The Infosys Prizes, 2015

I realized that it was the end of November the other day, and it somehow struck me that I should check out if there has been any news on the Infosys prizes for this year. I vaguely recalled that they make the yearly announcements sometime in the last quarter of a year.

Turns out that, although academic bloggers whose blogs I usually check out had not highlighted this news, the prizes had already been announced right in mid-November [^].

It also turns out also that, yes, I “know”—i.e., have in-person chatted (exactly once) with—one of the recipients. I mean Professor Dr. Umesh Waghmare, who received this year’s award for Engineering Sciences [^]. I had run into him in an informal conference once, and have written about it in a recent post, here [^].

Dr. Waghmare is a very good choice, if you ask me. His work is very neat—I mean both the ideas which he picks out to work on, and the execution on them.

I still remember his presentation at that informal conference (where I chatted with him). He had talked about a (seemingly) very simple idea, related to graphene [^]—its buckling.

Here is my highly dumbed down version of that work by Waghmare and co-authors. (It’s dumbed down a lot—Waghmare et al’s work was on buckling, not bending. But it’s OK; this is just a blog, and guess I have a pretty general sort of a “general readership” here.)

Bending, in general, sets up a combination of tensile and compressive stresses, which results in the setting up of a bending moment within a beam or a plate. All engineers (except possibly for the “soft” branches like CS and IT) study bending quite early in their undergraduate program, typically in the second year. So, I need not explain its analysis in detail. In fact, in this post, I will write only a common-sense level description of the issue. For technical details, look up the Wiki articles on bending [^] and buckling [^] or Prof. Bower’s book [^].

Assuming you are not an engineer, you can always take a longish rubber eraser, hold it so that its longest edge is horizontal, and then bend it with a twist of your fingers. If the bent shape is like an inverted ‘U’, then, the inner (bottom) surface has got compressed, and the outer (top) surface has got stretched. Since compression and tension are opposite in nature, and since the eraser is a continuous body of a finite height, it is easy to see that there has to be a continuous surface within the volume of the eraser, some half-way through its height, where there can be no stresses. That’s because, the stresses change sign in going from the compressive stress at the bottom surface to the tensile stresses on the top surface. For simplicity of mathematics, this problem is modeled as a 1D (line) element, and therefore, in elasticity theory, this actual 2D surface is referred to as the neutral axis (i.e. a line).

The deformation of the eraser is elastic, which means that it remains in the bent state only so long as you are applying a bending “force” to it (actually, it’s a moment of a force).

The classical theory of bending allows you to relate the curvature of the beam, and the bending moment applied to it. Thus, knowing bending moment (or the applied forces), you can tell how much the eraser should bend. Or, knowing how much the eraser has curved, you can tell how big a pair of fforces would have to be applied to its ends. The theory works pretty well; it forms of the basis of how most buildings are designed anyway.

So far, so good. What happens if you bend, not an eraser, but a graphene sheet?

The peculiarity of graphene is that it is a single atom-thick sheet of carbon atoms. Your usual eraser contains billions and billions of layers of atoms through its thickness. In contrast, the thickness of a graphene sheet is entirely accounted for by the finite size of the single layer of atoms. And, it is found that unlike thin paper, the graphen sheet, even if it is the the most extreme case of a thin sheet, actually does offer a good resistance to bending. How do you explain that?

The naive expectation is that something related to the interatomic bonding within this single layer must, somehow, produce both the compressive and tensile stresses—and the systematic variation from the locally tensile to the locally compressive state as we go through this thickness.

Now, at the scale of single atoms, quantum mechanical effects obviously are dominant. Thus, you have to consider those electronic orbitals setting up the bond. A shift in the density of the single layer of orbitals should correspond to the stresses and strains in the classical mechanics of beams and plates.

What Waghmare related at that conference was a very interesting bit.

He calculated the stresses as predicted by (in my words) the changed local density of the orbitals, and found that the forces predicted this way are way smaller than the experimentally reported values for graphene sheets. In other words, the actual graphene is much stiffer than what the naive quantum mechanics-based model shows—even if the model considers those electronic orbitals. What is the source of this additional stiffness?

He then showed a more detailed calculation (i.e. a simulation), and found that the additional stiffness comes from a quantum-mechanical interaction between the portions of the atomic orbitals that go off transverse to the plane of the graphene sheet.

Thus, suppose a graphene sheet is initially held horizontally, and then bent to form an inverted U-like curvature. According to Waghmare and co-authros, you now have to consider not just the orbital cloud between the atoms (i.e. the cloud lying in the same plane as the graphene sheet) but also the orbital “petals” that shoot vertically off the plane of the graphene. Such petals are attached to nucleus of each C atom; they are a part of the electronic (or orbital) structure of the carbon atoms in the graphene sheet.

In other words, the simplest engineering sketch for the graphene sheet, as drawn in the front view, wouldn’t look like a thin horizontal line; it would also have these small vertical “pins” at the site of each carbon atom, overall giving it an appearance rather like a fish-bone.

