Where are those other equations?

Multiple header images, and the problem with them:

As noted in my last post, I have made quite a few changes to the layout of this blog, including adding a “Less transient” page [^].

Another important change was that now, there were header images too, at the top.

Actually, initially, there was only one image. For the record, it was this: [^] However, there weren’t enough equations in it. So, I made another image. It was this [^]. But as I had already noted in the last post, this image was already crowded, and even then, it left out some other equations that I wanted to include.

Then, knowing that WordPress allows multiple images that can be shown at random, I created three images, and uploaded them. These are what is being displayed currently.

However, randomizing means that even after re-loading a page a couple of times, there still is a good chance that you will miss some or the other image, out of those three.

Ummm… OK.


A quick question:

Here is the problem statement:

There are three different header images for this blog. The server shows you only one of them during a single visit. Refreshing the page in the browser also counts as a separate visit. In each visit, the server will once again select an image completely at random.

Assume also that the PDF for the random sequence is uniform. That is to say, there is no greater probability for any of the three images during any visit. Cookies, e.g., play no role.

Now, suppose you make only three visits to this blog. For instance, suppose you visit some page on this blog, and then refresh the same page twice in the browser. The problem is to estimate the chances that you will get to see:

  • all of the three different images, but in only three visits
  • one and the same image, each time, during exactly three visits
  • exactly two different images, during exactly three visits

Don’t read further until you solve this problem, right now: right on-the-fly and right in your head (i.e. without using paper and pencil).

(Hint [LOL!]: There are three balls of different colors (say Red, Green, and Blue) in a box, and \dots.)


 

…No, really!

 


Ummm… Still with me?

OK. That tells me that you are now qualified to read further.

Just in case you were wondering what was there in the “other” header images, here is a little document I am uploading for you. Go, see it (.PDF [^]), but also note the caveat below.

Caveats: It is a work in progress. If you spot a mistake or even just a typo, then please do let me know. Also, don’t rely on this work.

For example, the definition of stress given in the document is what I have not so far read in any book. So, take it with a pinch of the salt—even if I feel confident that it is correct. Similarly, there might be some other changes, especially those related to the definition of the flux and its usage in the generic equation. Also, I am not sure if the product ansatz for the separation of variables technique began with d’Alembert or not. I vaguely remember its invention being attributed to him, but it was a long time ago, and I am no longer sure. May be it was before him. May be it was much later, at the hands of Fourier, or, even still later, by Lame. … Anyway let it be…

…BTW, the equations in the images currently being shown are slightly different—the PDF document is the latest thing there is.

Also, let me have your suggestions for any further inclusions, too, if any. (As to me: Yes, I would like to add a bit on the finite volume method, too.)

As usual, I may change the PDF document at any time in future. However, the document will always carry the date of compilation as the “version number”.


General update:

These days, I am also busy converting my already posted CFD snippets [^] into an FVM-based code.

The earlier posted code was done using FDM, not FVM, but it was not my choice—SPPU (Pune University) had thrust it upon me.

Writing an illustrative code for teaching purposes is fairly simple and straight-forward, esp. in Python—and especially if you treat the numpy arrays exactly as if they were Python arrays!! (That is, very inefficiently.) But I also thought of writing some notes on at least some initial parts of FVM (in a PDF document) to go with the code. That’s why, it is going to take a bit of time.

Once all this work is over, I will also try to model the Schrodinger equation using FVM. … Let’s see how it all goes…

…Alright, time to sign off, already! So, OK, take care and bye for now. …

 


A Song I Like:
(Hindi) “baharon, mera jeevan bhee savaron…”
Music: Khayyam
Singer: Lata Mangeshkar
Lyrics: Kaifi Aazmi


[The obligatory PS: In all probability, I won’t make any changes to the text of this post. However, the linked PDF document is bound to undergo changes, including addition of new material, reorganization, etc. When I do revise that document, I will note the updates in the post, too.]

 

Advertisements

Changes at this blog…

The changes at this blog:

In case you haven’t noticed it already, notice [what else?] that the layout of this blog has undergone a change. Hopefully for the better!

In particular, I’ve made the following changes:

  1. This blog is now concerned not only with the more transient writings of mine, but also with the less transient ones! … Accordingly, I have made a new page which holds links to my less transient writings, too, whether the write-ups were published here or elsewhere. See that page here [^].
  2. The tagline too now reflects the change in the purpose of this blog.
  3. I have added a header image, too. As of now, it holds some of the equations that have come to grab my attention for a long while. This may change in future. (See the separate section below.)
  4. A more minor change is the one made to the font.

A note for reading on the mobile:

In case you read this blog on a mobile phone, then to see the “less transient” page, you will have to press the menu button appearing at the top to get to the new page. On a desktop, however, the menu is by default seen as expanded.


The image at the top:

Just for the record, the equations in the top image, as of today (13 August 2018, 11:31 hrs), are the following:

  • The inner product and the outer product of two vectors, expressed using the more familiar notation of matrices.
  • Definitions of the grad of scalars and vectors, and the div of vectors and tensors.
  • The Taylor series expansion
  • The Fourier series expansion
  • The generic conservation equation for a scalar quantity, in the Eulerian form
  • The conservation equation for momentum, in the Eulerian form. (NB: The source term is in terms of \Phi i.e. the conserved quantity itself, whereas the rest of the terms have the mass-specific term \phi in them. This is correct.)
  • Definition of stress. (See the note for this equation below.)
  • Definitions of the displacement gradient tensor, the strain tensor, and the rotation tensor.
  • Cauchy’s formula (the relation between stress and the net force)
  • The Planck-Einstein relations
  • The most general form of the Schrodinger equation
  • The time-dependent Schrodinger equation in 1D
  • The inner product defined over a Hilbert space, and expansion of a function in terms of its basis set defined in a Hilbert space

An important note on the definition of stress as given in the header image:

I haven’t yet seen this definition in any solid/fluid/continuum mechanics text. So, please treat it with caution.

