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]

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