What happens when you bend the graphene sheet is that on the compression side, the orbital clouds for these vertical petals run into each other. Now, you know that an orbital cloud can be loosely taken as the electronic charge density, and that the like charges (e.g. the negatively charged electrons) repel each other. This inter-electronic repulsive force tends to oppose the bending action. Thus, it is the petals’ contribution which accounts for the additional stiffness of the graphene sheet.

I don’t know whether this result was already known to the scientific community back then in 2010 or not, but in any case, it was a very early analysis of bending of graphene. Further, as far as I could tell, the quality of Waghmare’s calculations and simulations was very definitely superlative. … You work in a field (say computational modeling) for some time, and you just develop a “nose” of sorts, that allows you to “smell” a superlative calculation from an average one. Particularly so, if your own skills on the calculations side are rather on the average, as happens to be the case with me. (My strengths are in conceptual and computational sides, but not on the mathematical side.) …

So, all in all, it’s a very well deserved prize. Congratulations, Dr. Waghmare!

 


A Song I Like:

(The so-called “fusion” music) “Jaisalmer”
Artists: Rahul Sharma (Santoor) and Richard Clayderman (Piano)
Album: Confluence

[As usual, may be one more editing pass…]

[E&OE]

Blogging some crap…

I had taken a vow not to blog very frequently any more—certainly not any more at least right this month, in April.

But then, I am known to break my own rules.

Still, guess I really am coming to a point where quite a few threads on which I wanted to blog are, somehow, sort of coming to an end, and fresh topics are still too fresh to write anything about.

So, the only things to blog about would be crap. Thus the title of this post.

Anyway, here is an update of my interests, and the reason why it actually is, and also would be, difficult for me to blog very regularly in the near future of months, may be even a year or so. [I am being serious.]

1. About micro-level water resources engineering:

Recently, I blogged a lot about it. Now, I think I have more or less completed my preliminary studies, and pursuing anything further would take a definitely targeted and detailed research—something that only can be pursued once I have a master’s or PhD student to guide. Which will only happen once I have a job. Which will only happen in July (when the next academic term of the University of Mumbai begins).

There is only one idea that I might mention for now.

I have installed QGIS, and worked through the relevant exercises to familiarize myself with it. Ujaval Gandhi’s tutorials are absolutely great in this respect.

The idea I can blog about right away is this. As I mentioned earlier, DEM maps with 5 m resolution are impossible to find. I asked my father to see if he had any detailed map at sub-talukaa level. He gave me an old official map from GSI; it is on a 1:50000 scale, with contours at 20 m. Pretty detailed, but still, since we are looking for check-dams of heights up to 10 m, not so helpful. So, I thought of interpolating contours, and the best way to do it would be through some automatic algorithms. The map anyway has to be digitized first.

That means, scan it at a high enough resolution, and then perform a raster to vector conversion so that DEM heightfields could be viewed in QGIS.

The trouble is, the contour lines are too faint. That means, automatic image processing to extract the existing contours would be of limited help. So, I thought of an idea: why not lay a tracing paper on top, and trace out only the contours using black pen, and then, separately scan it? It was this idea that was already mentioned in an official Marathi document by the irrigation department.

Of course, they didn’t mean to go further and do the raster-to-vector conversion and all.  I would want to adapt/create algorithms that could simulate rainfall run-offs after high intensity sporadic rains, possibly leading also to flooding. I also wanted to build algorithms that would allow estimates of volumes of water in a check dam before and after evaporation and seepage. (Seepage calculations would be done, as a first step, after homogenizing the local geology; the local geology could enter the computations at a more advanced stage of the research.) A PhD student at IIT Bombay has done some work in this direction, and I wanted to independently probe these issues. I could always use raster algorithms, but since the size of the map would be huge, I thought that the vector format would be more efficient for some of these algorithms. Thus, I had to pursue the raster-to-vector conversion.

So I did some search in this respect, and found some papers and even open source software. For instance, Peter Selinger’s POTrace, and the further off-shoots from it.

I then realized that since the contour lines in the scanned image (whether original or traced) wouldn’t be just one-pixel wide, I would have to run some kind of a line thinning algorithm.

Suitable ready made solutions are absent and building one from the scratch would be too time consuming—it can possibly be a good topic for a master’s project in the CS/Mech departments, in the computer graphics field. Here is one idea I saw implemented somewhere. To fix our imagination, launch MS Paint (or GIMP on Ubuntu), and manually draw a curve in a thick brush, or type a letter in a huge font like 48 points or so, and save the BMP file. Our objective is to make a single pixel-thick line drawing out of this thick diagram. The CS folks apparently call it the centerlining algorithm. The idea I saw implemented was something like this: (i) Do edge detection to get single pixel wide boundaries. The “filled” letter in the BMP file would now become “hollow;” it would have only the outlines that are single pixel wide. (ii) Do raster-to-vector conversion, say using POTrace, on this hollow letter. You would thus have a polygon representation for the letter. (iii) Run a meshing software (e.g. Jonathan Schewchuk’s Triangle, or something in the CGAL library) to fill the interior parts of this hollow polygon with a single layer of triangles. (iv) Find the centroids of all these triangles, and connect them together. This will get us the line running through the central portions of each arm of the letter diagram. Keep this line and delete the triangles. What you have now got is a single pixel-wide vector representation of what once was a thick letter—or a contour line in the scanned image.