Also, please do drop me a line if you find it erroneous, problematic, or simply not general enough.

On the other hand, if you run into this definition anywhere, then please do bring the reference to my attention; thanks in advance. [This definition is a part of my planned paper on stress and strain.]


Some of the equations that got left out:

The equations which I would have liked to have in the header, but which got left out for a lack of space, are the following (in no particular order):

  • Newton’s second law defining force
  • Definitions of action (as momentum-dot-displacement and energy-times-time); action as an integral; action as a functional
  • The general equation for the methods of the weighted residuals, and the particular equations for the commonly used test functions (i.e., the Galerkin, the pseudospectral, the least-squares, the method of moments, and the collocation)
  • The Euler identity

Perhaps also, things like:

  • The wavefunction normalization principle, and the Born equation for finding probabilities
  • Structure of probability: simultaneous vs. subsequent events
  • The wave, diffusion and potential equations (juxtaposed with the Schrodinger equation)

On the other hand, some of the equations that are generally of great importance, but which have not come to preoccupy me a lot, are the following:

  • The Euler-Lagrange equations for classical mechanics
  • The Maxwell equations of electrodynamics, supplemented with the “fifth” (i.e. the Lorentz) force equation
  • Boltzmann’s equation, and other equations from statistical mechanics

I must have left out quite a few more in both the lists.

However, I am sure that the three laws of thermodynamics probably would not make it to the header image, despite all their grandeur, their all-encompassing scope.

The reason is this: a computational modeler like me seldom works in a very direct manner with the laws of thermodynamics themselves. These laws do inform his theory; the derivation of the equations he uses indeed are based on them, even if only indirectly. However, the equations he works with happen to be much more detailed (and of far more delimited scope). For instance: the Navier-Stokes system (CFD)—an expression of the first law; the stress-strain fields (FEM)—which makes for merely a part of the internal energy; or the Maxwell system (FDTD)—ditto. Etc.


Further change may be coming:

All in all, I am not quite happy with the top image as it exists right now. … It is too crowded, and speaking from a visual aesthetics point of view, its layout is not well-balanced.

So, on both these counts (too much crowding already, and too many good equations being left out), I am thinking of a further idea: may be I should create a sequence of images, each containing only a few equations, and let the server show you one of them at random. Whaddaya think?


Do check out the “less transient” page:

But yes, if you are interested, check out the “less transient” page too, and let me know if something I wrote in the past should be there or not.


So… does that mean that I’ve gone “mathy”?

Though I exclusively include only equations in the header image—no pictures or visualizations at all, no code, and not much text either—it doesn’t mean that I have gone “mathy”. … Hell, no! Not at all! … Just check out my less transient page [^].


A song I like:

(Hindi) “aankhon aankhon mein hum tum, ho gaye…”
Music: Kalyanji-Anandji
Singers: Kishore Kumar, Asha Bhosale
Lyrics: Anand Bakshi


 

In maths, the boundary is…

In maths, the boundary is a verb, not a noun.

It’s an active something, that, through certain agencies (whose influence, in the usual maths, is wholly captured via differential equations) actually goes on to act [directly or indirectly] over the entirety of a [spatial] region.

Mathematicians have come to forget about this simple physical fact, but by the basic rules of knowledge, that’s how it is.

They love to portray the BV (boundary-value) problems in terms of some dead thing sitting at the boundary, esp. for the Dirichlet variety of problems (esp. for the case when the field variable is zero out there) but that’s not what the basic nature of the abstraction is actually like. You couldn’t possibly build the very abstraction of a boundary unless if first pre-supposed that what it in maths represented was an active [read: physically active] something!

Keep that in mind; keep on reminding yourself at least 10^n times every day, where n is an integer \ge 1.

 


A Song I Like:

[Unlike most other songs, this was an “average” one  in my [self-]esteemed teenage opinion, formed after listening to it on a poor-reception-area radio in an odd town at some odd times. … It changed for forever to a “surprisingly wonderful one” the moment I saw the movie in my SE (second year engineering) while at COEP. … And, haven’t yet gotten out of that impression yet… .]

(Hindi) “main chali main chali, peechhe peeche jahaan…”
Singers: Lata Mangeshkar, Mohammad Rafi
Music: Shankar-Jaikishan
Lyrics: Shailendra


[May be an editing pass would be due tomorrow or so?]

 

A prediction. Also, a couple of wishes…

The Prediction:

While the week of the Nobel prizes always has a way to generate a sense of suspense, of excitement, and even of wonderment, as far as I am concerned, the one prize that does that in the real sense to me is, of course, the Physics Nobel. … Nothing compares to it. Chemistry can come close, but not always. [And, Mr. Nobel was a good guy; he instituted no prize for maths! [LOL!]]. …

The Physics Nobel is the King of all awards in all fields, as far as I am concerned.

That’s why, this year, I have this feeling of missing something. … The reason is, this year’s Physics Nobel is already “known”; it will go to Kip Thorne and pals.

[I will not eat crow even if they don’t get it. [… Unless, of course, you know a delicious recipe or two for the same, and also demonstrate it to me, complete with you sampling it first.]]

But yes, Kip Thorne richly deserves it, and he will get it. That’s the prediction. I wanted to slip it in even if only few hours before the announcement arrives.

I will update this post later right today/tonight, after the Physics Nobel is actually announced.


