A bit about the Dirac delta (and the SPH)

I have been thinking about (and also reading on!) SPH recently.

“SPH” here means: Smoothed Particle Hydrodynamics. Here is the Wiki article on SPH [^] if all you want is to gain some preliminary idea (or better still, if that’s your purpose, just check out some nice YouTube videos after googling on the full form of the term).


If you wish to know the internals of SPH in a better way: The SPH literature is fairly large, but a lot of it also happens to be in the public domain. Here are a few references:

  • A neat presentation by Maneti [^]
  • Micky Kelager’s project report listed here [^]. The PDF file is here [(5.4 MB) ^]
  • Also check out Cossins for a more in-depth working out of the maths [^].
  • The 1992 review by Monaghan himself also is easily traceable on the ‘net
  • The draft of a published book [(large .PDF file, 107 MB) ^] by William Hoover; this link is listed right on his home page [^]. Also check out another book on molecular dynamics which he has written and also put in the public domain.

For gentler introductions to SPH that come with pseudo-code, check out:

  • Browne and Lewinder [(.PDF, 5.2 MB) ^], and
  • David Bindel’s notes [(.PDF, ) ^].

I have left out several excellent introductory articles/slides by others, e.g. by Mathias Muller (and may expand on this list a day or two later).


The SPH theory begins with the identity:

f(x) = \int\limits_{\Omega} \text{d}\Omega_{x'}\,f(x')\,\delta(x- x')

where \delta(x- x') is Dirac’s delta, and x' is not a derivative of x but a dummy variable mimicking x; for a diagrammatic illustration, see Maneti’s slides mentioned above.

It is thus, in connection with SPH (but not of QM) that I thought of going a little deeper with Dirac’s delta.

After some searches, I found an article by Balki on this topic [^], and knowing the author, immediately sat reading it. [Explanations and clarifications: 1. “Balki” means: Professor Balakrishnan of the Physics department of IIT Madras. 2. I know the author; the author does not know me. 3. Everyone on the campus calls him Balki (though I don’t know if they do that in his presence, too).] The link given here is to a draft version; the final print version is available for free from the Web site of the journal: [^].

A couple of days later, I was trying to arrange in my mind the material for an introductory presentation on SPH. (I was doing that even if no one has invited me yet to deliver it.) It was in this connection that I did some more searches on Dirac’s delta. (I began by going one step“up” the directory tree of the first result and thus landed at this directory [^] maintained by Dr. Pande of IIT Hyderabad [^]. … There is something to be said about keeping your directories brows-able if you are going share the entire content one way or the other; it just makes searching related contents easier!)

Anyway, thus, starting there, my further Google searches yielded the following articles/essays/notes: [^], [^], [^], [^], [^], [^], [^], and [^] . And, of course, the Wiki [^].

As any one would expect, some common points were of course repeated in each of these references. However, going through the articles/notes, though quite repetitive, didn’t get all that boring to me: each individual brings his own unique way of explaining a certain material, and Dirac’s delta being a concept that is both so subtle and so abstract, any person who [dare] attempts explaining it cannot help but bring his own individuality to that explanation. (Yes, the concept is subtle. The gifted American mathematician John von Neumann had spent some time showing how Dirac’s notions were mathematically faulty/untenable/not rigorous/something similar. … Happens.)

Anyway, as I expected, Balki’s article turned out to be the easiest and the most understanding-inducing a read among them all! [No, my attending IIT M had nothing to do with this expectation.]

Yet, there remained one minor point which was not addressed very directly in the above-mentioned references—not even by Balki. (Though his treatment is quite clear about the point, he seems to have skipped one small step I think is necessary.) The point I was looking for, is concerned with a more complete answer to this question:

Why is it that the \delta is condemned to live only under an integral sign? Why can’t it have any life of its own, i.e., outside the integral sign?

The question, of course is intimately related to the other peculiar aspects of Dirac’s delta as well. For instance, as the tutorial at Pande’s site points out [^]:

The delta functions should not be considered to be an infinitely high spike of zero width since it scales as: \int_{-\infty}^{\infty} a\,\delta(x)\,\text{d}x = a .

Coming back to the caged life of the poor \delta, all authors give hints, but none jots down all the details of the physical (“intuitive”) reasoning lying behind this peculiar nature of the delta.

Then, imagining as if I am lecturing to an audience of engineering UG students led me to a clue which answers that question—to the detail I wanted to see. I of course don’t know if this clue of mine is mathematically valid or not. … It’s just that I “day-dreamt” one form of a presentation, found that it wouldn’t be hitting the chord with the audience and so altered it a bit, tried “day-dreaming” again, and repeated the process some 3–4 times over the past week. Finally, this morning, I got to the point where I thought I now have got the right clue which can make the idea clearer to the undergraduates of engineering.

I am going to cover that point (the clue which I have) in my next post, which I expect to write, may be, next week-end or so. (If I thought I could write that post without drawing figures, I would have written the answer right away.) Anyway, in the meanwhile, I would like to share all these references on SPH and on Dirac’s delta, and bring the issue (i.e., the question) to your attention.

… No, the point I have in mind isn’t at all a major one. It’s just that it leads to a presentation of the concept that is more direct than what the above references cover. (I can’t better Balki, but I can fill in the gaps in his explanations—at least once in a while.)

Anyway, if you know of any other direct and mathematically valid answers to that question, please point them out to me. Thanks in advance.

 


A Song I Like:

(Marathi) “mana chimba paavasaaLi, jhaaDaat rang ole…”
Music: Kaushal Inamdar
Lyrics: N. D. Mahanor
Singer: Hamsika Iyer

 

[E&OE]

 

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3 thoughts on “A bit about the Dirac delta (and the SPH)

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