Sine this algorithm seemed too complicated, I thought whether it won’t be possible to simply apply a suitable diffusion algorithm to simply erode away the thickness of the line. For instance, think that the thick-walled letter is initially made uniformly cold, and then it is placed in uniformly heated surroundings. Since the heat enters from boundaries, the outer portions become hotter than the interior. As the temperature goes on increasing, imagine the thick line to begin to melt. As soon as a pixel melts, check whether there is any solid pixel still left in its neighbourhood or not. If yes, remove the molten pixel from the thick line. In the end, you would get a raster representation one pixel thick. You can easily convert it to the vector representation. This is a simplified version of the algorithm I had implemented for my paper on the melting snowman, with that check for neighbouring solid pixels now being thrown in.

Pursuing either would be too much work for the time being; I could either offload it to a student for his project, or work on it at a later date.

Thus ended my present thinking line on the micro-level water-resources engineering.

2. Quantum mechanics:

You knew that I was fooling you when I had noted in my post dated the first of April this year, that:

“in the course of attempting to build a computer simulation, I have now come to notice a certain set of factors which indicate that there is a scope to formulate a rigorous theorem to the effect that it will always be logically impossible to remove all the mysteries of quantum mechanics.”

Guess people know me too well—none fell for it.

Well, though I haven’t quite built a simulation, I have been toying with certain ideas about simulating quantum phenomena using what seems to be a new fluid dynamical model. (I think I had mentioned about using CFD to do QM, on my blog here a little while ago).

I pursued this idea, and found that it indeed should reproduce all the supposed weirdities of QM. But then I also found that this model looks a bit too contrived for my own liking. It’s just not simple enough. So, I have to think more about it, before allocating any specific or concrete research activities about it.

That is another dead-end, as far as blogging is concerned.

However, in the meanwhile, if you must have something interesting related to QM, check out David Hestenes’ work. Pretty good, if you ask me.

OK. Physicists, go away.

3. Homeopathy:

I had ideas about computational modelling for the homeopathic effect. By homeopathy, I mean: the hypothesis that water is capable of storing an “imprint” or “memory” of a foreign substance via structuring of its dipole molecules.

I have blogged about this topic before. I had ideas of doing some molecular dynamics kind of modelling. However, I now realize that given the current computational power, any MD modelling would be for far too short time periods. I am not sure how useful that would be, if some good scheme (say a variational scheme) for coarse-graining or coupling coarse-grained simulation with the fine-grained MD simulation isn’t available.

Anyway, I didn’t have much time available to look into these aspects. And so, there goes another line of research; I don’t have much to do blogging about it.

4. CFD:

This is one more line of research/work for me. Indeed, as far as my professional (academic research) activities go, this one is probably the most important line.

Here, too, there isn’t much left to blog about, even if I have been pursuing some definite work about it.

I would like to model some rheological flows as they occur in ceramics processing, starting with ceramic injection moulding. A friend of mine at IIT Bombay has been working in this area, and I should have easy access to the available experimental data. The phenomenon, of course, is much too complex; I doubt whether an institute with relatively modest means like an IIT could possibly conduct experimentation to all the required level of accuracy or sophistication. Accurate instrumentation means money. In India, money is always much more limited, as compared to, say, in the USA—the place where neither money nor dumbness is ever in short supply.

But the problem is very interesting to a computational engineer like me. Here goes a brief description, suitably simplified (but hopefully not too dumbed down (even if I do have American readers on this blog)).

Take a little bit of wax in a small pot, melt it, and mix some fine sand into it. The paste should have the consistency of a toothpaste (the limestone version, not the gel version). Just like you pinch on the toothpaste tube and pops out the paste—technically this is called an extrusion process—similarly, you have a cylinder and ram arrangement that holds this (molten wax+sand) paste and injects it into a mould cavity. The mould is metallic; aluminium alloys are often used in research because making a precision die in aluminium is less expensive. The hot molten wax+ceramic paste is pushed into the mould cavity under pressure, and fills it. Since the mould is cold, it takes out the heat from the paste, and so the paste solidifies. You then open the mould, take out the part, and sinter it. During sintering, the wax melts and evaporates, and then the sand (ceramic) gets bound together by various sintering mechanism. Materials engineers focus on the entire process from a processing viewpoint. As a computational engineer, my focus is only up to the point that the paste solidifies. So many interesting things happen up to that point that it already makes my plate too full. Here is an indication.

The paste is a rheological material. Its flow is non-Newtonian. (There sinks in his chair your friendly computational fluid dynamicist—his typical software cannot handle non-Newtonian fluids.) If you want to know, this wax+sand paste shows a shear-thinning behaviour (which is in contrast to the shear-thickening behaviour shown by, say, corn syrup).

Further, the flow of the paste involves moving boundaries, with pronounced surface effects, as well as coalescence or merging of boundaries when streams progressing on different arms of the cavity eventually come together during the filling process. (Imagine the simplest mould cavity in the shape of an O-ring. The paste is introduced from one side, say from the dash placed on the left hand side of the cavity, as shown here: “-O”. First, after entering the cavity, the paste has to diverge into the upper and lower arms, and as the cavity filling progresses, the two arms then come together on the rightmost parts of the “O” cavity.)