Now let me come to the couple of wishes, as mentioned in the title. I will try to be brief. [Have been too busy these days… OK. Will let you know. We are going in for accreditation, and so, it’s been all heavy documentation-related work for the past few months. Despite all that hard-work, we still have managed to slip a bit on the progress, and so, currently, we are working on all week-ends and on most public holidays, too. [Yes, we came to work yesterday.] So, it’s only somehow that I manage to find some time to slip in this post—which is written absolutely on the fly, with no second thoughts or re-reading before posting. … So excuse me if there is a bit of lack of balance in the presentation, and of course, typos etc.]


Wish # 1:

The first wish is that a Physics Nobel should go, in a combined way, to what actually are two separate, but very intimately related, and two most significant advances in the physical understanding of man: (i) chaos theory (including fractals) and (ii)catastrophe theory.

If you don’t like the idea of two ideas being given a single Nobel, then, well, let me put it this way: the Nobel should be given for achieving the most significant advancements in the field of the differential nonlinearities, for a very substantial progress in the physical understanding of the behaviour of nonlinear physical systems, forging pathways for predictive capacity.

Let me emphasize, this has been one of the most significant advances in physics in the last century. No, saying so is emphatically not a hyperbole.

And, yes, it’s an advance in physics, primarily, and then, also in maths—but only secondarily.

… It’s unfortunate that an advancement which has been this remarkable never did register as such with most of the S&T “manpower”, esp., engineers and practical designers. It’s also unfortunate that the twin advancement arrived on the scene at the time of bad cultural (even epistemological) trends, and so, the advancements got embedded in a fabric of hyperbole, even nonsense.

But regardless of the cultural tones in which the popular presentations of these advancements (esp. of the chaos theory) got couched, taken as a science, the studies of nonlinearity in the physical systems has been a very, very, original, and a very, very creative, advancement. It needs to be recognized as such.

That way, I don’t much care for what it helped produce on the maths side of it. But yes, even a not very extraordinarily talented undergraduate in CS (one with a special interest in deterministic methods in cryptography) would be able to tell you how much light got shone on their discipline because of the catastrophe and chaos theories.

The catastrophe theory has been simply marvellous in one crucial aspect: it actually pushed the boundaries of what is understood by the term: mathematics. The theory has been daring enough to propose, literally for the first time in the entire history of mankind, a well-refined qualitative approach to an infinity of quantitative processes taken as a group.

The distinction between the qualitative and the quantitative had kept philosophers (and laymen) pre-occupied for millenia. But the nonlinear theory has been the first theoretical approach that tells you how to spot and isolate the objective bases for distinguishing what we consider as the qualitative changes.

Remove the understanding given by the nonlinear theory—by the catastrophe-theoretical approach—and, once in the domain of the linear theory, the differences in kind immediately begin to appear as more or less completely arbitrary. There is no place in theory for them—the qualitative distinctions are external to the theory because a linear system always behaves exactly the same with any quantitative changes made, at any scale, to any of the controlling parameters. Since in the linear theory the qualitative changes are not produced from within the theory itself, such distinctions must be imported into it out of some considerations that are in principle external to the theory.

People often confuse such imports with “applications.” No, when it comes to the linear theory, it’s not the considerations of applications which can be said to be driving any divisions of qualitative changes. The qualitative distinctions are basically arbitrary in a linear theory. It is important to realize that that usual question: “Now where do we draw the line?” is basically absolutely superfluous once you are within the domain of the linear systems. There are no objective grounds on the basis of which such distinctions can be made.

Studies of the nonlinear phenomena sure do precede the catastrophe and the chaos theories. Even in the times before these two theories came on the scene, applied physicists would think of certain ideas such as differences of regimes, esp. in the areas like fluid dynamics.

But to understand the illuminating power of the nonlinear theory, just catch hold of an industrial CFD guy (or a good professor of fluid dynamics from a good university [not, you know, from SPPU or similar universities]), and ask him whether there can be any deeper theoretical significance to the procedure of the Buckingham Pi Theorem, to the necessity, in his art (or science) of having to use so many dimensionless numbers. (Every mechanical/allied engineering undergraduate has at least once in life cursed the sheer number of them.) The competent CFD guy (or the good professor) would easily be at a loss. Then, toss a good book on the Catastrophe Theory to him, leave him alone for a couple of weeks or may be a month, return, and raise the same question again. He now may or may not have a very good, “flowy” sort of a verbal answer ready for you. But one look at his face would tell you that it has now begun to reflect a qualitatively different depth of physical understanding even as he tries to tackle that question in his own way. That difference arises only because of the Catastrophe Theory.

As to the Chaos Theory (and I club the fractal theory right in it), more number of people are likely to know about it, and so, I don’t have to wax a lot (whether eloquently or incompetently). But let me tell you one thing.

Feigenbaum’s discovery of the universal constant remains, to my mind, one of the most ingenious advancements in the entire history of physics, even of science. Especially, given the experimental equipment with which he made that discovery—a handheld HP Calculator (not a computer) in the seventies (or may be in the sixties)! … And yes, getting to that universal constant was, if you ask me, an act of discovery, and not of invention. (Invention was very intimately involved in the process; but the overall act and the end-product was one of discovery.)

So, here is a wish that these fundamental studies of the nonlinear systems get their due—the recognition they so well deserve—in the form of a Physics Nobel.

…And, as always, the sooner the better!


Wish # 2:

The second wish I want to put up here is this: I wish there was some commercial/applied artist, well-conversant with the “art” of supplying illustrations for a physics book, who also was available for a long-term project I have in mind.

To share a bit: Years ago (actually, almost two decades ago, in 1998 to be precise), I had made a suggestion that novels by Ayn Rand be put in the form of comics. As far as I was concerned, the idea was novel (i.e. new). I didn’t know at that time that a comics-book version of The Fountainhead had already been conceived of by none other than Ayn Rand herself, and it, in fact, had also been executed. In short, there was a comics-book version of The Fountainhead. … These days, I gather, they are doing something similar for Atlas Shrugged.

