relativistic heat equation, but with limited success.”

The correct statement is:

“There is IAD in Newton’s theory of gravity. There also is IAD in the Fourier theory. But there is no IAD in the diffusion equation itself. Following the commonly accepted way of taking it, the kinetic molecular theory may be taken not to have any IAD in it. However, it would be easily possible to introduce IAD also into it. The relativistic wave equation is not at all relevant to this set of basic observations.”

There is quite a difference between the two sets of statements.

I am still going through Narasimhan’s other papers, but at least after a cursory look, these seem more like just restatements being made to different audiences of what essentially are the same basic positions.

Apart from it all, here are a few other papers, now on the Brownian movement side of it:

Hanggi, Peter and Marchesoni, Fabio (2005) “100 years of Brownian motion,” arXiv:cond-mat/0502053v1. 2 Feb. 2005

Chowdhury, Debashish (2005) “100 years of Einstein’s theory of Brownian motion: from pollen grains to protein trains,” arXiv:cond-mat/0504610v1. 24 April 2005

Gillespie, Daniel T. (1996) “The mathematics of Brownian motion and Johnson noise,” American Journal of Physics, vol. 64, no. 3, pp. 225–240

As you can see by browsing through all these papers, few people seem to have appreciated the aspect of IAD or locality, in these two theories. Perhaps, that’s a part of the reason why quantum conundrums continue to flourish. Yet, it is important to isolate this particular aspect, if we are to be clear concerning our fundamentals. As someone said, Fourier’s theory has by now become a part of the very culture of science. So deep is its influence. It’s time we stopped being nonchalant about it, and began re-examining its premises and implications.

Ok. Enough for today. If you are interested, go through these papers, and I will be back with some further comments on them. Hopefully soon. I anyway need to finish this paper. Without getting a couple of papers or so published in journals, I cannot guide PhD students. But, that doesn’t mean I will deliberately send this diffusion paper to a sub-standard or even a low impact journal. I will try to get it published in as high quality (but fitting sort of) a journal as possible. And, if you have any suggestions as to which journal I should send this diffusion paper, please do not hesitate in dropping me a line.

* * * * *   * * * * *   * * * * *

I still remain jobless.

However, on a whim, for this post, I decided to add the “A Song I Like” section.

The selection, as usual, is more or less completely random. How random? Well, here is the story about the selection of this particular song, this time around…

As it so happens, sometimes, you “get” or “catch” some tune right in the morning, and then it stays with you for the whole day. The harder you try to get it out of your mind, the more lightly but more firmly it keeps returning to you throughout the day. It doesn’t even have to be a good tune; it simply keeps returning back. That’s what happened with this song, though, the song happens to be a better one. The song happened to so fleetingly alight on my mind in a recent short journey, that I had not realized that I was silently humming it almost halfway down in that journey. (I was not playing any song/music at that time.) So, even though there is another Lata number (“yeh kaun aayaa”) from the same movie (“saathee”) which I perhaps would have chosen if I were deliberately to make a selection, in view of the lightness with which it had come to me—almost as if entirely by itself—I decided to keep this particular song (“mere jeevan saathee”). (BTW, I haven’t seen the movie, and as usual, the video and other aspects don’t count.) Another point. Neither of these two songs looks like it was composed by Naushad—and, in my books, that’s a plus. When truly in his elements, Naushad feels—to me at least—too traditional, perhaps a bit too melancholic, and, what’s the word… too conforming and invention-less?… Yes, that’s it. He feels too much of a conformist and too invention-less, as far as I am concerned. (Even if some tunes of his might have actually been inventive or of high quality, they follow the groove of the traditional song composition, the traditional guideposts so faithfully, that upon listening to the song, he feels invention-less, anyway. And then, I have my own doubts as to how many times he actually was being inventive, anyway!) Alright. Here is that song—an exception for Naushad, as far as I am concerned. And then, Lata, as usual, takes what is only a first class tune, and manages to take it to an altogether different, higher plane, imparting it with, say, a distinction class:

A Song I Like:
(Hindi) “mere jeevan saathee, kalee thee main to pyaasee…”
Singer: Lata Mangeshkar
Music: Naushad
Lyrics: Majrooh Sultanpuri

[May be, a minor editing is due, though I would not spend much time on it when I return.]

[E&OE]