Modelling moving boundaries is a challenge. No textbook on CFD would even hint at how to handle it right, because all of them are based on rocket science (i.e. the aerodynamics research that NASA and others did from fifties onwards). It’s a curious fact that aeroplanes always fly in air. They never fly at the boundary of air and vacuum. So, an aeronautical engineer never has to worry about a moving fluid boundary problem. Naval engineers have a completely different approach; they have to model a fluid flow that is only near a surface—they can afford to ignore what happens to the fluid that lies any deeper than a few characteristic lengths of their ships. Handling both moving boundaries and interiors of fluids at the same time with sufficient accuracy, therefore, is a pretty good challenge. Ask any people doing CFD research in casting simulation.

But simulation of the flow of the molten iron in gravity sand-casting is, relatively, a less complex problem. Do dimensional analysis and verify that molten iron has the same fluid dynamical characteristics as that of the plain water. In other words, you can always look at how water flows inside a cavity, and the flow pattern would remain exactly the same also for molten iron, even if the metal is so heavy. Implication, surface tension effects are OK to handle for the flow of molten iron. Also, pressures are negligibly small in gravity casting.

But rheological paste being too thick, and it flowing under pressure, handling the surface tensions effect right should be even bigger a challenge. Especially at those points where multiple streams join together, under pressure.

Then, there is also heat transfer. You can’t get away doing only momentum equations; you have to couple in the energy equations too. And, the heat transfer obviously isn’t steady-state; it’s necessarily transient—the whole process of cavity filling and paste solidification gets over within a few seconds, sometimes within even a fraction of a second.

And then, there is this phase change from the liquid state to the solid state too. Yet another complication for the computational engineer.

Why should he address the problem in the first place?

Good question. Answer is: Economics.

If the die design isn’t right, the two arms of the fluid paste lose heat and become sluggish, even part solidify at the boundary, before joining together. The whole idea behind doing computational modelling is to help the die designer improve his design, by allowing him to try out many different die designs and their variations on a computer, before throwing money into making an actual die. Trying out die designs on computer takes time and money too, but the expense would be relatively much much smaller as compared to actually making a die and trying it. Precision machining is too expensive, and taking a manufacturing trial takes too much time—it blocks an entire engineering team and a production machine into just trials.

So, the idea is that the computational engineer could help by telling in advance whether, given a die design and process parameters, defects like cold-joins are likely to occur.

The trouble is, the computational modelling techniques happen to be at their weakest exactly at those spots where important defects like cold-joins are most likely. These are the places where all the armies of the devil come together: non-Newtonian fluid with temperature dependent properties, moving and coalescing boundaries, transient heat transfer, phase change, variable surface tension and wall friction, pressure and rapidity (transience would be too mild a word) of the overall process.

So, that’s what the problem to model itself looks like.

Obviously, ready made software aren’t yet sophisticated enough. The best available are those that do some ad-hoc tweaking to the existing software for the plastic injection moulding. But the material and process parameters differ, and it shows in the results. And, that way, validation of these tweaks still is an on-going activity in the research community.

Obviously, more research is needed! [I told you the reason: Economics!]

Given the granular nature of the material, and the rapidity of the process, some people thought that SPH (smoothed particle hydrodynamics) should be suitable. They have tried, but I don’t know the extent of the sophistication thus far.

Some people have also tried finite-differences based approaches, with some success. But FDM has its limitations—fluxes aren’t conserved, and in a complex process like this, it would be next to impossible to tell whether a predicted result is a feature of the physical process or an artefact of the numerical modelling.

FVM should do better because it conserves fluxes better. But the existing FVM software is too complex to try out the required material and process specific variations. Try introducing just one change to a material model in OpenFOAM, and simulating the entire filling process with it. Forget it. First, try just mould filling with coupled heat transfer. Forget it. First, try just mould filling with OpenFOAM. Forget it. First, try just debug-stepping through a steady-state simulation. Forget it. First, try just compiling it from the sources, successfully.

I did!

Hence, the natural thing to do is to first write some simple FVM code, initially only in 2D, and then go on adding the process-specific complications to it.

Now this is something about I have got going, but by its nature, it also is something about you can’t blog a lot. It will be at least a few months or so before even a preliminary version 0.1 code would become available, at which point some blogging could be done about it—and, hopefully, also some bragging.

Thus, in the meanwhile, that line of thought, too comes to an end, as far as blogging is concerned.

Thus, I don’t (and won’t) have much to blog about, even if I remain (and plan to remain) busy (to very busy).

So allow me to blog only sparsely in the coming weeks and months. Guess I could bring in the comments I made at other blogs once in a while to keep this blog somehow going, but that’s about it.

In short, nothing new. And so, it all is (and is going to be) crap.

More of it, later—much later, may be a few weeks later or so. I will blog, but much more infrequently, that’s the takeaway point.

* * * * *   * * * * *   * * * * *

(Marathi) “madhu maagashee maajhyaa sakhyaa pari…”
Lyrics: B. R. Tambe
Singer: Lata Mangeshkar
Music: Vasant Prabhu

[I just finished writing the first cut; an editing pass or two is still due.]