If you think about it, my idea was not at all a leap of imagination. Newspapers (even those in India) have been carrying comic strips for decades (right since before my own childhood), and Amar Chitrakatha was coming of age just when I was. (It was founded in 1967 by Mr. Pai.)

Similarly, conceiving of a comics-like book for physics is not at all a very creative act of imagination. In fact, it is not even original. Everyone knows those books by that Japanese linguistics group, the books on topics like the Fourier theory.

So, no claim of originality here.

It’s just that for my new theory of QM, I find that the format of a comics-book would be most suitable. (And what the hell if physicists don’t take me seriously because I put it in this form first. Who cares what they think anyway!)

Indeed, I would even like to write/produce some comics books on maths topics, too. Topics like grads, divs, curls, tensors, etc., eventually. … Guess I will save that part for keeping me preoccupied during my retirement. BTW, my retirement is not all that far away; it’s going to be here pretty soon, right within just five years from now. (Do one thing: Check out what I was writing, say in 2012 on this blog.)

But the one thing I would like write/produce right in the more immediate future is: the comics book on QM, putting forth my new approach.

So, in the closing, here is a request. If you know some artist (or an engineer/physicist with fairly good sketching/computer-drawing skills), and has time at hand, and has the capacity to stay put in a sizeable project, and won’t ask money for it (a fair share in the royalty is a given—provided we manage to find a publisher first, that is), then please do bring this post to his notice.

 


A Song I Like:

And, finally, here is the Marathi song I had promised you the last time round. It’s a fusion of what to my mind is one of the best tunes Shrinivas Khale ever produced, and the best justice to the words and the tunes by the singer. Imagine any one else in her place, and you will immediately come to know what I mean. … Pushpa Pagdhare easily takes this song to the levels of the very best by the best, including Lata Mangeshkar. [Oh yes, BTW, congrats are due to the selection committe of this year’s Lata Mangeshkar award, for selecting Pushpa Pagdhare.]

(Marathi) “yeuni swapnaat maajhyaa…”
Singer: Pushpa Pagdhare
Music: Shrinivas Khale
Lyrics: Devakinandan Saraswat

[PS: Note: I am going to come back and add an update once this year’s Physics Nobel is announced. At that time (or tonight) I will also try to streamline this post.

Then, I will be gone off the blogging for yet another couple of weeks or so—unless it’s a small little “kutty” post of the “Blog-Filler” kind or two.]

 

A flip, but not a flop…

“Why is it that when you look in the mirror, the left and right directions appear flipped, but not the up and down?”


Stop reading!

Do not read further until you have honestly tried answering that question!


The question was asked at the Physics StackExchange.

As often is the case, using only text is not at all good when it comes to explaining physics [^]; adding figures does help [^]. And then, animations are even better at it than having just “dead” (static) figures. Going further, interactive graphics, which let the user participate in manipulating the presentation of information, of course beats those mere animations. Better than that, if possible, is an actual demonstration in real life, accompanied by an explanation using simple words.

…As far as the above question is concerned, the Physics Girl [^] does a fairly good job [^].

The best mode of teaching-learning, of course, is an actual and immediate interaction with a person, who in turn might use (and allow you to use) any and all of the above options!

And that’s the reason why, regardless of how much technology progresses, the actual person-to-person type of teaching will never go out of business.


A Video I Liked:

A `Thought Leader’ gives a talk that will inspire your thoughts: [^]

 

 

Shaken, because of a stir

We have demonstrably been shaken here on earth, because of a stir in the cosmos.

The measured peak strain was 10^{-21} [^].

For comparison: In our college lab, we typically measure strains of magnitude like 10^{-3} or at the most 10^{-4}. (Google search on “yield strain of mild steel” does not throw up any directly relevant page, but it does tell you that the yield strength of mild steel is 450 MPa, and all mechanical (civil/metallurgical/aero/etc.) engineers know that Young’s modulus for mild steel is 210 GPa. … You get the idea. …)


Einstein got it wrong twice, but at least eventually, he did correct himself.

But other physicists (and popular science writers, and blog-writers), even after getting a full century to think over the issue, still continue to commit blunders. They continue using terms like “distortions of spacetime.” As if, space and time themselves repeatedly “bent” (or, to use a euphemism, got “distorted”) together, to convey the force through “vacuum.”

It’s not a waving of the “spacetime” through a vaccum, stupid! It’s just the splashing of the aether!!


The Indian credit is, at the most, 1.3%.

If it could be taken as 3.7%, then the number of India’s science Nobels would also have to increase dramatically. Har Gobind Singh Khorana, for instance, would have to be included. The IAS-/MPSC-/scientist-bureaucrats “serving” during my childhood-days had made sure to include Khorana’s name in our school-time science text-books, even though Khorana had been born only in (the latter-day) Pakistan, and even if he himself had publicly given up on both Pakistan and India—which, even as children, we knew! Further, from whatever I recall of me and all my classmates (from two different schools), we the (then) children (and, later, teen-agers) were neither inspired nor discouraged even just a tiny bit by either Khorana’s mention or his only too willing renunciation of the Indian citizenship. The whole thing seemed too remote to us. …

Overall, Khorana’s back-ground would be a matter of pride etc. only to those bureaucrats and possibly Delhi intellectuals (and also to politicians, of course, but to a far lesser extent than is routinely supposed). Not to others.

Something similar seems to be happening now. (Something very similar did happen with the moon orbiter; check out the page 1 headlines in the government gazettes like Times of India and Indian Express.)

Conclusion: Some nut-heads continue to run the show from Delhi even today—even under the BJP.