[E&OE]

 

Certain features of Dirac’s notation and a physical analog

Important Update on 2015.03.20:

tl;dr version: Don’t bother with this post. It’s in error.

Long version:

On second [and third…] thoughts, I think that this post has turned out to be just bad. (I am being serious here.) Regardless of whatever seeds of some good or promising ideas there may be in it (and I do think there are some), there also are far too many errors or wrong ideas in it, and the errors make the overall description just plain wrong.

If you are interested in knowing which ones I now think are bad or very bad, drop me a line. That is, should you decide to read this post at all, in the first place—something I won’t recommend. The only reason I am keeping this post is to keep a record of how crazy QM can sometimes get to get, especially to me. [Yes, even if I have published a paper on some aspects of the foundations of QM.]

Yet, if you still choose to go through this post, then I would say: OK, go through it, finish reading it, and then come back to this point once again, and think about points like these: (i) Why two chambers? Ideally, there should be only one chamber. (ii) Does the system really model a complex-valued vector and its conjugate correctly? Answer: no. (iii) Does the system model the vector-matrix-vector multiplication right? Answer: no. (iv) Does it even model the multiplication? Answer: no, not really. (v) There also are other inconsistencies.

Of course it’s a fact that as far as QM is concerned, I don’t get to discuss ideas with any one—there is absolutely no informal tossing of ideas back and forth with any one—no fleshing out (or thrashing out) of ideas at the blackboard, gaining clarity as you go on explaining them to someone else (say to a student), nothing. … So, things do get a bit crazy. … Yesterday, I met an engineer friend, and thus had my very first chance to speak with anyone else about the ideas of this post. I could not discuss the QM aspects of it because he hasn’t studied it, but I could at least discuss phasors and conjugates, vectors and matrices, Fourier transforms and waves, etc. I told him the kind of error I thought I was making, and asked him to confirm it. Frankly speaking, he was not sure. He could give me a benefit of doubt because of symmetries, though, being an informal discussion (over a small drink), we let it go at that. But whatever he happened to mention also brought phasors into full focus for me. That was enough to confirm my suspicions. … Finally, today, I decided to put on record the bad points, too.

No, I will not give up attempting to model the Dirac notation via some easily understandable physical analogs. And if I get to something right, I will sure post about it.

That way, these days, I hardly even look at QM (except for browsing of others’ blogs now and then). I am mostly thinking or reading or working something about my other researches—water conservation, CFD, FEM, etc. So, it will be a long while before I could possibly take out some time to get down to thinking about the Dirac notation and all, as my primary thinking goal. And, it can only be after that, that if I at all get something about it consistently right, I could post something about it.

All that I am saying, in the meanwhile, is that no matter how many seeds of some workable ideas this post might otherwise have, the system description in this post is in error. It is bad—bad, even as an analogy. Treat it that way.

Let me not bother with this post any further.

* * * * *   * * * * *   * * * * *

[Note: I have added a significant update (more like an extension) on 2015.03.19]

* * * * *   * * * * *   * * * * *

This post follows my browsing of Piotr Migdal’s guest post on John Baez’ blog, here [^], yesterday. Migdal’s aim is make QM simple to understand. He somehow begins with Dirac’s notation, and rapidly comes to stating this formalism:

E = \langle \psi | H | \psi \rangle

I read through about half of this post, and then rapidly browsed through the remaining part, before returning to this formalism and begin thinking a bit about it. … After all, he was doing something about presenting the QM ideas as simply as possible, you know…

Then, an analogy struck me. It’s based on my ideas of QM, of course—remember those pollen grains and the bumping particles and all that stuff which I had written a couple of months ago or so? (On second thoughts, here it is: [^].)

Anyway, let me share with you the analogy that struck me today. If you find something objectionable with it, sure feel free to drop me a line.

* * * * *   * * * * *   * * * * *

A physical system with a gas-filled cylinder:

Consider a cylinder with two pistons, one at each end, and a rigid, impermeable but movable partition in the middle. Assume that the system is frictionless.

Suppose that both the chambers of the cylinder are of the same length and that both are filled with the ideal gas to the same pressure—some sufficiently low pressure.

Now suppose that the piston on the right hand side (RHS for short) is moved to and fro at a constant angular frequency \nu, a certain maximum displacement A, and a certain initial phase \theta_0. This motion can be specified using a phasor, i.e. a complex number; the phasor rotates in the CCW sense in the abstract phasor plane.

The RHS piston imparts momentum to the gas molecules in the right chamber. The generated sound waves hit the central partition, impart it the momentum, and thus tend to make it move back and forth as well.

But suppose we wish to ensure that the partition in the middle remained stationary. How could we accomplish this goal?

If you were allowed to move the piston on the left, in precisely what way would you move it so that the central partition remained motionless at all times?

Obviously, you would have to move the LHS piston in such a way that its frequency and maximum amplitude are the same as for the RHS piston, viz., the same values as \nu and A. However, the initial phase of the phasor for the LHS piston must be made  -\theta_0 (opposite to that of the RHS piston), and the sense of rotation of the phasor for the LHS piston must be made CW (whereas that for the RHS piston had the CCW sense).

If the pistons were to be linked to the central partition via ideal continuous springs, then the central partition would always remain perfectly standstill.