Anyway, the reason I said “at most” 1.3 % is because, even though I lack a knowledge of the field, I do know that there’s a difference between 1976, and, say, 1987. This fact by itself sets a natural upper bound on the strength of the Indian contribution.

BTW, I don’t want to take anything away from Prof. Dhurandhar (and from what I have informally gathered here in Pune, he is a respectable professor doing some good work), but reading through the media reports (about how he was discouraged 30 years ago, and how he has now been vindicated today etc.) made me wonder: Did Dhurandhar go without a job for years because of his intellectual convictions—the way I have been made to go, before, during and after my PhD?

As far as I am concerned, the matter ends there.

At least it should—I mean, this post should end right here. But, OK, let me make an exception, and note a bit about one more point.


The experimental result has thrown the Nobel bookies out of business for this year—at least to a great part.

It is certain that Kip Thorne will get the 2016 Physics Nobel. There is no uncertainty on that count.

It is also nearly as certain that he will only co-win the prize—there will be others to share the credit (and obviously deservingly so). The only question remaining is, will it be just one more person or will it be two more (Nobel rules allow only max 3, I suppose), what will be their prize proportions, and who those other person(s) will be (apart from Thorne). So, as far as the bettors and the bookies are concerned, they are not entirely out of the pleasure and the business, yet.

Anyway, my point here was twofold: (i) The 2016 Physics Nobel will not be given for any other discovery, and (ii) Kip Thorne will be one of the (richly deserving) recipients.


[E&OE]

 

Yo—4: The 2014 Physics Nobel

The physics Nobel for this year has gone to Isamu Akasaki, Hiroshi Amano and Shuji Nakamura, for the invention of the blue LED. Check out the official Nobel prize page to see the huge cost savings the invention implies [^]. Very, very, very well deserved!

Congratulations to all the winners.

Their invention has already begun transforming our lives, and, definitely, much more is slated to come. Just to name two: (i) flexible LEDs, (ii) lost-cost designer walls smartly emitting diffuse light (so that the light (in the sense lamp) is not different from the wall, the wall itself is light)… The possibilities are just astounding… For instance, think how it might affect (i) buildings architecture and interior designing, (ii) lighting inside tunnels/underground transportation…. You couldn’t wish for more…

Or, may be, you could! (Hey, this is science and this is life… There is always scope for more!) … If you ask me to single out just one thing, as far as these light- and energy-related matters go, I would choose: cost-effective (i.e. scalable and high energy-density) artificial photo-synthesis. … But then, that is strictly for another day.

As of today, do pause to note here that the realization of the blue LED also took something like three decades! Enormous achievement, that!!

Congratulations to the winners, once again!

* * * * *   * * * * *   * * * * *

A Song I Like:
(Hindi) “pyaar baanTate chalo…”
Singer: Kishore Kumar
Music: Laxmikant Pyarelal
Lyrics: Asad Bhopali
[BTW, if some of you don’t like some part of the lyrics, or of the video, then that can be OK by me, I can understand. Just don’t let it spoil this song itself for you, though. [And, also avoid the cynical temptation to associate this song with the money-distribution that does often go on, at the time of elections.] Instead, just take this song as a song, and appreciate that joyous sense which it carries, the sense of innocence and benevolence which comes so abundantly overflowing from it.]

[E&OE]

 

The most economic particles model of a[n utterly] fake fluid—part 1

Real fluids are viscous.

Newton was the first to formulate a law of viscosity; his law forms an essential part of the engineering fluid mechanics even today.

The way the concept of viscosity is usually presented to undergraduates is in reference to a fluid moving over a horizontal solid surface, e.g., water flowing over a flat river-bed. The river-bed itself is, of course, stationary. The students are then asked to imagine a laminar flow in which the horizontal layers of fluid go slipping past each other with different velocities. The viscous forces between the fluid layers tend to retard their relative motion. Now, under the assumption that the layer adjacent to the stationary solid surface has zero velocity, and that the flow is laminar, a simple parabolic profile is obtained for the velocity profile. The velocity progressively increases from 0 at the solid surface to some finite mainstream value as you go up and away from the horizontal solid surface. Newton’s law is then introduced via the equation:

\tau \propto \dfrac{dU}{dy}

where \tau is the shear stress between the fluid layers slipping past each other, and \frac{dU}{dy} is the velocity gradient along the vertical direction. The constant of proportionality is viscosity, \mu:

\mu \equiv \dfrac{\tau}{\left(\dfrac{dU}{dy}\right)}

This picture of layers of fluids slipping past with progressively greater velocities, as in a deck of card given a gentle horizontal push, is easy to visualize; it helps people visualize what otherwise is not available to direct perception.

That’s quite fine, but then, as it happens, sometimes, concrete pictures also tend to over-concretize the abstract ideas. The above mentioned picture of viscosity is one of these.

You see, the trouble is, people tend to associate viscosity to be operative only in this shear mode. They can’t readily appreciate the fact that viscous forces also arise in the normal direction. The reason is, they can’t as easily imagine velocity gradients along the flow direction. No engineering (or physics) text-book ever shows a diagram illustrating the action of viscosity along the direction of the flow.

One reason for that, in turn, is that while in solids stress depends on the extent of deformation, in fluids, it depends on the rate of deformation. Indeed the extent of possible deformation, in fluids, is theoretically undefined (or infinite, if you wish). A fluid will continually go on changing its shape so long as a shear stress is applied to it… It’s easily possible to pour water from a tap onto a tilted plate, and then, from that plate onto the bottom of a kitchen sink, without any additional stress coming into picture as the water continuously goes on deforming in the act of pouring. The fact that water has already suffered deformation while being poured from the tap to the plate, does not hinder the additional deformation that it further suffers while falling off from the plate. And, all this deformation, inasmuch as it involves a change of the initial shape, involves only shear. And, as to the stress, when it comes to fluids, the extent of deformation does not matter; the rate of deformation—or the velocity gradient—does. Stress in fluids is related to the velocity gradient, not to the deformation gradient (as in solids).