However, if instead of springs, a gas is used for filling the chambers, then since a gas is made of only a finite number of discrete molecules, the transmission of momentum to the central partition acquires a discrete character. Further, if the molecules are randomly distributed (in terms of either positions, momenta, or both), then the momentum transmission acquires a stochastic character.

As a result, the partition does not remain perfectly standstill at all times, but undergoes a small, random, vibratory motion.

In the terminology deployed by QM, the position of the partition is said to be, you know, uncertain.

* * * * *   * * * * *   * * * * *

Update the next day (on 2015.03.19)

Let me rapidly note down a few additional points (some of which should be very obvious to many):

(i) Irregular pulses instead of a regular (single) sine wave:

The motion of the RHS piston doesn’t have to be perfectly sinusoidal. Even if the motion is a rather irregular wave (as is the case when one side of a drum is banged), such a motion can always be analysed via the Fourier transform. In other words, |\psi\rangle now has several basis components of different frequencies. Doesn’t matter; just make sure that for each frequency component, the LHS piston too perfectly opposes the motion.

(ii) A system with parallel grooves:

For illustration via a working physical model (or for implementation in a C++ program), I think it could be better to think of the following situation.

Suppose there are ten (/hundred/thousand) straight-line grooves smoothly carved into a horizontal platform. All the grooves are of same width and lie parallel to each other. Suppose, there are several ball-bearing balls placed in each groove (the number per groove may or may not be constant). At the initial time, the balls are placed at randomly different distances. Instead of the RHS piston, we now have a rigid plunger normal to the grooves; it simultaneously moves through the same distance over all the grooves—something like a comb going over some parallel scratches. The middle partition and the LHS piston, too, of course are something like this “comb.” The balls represent the gas molecules. This mechanism makes the one-dimensionality of motion (positions and momenta) inescapable. You can figure out the rest. (For instance, ask yourself what role does the initial speed of a ball has? Does it imply anything towards an independent frequency component, energy, basis vector? Can all balls in a given groove have random initial positions but the same initial speeds, with balls from different grooves differing in speeds? Etc.). You can more easily implement a software program than a build a physical model, to study the behaviour.

(iii) Trying something for the quantum discreteness:

If you wish to go even further, think of having side-walls parallel to and outside of the extreme grooves, and suppose that these walls carry some serrations. Suppose also that the middle partitioning “comb” carries a small ball and a spring (lying in the plane of the comb) in such a way that the comb successively halts only in the valleys of the serrations, The middle partition thus snaps in at discrete positions, say, 0, \pm \Delta x, \pm 2 \Delta x, \cdots, etc., thereby imparting the motion of the partition something like a discrete character.

Finally, if you must have something to stand in for that H symbol, think of a system with two symmetrically placed middle partitions instead of just one—say, one each at \pm x. This gives rise to a system of three chambers. For a system with the ideal gas, insert a sensitive thermometer in the central chamber. It will measure the level of the kinetic energy contained within the central chamber. …

Honestly, though, at least to me, this idea looks like an overkill. After all, the entire system still remains only classical. It merely serves to highlight some of the features of QM—not all.

(iv) What all these systems are good for:

Realize, all the above models are purely classical. None is fully quantum. They do, however, help simplify and bring out certain features of QM.

As far as I am concerned, even a simple C++ program with just two chambers (or parallel grooves with just one partition) might be enough—it will still bring out the the discrete and stochastic momentum-transmission events, and the 1D random walk undergone by the middle partition.

And even this simple a system should bring out many more features of the quantum formalism pretty well… Features like: the necessity of complex numbers in the Dirac notation, the necessity to define the row vectors with complex conjugates, the idea of basis vectors for the column and row vectors, etc.

This is good enough. It is much better than letting your ideas float in an abstract Dirac sea the thin air—thereby making you susceptible for recruitment by many quantum interpretations [^]. The chance that irrational ideas have to grab or overpower your mind is inversely proportional to the clarity which you derive about even simple-looking, basic, concepts. Even a partial clarity can be sometimes good enough. I mean not some half-baked knowledge, but a full clarity on some aspects of a very complex phenomenon. You can always build on it, later.

Bye for now. In the next post, I will return to some notes from my studies of the micro-level water resources engineering.

* * * * *   * * * * *   * * * * *

A Song I Like:
(Marathi) “jaambhuL pikalyaa jhaaDaakhaali…”
Music: Hridaynath Mangeshkar
Lyrics: N. D. Mahanor
Singer: Asha Bhosale

[E&OE]

Many Quantum Interpretations

Suppose you are a student of engineering—say, of mechanical engineering or materials engineering (of perhaps even of computer engineering). You are taking a course on statistics or experimental methods, and your professor has suggested that you could easily create an interesting experimental apparatus: you could build a physical, particles-based model that illustrates the kind of process lying at the roots of the normal distribution. In other words, you could construct Galton’s board [^]. The professor happens to mention this point in your class only in the passing.