Another, related, reason for the difficulty in visualizing viscosity appearing in the normal direction is that, in our usual imagination, we can’t visualize fluids being gripped from its ends and pulled apart, the way solids (e.g. rubber band) can be. The trouble is not in the stretching part of it; the trouble is in the “being held” part of it: you can’t grab of a piece of a fluid in exactly the same way as you can, say, a bite of food. There is no bite of water, only a gulp of it. But the practical impossibility of holding fast onto an end of a fluid also carries over when it comes to imagining fluids being stretched purely along the normal direction, i.e., without involving shear.

Of course, as far as exerting a normal force to a fluid is concerned, people have no difficulty imagining that. You can always exert a compressive normal force on a fluid, by applying a pressure. But then, that is only a compressive force, and, a static situation. You don’t have to have spatially varying velocities to arrive at the concept of pressure—indeed, you don’t make any reference to the very idea of velocity, in that concept. Pressure refers to static forces.

Now, when people try to visualize velocity gradients in the normal direction, they unwittingly tend to take the visualization on the lines parallel to the viscosity-defining picture. So, they take, say, a 10 m/s velocity vector at origin, an 8 m/s velocity vector at the point x = 1, a 6 m/s vector at x = 2, and so on. Soon, they end up imagining having a zero magnitude velocity vector.

But this is a poorly imagined situation because it can never be realized in one-dimension—quantitatively, it violates the mass conservation principle, i.e. the continuity equation (at least for the incompressible 1D flow without sources/sinks, it does).

Now, when pushed further, people do end up imagining an `L’ kind of bend in a pipe (or a fluid bifurcating at a`T’ joint), i.e., taking velocity vectors to be just x-components of a 2D/3D velocity field.

But, speaking in general terms, at least in my observation, people still can’t easily imagine viscosity being defined in reference to velocity gradients along the direction of the flow. Many engineers in fact express a definite surprise at such a definition of viscosity. The only picture ever presented to them refers to the shear deformation, and given the peculiar nature of fluids, velocity gradients in the normal direction (i.e. along the flow) are not as easy to visualize unless you are willing to break continuity.

Recently when Prof. Suo wrote an iMechanica post about viscosity (in reference to a course he is currently teaching at Harvard), the above-mentioned observations came rushing to my mind, and that’s how I had a bit of discussion with him on this topic, here [^].

As mentioned in that discussion, to help people visualize the normal viscosity, I then thought of introducing a particles model of fluid, specifically, the Lennard-Jones (LJ) fluid [^]. It also goes well with my research interests concerning the particles approaches to fluids.

But then, of course, I have been too busy just doing the class-room teaching this semester, and find absolutely no time to pursue anything other than that—class-room teaching, or preparation for the same, or follow-up activities concerning the same (e.g. designing assignments, unit tests, etc.). But no time at all is left for research, blogging, or why, even just building a few software toys at home. (As a matter of fact, I find myself hard-pressed to find time even for just grading of unit-test answer-books.)

Therefore, writing some quick-n-simple illustrative software (actually, completing writing this software—something which I had began last summer) was out of the question. Still, I wanted to steal some time, to think about this question.

I therefore decided to drastically simplify the matters. I would work on the problem, but only to the extent that I can work on it off my head (i.e. without using even paper and pencil, let alone a computer or a software)—that’s what I decided.

So, instead of taking the (1/r)^{12} - (1/r)^6 potential, I began wondering what if I take a simple 1/r attractive potential (as in Newtonian gravity). After all, most every one knows about the inverse-square law, and so, it would be easier for people to make the conceptual connections if a fluid could be built also out of the plain inverse-distance potential.

So, the question was: (Q1) if I take a few particles with (only attractive) gravitational interactions among them—would they create a fluid out of them, just the way the LJ potential does? And if the answer is yes, then would these particles also create a solid out of them, too, just the way the LJ potential does?

Before you rush into an affirmative answer, realize here that the LJ potential carries both attractive and repulsive terms, whereas the gravitational interaction is always only attractive.

But, still, suppose such a hypothetical fluid is possible, then, (Q2) what would distinguish this hypothetical fluid from its corresponding solid? How precisely would the phase transition between the solid and fluid occur? For instance, how would the fluid consisting of only gravity-interacting particles, melt or solidify?

And, (Q3) what is the minimum number of such particles that must be present before they can create a solid? a fluid? a liquid? a gas?

Of course, answering these questions is not a big deal (neither is thinking up these questions). The point is, I had some fun thinking along these lines, in whatever time I could still find.

However, since this post is already more than a thousand words-long, let me stop here, and ask you to think about the above mentioned questions. In my next post, I will give my answers to them. In the meanwhile, think about it, have fun, and if you think you have got an answer that you could share with me, feel free to drop a comment or an email.

* * * * *   * * * * *   * * * * *

A Song I Like:
(Marathi) “waaT ithe swapnaatil sampali jaNu…”
Singer: Suman Kalyanpur
Music: Ashok Patki
Lyrics: Ashok Paranjape

[E&OE]

 

 

Some Interesting Reading (October 2013)

Concerning our series of posts on the concept of space, I have jotted down my thoughts on paper, but not yet made any progress on creating the diagrams to go with them. (Plain laziness.) Hence, the delay in posting it.

In the meanwhile, here are a few links to some reading that I found interesting over the past few days (in no particular order).

1. R. J. Lipton of GeorgiaTech on how “Teaching helps research” [^]

2. Ricardo Heras, “Individualism: The legacy of great physicists,” arXiv:1310.7326 [physics.pop-ph] [^]. Heras is a first year graduate student at University College, London. Check out the Fermi quote at the end of this paper. (And, also, the quote by Max Planck at the opening.)

3. Roger Schlafly puts in one place all the links to his blog posts updating his book “How Einstein Ruined Physics,” [^].

4. Tony Rothman, “Lost in Einstein’s shadows” [^]

5. Physics World, 25th Anniversary Issue, available for free downloads [^] (HT to QuantumFrontiers [^]). This special issue has the magazine’s lists of 5 images, 5 discoveries, 5 questions, 5 spin-offs, and 5 people, that mattered over the last 25 years.

6. Paul G. Kwiat, on what he calls it “interaction-free measurement” [^]. You think it’s mysterious? (LOL!)

And, in place of the usual “A Song I Like” section, yet another link!:

7. Tom Swanson, a physicist himself, offers physicists a horoscope [^]

[E&OE]

The 2012 Physics Nobel

I happened to have been online already, already browsing here and there, at the time that the 2012 Nobel for Physics was about to be announced, though I didn’t know about the precise timing of it. Still, the very first time yesterday when I browsed their official Web site, I saw the counter go down in the seconds!: 6, 5, 4, 3, 2, 1, 0! I couldn’t believe the sheer timing of it. But, then, nothing happened, not for quite some time, may be for more than half a minute or so. The Web site froze, in a way. That could have happened because I had clicked on that video link, I thought. I checked elsewhere on the Web, but there wasn’t anything yet on Yahoo!, Google, or even on Twitter! Anyway, to try something, I did a quick Google search, found another Web page (at the same official server) for the prize announcements, opened that link, and even before noticing that it was a text-based Web page without any video embedded in it, it so happened that I had already read the names: Haroche and Wineland. … Wow!

Actually, I don’t have a right to squeak out as much excited a “wow” as the above description seems to suggest. … Frankly, I was not directly familiar with either’s work. Despite quantum foundations being a part of my direct research interests, I had not read their papers in original, nor had come across any expert commentaries written with a great flourish by someone else. As a matter of a definite fact, Wineland’s name had only a vague recall in my mind; Haroche’s had almost none.

I had to go through the Nobel award’s description and visit their Web sites before getting to know about the nature of their work. When I did that, I began to think that the Nobel committee’s choice this time round seems to be exceptionally good. The more I thought about it, the more convinced I grew about my conclusion.

Which is unlike the last year’s physics Nobel. Last year, I had thought that Steinhardt should have shared the Nobel with Shechtman. The following is what I had mentioned about it, at Prof. Scott Aaronson’s blog [^] last year (a comment which went without any reply to it):

Re. The quasicrystals Nobel.

This happened to be the very first time (and it could easily remain the only time) that I could understand a Nobel-winning work right on the day of its announcement, i.e., without having to have someone else explain anything about the work to me. … Crystal structures is something I had studied as an UG (82–83), and I had courses on diffraction techniques (X rays and TEM) during my graduate studies (90–93).

I do think that Prof. Schechtman’s Nobel was well deserved.

However, I also wonder if others, esp. Steinhardt (perhaps together with someone else like Levine) might not have shared the second half of it (with Shechtman receiving the first half).

A Nobel to a mathematician (like Penrose, or even Alan Mackay) is out of the question, of course. However, I do think that the connections which Steinhardt and Levine made, right in 1984, were germane enough and early enough—and therefore, significant enough. Even later on, Steinhardt continued producing high-quality results relevant to this overall discovery. I think his contributions could have been recognized. I am not sure why the committee left him out. (BTW, do the Nobel rules allow for a 2/3 + 1/3 division between just two people?)

Ajit
[E&OE]

In contrast to the last year’s physics Nobel, the more I gathered about Haroche and Wineland’s respective works, the more confident I grew about the astuteness of the choice made by this year’s Nobel committee. (Oh, BTW, that famous three berillium atoms pic was, of course, an immediately recognizable sight; it’s just that the researcher’s name had somehow slipped into the background.)

OK. An aside. I had planned writing this post without talking about this side-matter, which I am going to do now. I decided to include it because I did a quick check on how other bloggers were reacting to the news. I noticed that Dr. Ashutosh Joglekar (who maintains the blog “Curious Wavefunction” at the Scientific American’s Web site) has brought out the relevant issue at Prof. Aaronson’s blog [^], and that the latter has responded to it the way he did [^].

Here is Ashutosh Joglekar’s comment:

It’s interesting that this well-deserved prize went to Heroche and Wineland when so many predictions centered on another experimental quantum mechanics paradigm; the work of Aspect, Zeilinger and Clauser. I wonder if this just means that the committee was not as confident of their work.

Here is the relevant excerpt from Scott Aaronson’s reply:

Curious #1: I’ve long thought that a prize should go to Aspect, Zeilinger and Clauser for experimental demonstration of the Bell violations. Maybe it will eventually.

My opinion is different.

I think that Zeilinger et al’s work does not necessarily deserve a Nobel. Theirs is a very good, very high quality, research work. They may even get a physics Nobel some day in future (and if they do, I might even congratulate them). Yet, the fact remains, in the heart of my hearts, if I try to take a sweeping survey of all the physics Nobels thus far, and thereby try to get some idea, form some implicit judgment, about the kind of standards that the prize is expected to keep, I would think that Zeilinger et al’s work falls short of it. Also Aspect’s. Or Clauser’s.

I, of course, had some good idea about the works of the last three. (Not a comprehensive idea, but still, a definite idea about their main results.) Like Joglekar and Aaronson (and many others), I, too, had this feeling that Zeilinger/Aspect/Clauser may be the front-runners in the race for this year’s Nobel. (The discovery of the Higgs boson was already pointed out by many people as not expected to be a front-runner right this year, out of many reasons: one being that the discovery is too new; and the other, that the prize could be given to only three people at the most, whereas there would be many more proper claimants for that discovery.) So, Aspect/Zeilinger/Clauser could be expected to be in the forefront, and this was a feeling I did share with others.