And so, on the next weekend, you go out shopping to the (Hindi) “junaa/chor bazaar” (English: flea market), get a few round rubber pieces, a discarded carom board, and a few ball-bearing balls. You affix the round rubber pieces onto the carom board following that Pascal’s triangle kind of arrangement. At the bottom, you affix a few wooden batten strips so as to collect the rolling balls into the compartmentalized collection bins. In the experiment, you would let the balls roll from the top of the triangle via an input channel, and after they have finished bumping into those various rubber pieces, and then rebounding and rolling down, you collect these balls into those various collection bins at the bottom. As the number of rows and the number of balls goes on increasing, the relative fractions of the balls cumulatively collected in the bottom bins tends towards the normal distribution [^].

Then, you think of an idea. You realize that what the mathematics requires is not this entire physical apparatus in all its physicality, but only certain quantitative aspects of it: the number of balls passing through the different places. And, focusing on the input and output of the system, you decide that the number of balls passing through the input channel at the top and the output channels at the bottom is all you are interested in.

Therefore, you think of some simple spring-loaded hammer-and-bell arrangement (or, on second thoughts, just some simple chiming cylinders of the Feng Shui sort) such that, whenever a ball rolls down through a given channel (input or output), it triggers a bell into chiming. To distinguish the various channels, you arrange to have each bell produce a different musical note. The advantage of this arrangement is that you don’t have to observe a ball as it goes rolling through your apparatus. You can simply hear it the moment it enters the apparatus, and you can hear its collection into each of the distinctive collection bins. Therefore, the only record that you need to keep is that of the musical notes: the input note, and the various output notes, say, Saa, Re, Ga, Ma… etc. (To the Western readers: Do, Re, Mi… or C, D, E…(with the appropriate sharps or flats as necessary)).

You demonstrate your working model in the class. Every one is impressed. Yes, even the professor. Not just him, but in fact, even the girls! They all have liked this idea of the bells…

Once the demonstration is over, as you head back to the hostels whistling, you find yourself toying with some ideas: would it be possible for you to collect all those appreciative glances coming from all those girls together, and use the collection to buy that super-bike with that oil-cooled twin-cylinder engine. … You continue walking, whistling happily over the bell idea…

Just then, you run into this budding physicist who lives in the adjacent hostel block. … He is a bit senior to you. You have always thought that a “wilting intellectual” would be a much more fitting term, but in this moment at least, that one seems to be an unnecessary kind of a detail if not a digression…

This guy—the budding etc. physicist—always carries an expression that is a linear combination of the following orthonormal components: (i) sleepy, (ii) sullen, (iii) dazed, (iv) abstract, (v) disturbed, and (vi) smug. The scalar multipliers along the individual dimensions do change more or less randomly, but the expression vector is always observed to span this six-dimensional space, you know by now. There is no change in the dimensionality of the space as it approaches you, not even on this bright, breezy and cool afternoon, you notice.

By now, you have had enough time to conclude that girls’ appreciative glances won’t buy you that bike. But even this realization wouldn’t hamper your aforementioned mood of utter joy and swelling confidence. You could solve any problem in the world, you are absolutely certain. Even a physicist’s problem. … Even a quantum physicist’s problem….

And so, you decide not to ignore the physicist the way you normally do. Instead, you approach him and offer if you could be of any help to him. … The expression vector collapses from (i) + (iv) + (vi) to mostly (v).  “I am interested in resolving the riddles of QM, you know,” you tell him. The expression vector undergoes some very rapid changes, and then settles down to (v) + (vi). … “Drop by my lab, tomorrow,” he asks you. And, without a single further word, walks away. The expression vector now, you guess, is: (ii) + (iii) + (iv). But neither (v) nor (vi) makes too big a presence in the linear combination. Not bad, you say to yourself… It is yet another affirmation that this is a great day, you conclude.

* * * * *   * * * * *   * * * * *

Next day, you land up in his laboratory in the physics department. His prof is a big shot. And, young. It was only in the last semester that he had joined here on a contract position, after a very successful post-doc at one of top five US schools. He has also managed to bring in a lot of funding and contacts with him, as he came. His lab has acquired some brand new equipment for some new quantum experiments; the equipment has cost millions. The funds even came from the alumni association, you know. …

Your friend isn’t exactly the local guru in the lab—his aspiration is to be a theoretical physicist. But no one objects to his hanging around in the lab—every one knows that the prof may be a big shot, but because he is so young and has arrived only on a contract position, he can’t possibly arrange for separate, cosy, air-conditioned cabins for his theoretical physics students. And therefore, this friend of yours has no option but to make do with an old wooden desk, one that is covered with that government-green felt cloth (but without the glass on its top). The desk is placed in a side-corner in this otherwise new and swanky lab. Even as the two of you settle down at his desk, no one in the lab seems to notice your presence—your own, or, for that matter, even that of your friend! No greetings, no inquiring glances, not even raised eyebrows—nothing. They seem to carry on business as usual….

Your friend steps out to grab a cup of coffee, and then, as you get a bit restless, you try chatting with a few lab folks. There is a shade of respect for you as they come to know that you are a student of that engineering department. The campus-wide workshop [/lab resource/computer centre] comes under your department. In between their daily routine in the lab, they answer your queries about the lab and your friend. “No, we don’t understand the theory he is working on all that well,” they say, “but no matter, he just can’t be a very successful theorist, to be sure,” they tell you in a matter-of-fact tone. “Not a single experiment has yet gone wrong since he began sitting here,” they explain. … And no, they wouldn’t at all mind showing you how their equipment works.