But, unlike most others, I also had the feeling that if this trio got the Nobel, it somehow wouldn’t be right. That the choice would somehow be tantamount to showing an understanding of progress in physics that is more superficial, less profound, than what is required by its history. That it therefore would be tantamount to giving a wrong direction to the future physics research. That it would mean that the results are not too unexpected, the researches that do not truly probe deeply into the unknown, the ideas that do not cover something truly profound and fundamental, could also get rewarded.

Just the way Einstein must not have been given the Nobel for his relativity, or Eddington, for its “verification” (to whatever degree of accuracy and precision he did “accomplish” it), similarly, I have come to believe that Bell must not have been given a Nobel for his proposal of that inequalities-related test, and that Aspect/Zeilinger/others should not be given a Nobel regarding its verification—or the related aspects.

Speaking at least only of scientists (and not necessarily of the common man), just like Einstein, these quantum computing guys certainly have managed to garner a lot of good “press” among the scientists. And, just like with Einstein’s relativity, theirs does not represent a really fundamental advancement of science—only a smart repackaging of the already known theoretical fundamentals, no matter how unintuitive or surprising the repackaging might seem to some.

Another example: Newton’s identification of the second-order differential with force, was an astounding, fundamental, result. Galileo had tried to reach it, but could not do so because he lacked the mathematical rigor of Newton. And, Newton didn’t simply stop  there. He applied the principle to the particle mechanics, and then went further to formulate the theory of universal gravitation. The idea of the force was thus well isolated and clearly established—by Newton. Later, when d’Alembert came on the scene and formulated his principle, it certainly was a welcome development. The conceptual shift that d’Alembert initiated was enormous—it remains at the bedrock on which the Lagrangian mechanics, and hence the Hamiltonian mechanics, and hence all the mathematical formulations of the mainstream quantum mechanics rest (or, at any rate, at least these do: Schrodinger’s, Dirac’s, Feynman’s). Even then, it’s clear that d’Alembert’s work is only a repackaging of Newton’s mechanics—though not as unimportant as being just a footnote to it (as some have argued).

I do believe that Bell’s theorem is something similar. It might encapsulate an enormously useful shift of a perspective, but it is nothing more than that—a shift of a viewpoint. The thing that is being viewed is still the same—our good old quantum mechanics. It’s still just a repackaging of the same theory. And so are the experimental works connected more specifically with only Bell’s theorem.

In contrast, I do think that the Nobel committee has done an exceptionally great job in picking up these two people, who, by way of mere technical classification, come from the same broad field: quantum optics.

From whatever I have found about their work so far, the committee seems to have done a wonderful job right up to the order in which their names appear—if the order is significant in any way. Reason? While studying the interaction of particles matter and light at the single particle level, there obviously are only two choices: choose matter as the main object of the study, or choose light. In between the two, choosing photons is more audacious. Why? Since photons obey the Bose statistics, they are, following the mainstream QM, “indistinguishable” entities. Which is unlike the fermions. To think of according an indistinguishable particle, a mere “quantized normal mode,” the status of a first-class object would have required, I guess, a slightly more audacious thought process. That’s why. Of course, if at all so fine a distinction at all matters anywhere in the ingenuity of their whole research programs!

Anyway, congratulations, Dr. Haroche and Dr. Wineland! Your researches obviously lie at the metaphorical cross-roads of so many different things; it can be described by multiple attributes like: long-range in terms of the overall design of research program, refined and inspired in terms of its execution, painstaking and ingenious in terms of experimental design, stunning in terms of the kind of results that have been obtained, and definitely promising in terms of the potential impact of the findings in the long-range future. Why, forget the watchmakers, there wouldn’t be a single quantum computing guy who would fail to acknowledge your precedence.

And, your research programs have not been too concerned with getting a good press, a policy which seems to have been effectively maintained over many long years. Well done!

Congratulations, again!

* * * * *    * * * * *    * * * * *

BTW, I noticed that Dr. Wineland has a BS from Berkeley. … It only goes on to show that no matter what kind of educational background you come from, if you are good and persistent and continuously engaged in improving yourself, then, one day, you will surely succeed!

* * * * *    * * * * *    * * * * *

Last Sunday, I appeared for an interview for a professor’s post in a college affiliated with the University of Pune, but the interview got aborted out of the same, old, tiresome, metallurgy-vs-mechanical issue. A separate post is due.

Yes, I still go jobless.

* * * * *    * * * * *    * * * * *

A Song I Don’t Like:
Since I go jobless, the section on a song I like was simply out of the question. So, I thought of adding this alternative section, which will be included on an irregular basis henceforth so long as I go without a job. It will mention the songs I don’t like. In fact, these are among the songs I never liked. Here is a song which falls into that category. I liked nothing about it—neither the audio nor the video. Neither the words nor the music nor the settings nor the choreography nor the actress nor her gestures nor her expressions nor the movie (which I haven’t watched, obviously) nor… you get the idea. Indeed, if you think that here is a song I hate, you wouldn’t be too off the mark. …

And the song is:
(Hindi) “mere haathon mein nau nau chuDiyaan…”

(BTW, in such a section, for the songs I don’t like, for obvious reasons, I won’t bother to find out and provide all the “credit”s.)

[First published on October 10, 2012, 6:50 PM IST. Streamlined a bit and updated (with just a few additions) on October 11, 2012, 1:45 PM IST. Another minor revision, on October 12, 2012, 6:15 PM IST. I guess I will now really leave it as is, regardless of grammatical errors etc.]

[E&OE]