There is a thick, black, metallic table with a lot of regularly drilled holes, serving as some kind of a platform, quite a few dazzlingly shiny steel bars/columns/tubes, looking glasses, flanges complete with gaskets, nuts and bolts, precision-built black enclosures, electronics, and wires, and also a couple of high-end workstations with 24″ monitors.

“What happens,” the lab fellows explain to you, “is that there is this central box in the middle of it all. There is a single quantum source—well not, single quantum, it actually is a stream, but the rate is so low that there is statistically very low chance that more than one quantum could be in the length of the box at any given instant of time. The stream of the statistically single quanta enters the box from this side. Then, there are these seven detectors on the other side. As the detectors detect the quanta, they generate a very small signal. We use this big imported amp, and a high-end data acquisition system, to capture these quantum events of interest to us, and the cables feed the data into these computers here.” They then show you the GUI of the software program. “Here, you see these seven circles in this GUI? Each circle represents one detector. For convenience, the circles carry different colours, in the VIBGYOR sequence. Whenever a detector event occurs, the circle lights up momentarily. It also adds the event to this large, terrabyte database that we maintain. Yes, we also do daily data backups. The software automatically shows you the fractions detected in the various detectors.”

“And what distribution is it? It looks something like the bell curve,” you wonder aloud.

“Wow! You know that, too, huh? … Well, yes, it is the normal curve,” they affirm in delight.

“And, what is inside that box?” you ask.

“That is an invalid question!” Your friend has returned, with only one cup of coffee—the one he is sipping from. All the friendly lab folks somehow begin to disperse in no time, and you follow your friend back to his desk.

Your friend resumes the discussion. He proceeds to cite the Solvay conference, the Bell inequalities, Schrodinger’s dead+alive cat, the EPR debate, Dirac’s anti-matter bubbles, the Stern-Gerlach experiment, the Bohr-Einstein debates, and so on and so forth. All of which proves, he says, that you cannot raise a question like that.

“We can talk meaningfully only of the observable quantum events.”

“That means, the lighting up of those seven VIBGYOR circles?”

Your friend ignores your interjection, and continues. “We can talk meaningfully only of the observable quantum events. But not of what can be there inside that box. That is just a hidden-variables nonsense. But hidden variables, by definition, cannot at all be observed. Ever. Hence, they can have no place in a theory of physics.”

He continues: “Quantum mechanics is a complete theory, an accurate theory. It has been experimentally tested for accuracy to the levels of one part in 1000(followed by many more zeroes), and it has always been found that the theory always gives results that are in complete agreement with the experiment.”

At this point of time, there is an increase in the dimensionality of the expression space; it has now acquired an additional dimension of “triumphant,” and the all the other scalar multipliers have become zero. You know that it is time to leave.

You decide to check out some books from the library before getting back to your hostel. At night, you begin to read them. You also do a lot of Web browsing, well into very late night. You are nowhere.

One day turns into one week, the one week turns into many weeks, then months, then years, and you still are nowhere. But you keep at it—at least intermittently. And then, finally, some realization descends on you. You switch on your computer, log in to your blogging account, and start writing a blog post.

* * * * *   * * * * *   * * * * *

The Copenhagen Interpretation:

The quantum shows the particle character as it enters the box. It shows a field character once in the box. The field collapses into a particle at the time of detection at one of those seven detectors. Thus, when the quantum is not observed, it exists as a field; when it is observed, it exists as a particle. This is called the Field-Particle Duality.

We cannot arrange the experimental apparatus of the triangular box in such a way that we could simultaneously observe both the field and the particle characters. This is called the Complementarity Principle.

We cannot ever hope to come to know how the quantum collapse occurs—how a field, an entity that is continuously spread over the entire triangular domain, suddenly localizes to a discretely observed particle, i.e., a spatially discontinuous entity or phenomenon.

There is an inherent uncertainty as to which detector a given quantum will hit. This is called the Uncertainty Principle.

However, the relative fraction of the times that quanta will be detected at a given detector, can be mathematically predicted, even if such a prediction can only be in  the probabilistic terms.

The math [sic] is the same as the Newtonian gravity field + the theory of bifurcation points, apart from, of course, the theory of probability.

Quantum mechanics refutes the classical idea that we can measure anything with as much precision as we like. The Uncertainty and the Complementarity Principles in fact imply much more.

The idea is not just that we don’t know how the field-collapse occurs; it is that we cannot ever come to know anything about it. The nature of the empirical facts thrown up by quantum mechanics is like that. Quantum mechanics places a limitation on human knowledge, by introducing uncertainty at its most fundamental level.

The Feynman Interpretation Reformulation:

All that fields vs particles is humbug. It’s a bunch of baloney. Real quantum does not behave that way at all. Real quantum is a particle. Yes, you got it right. This is what we know about quantum mechanics: The real quantum is a particle. But it’s bizarre! You have to construct those nice jazzy diagrams. In this case, the quantum undergoes these processes: a quantum goes from one place to another under the gravity field, or a quantum is absorbed and re-emitted with some momentum. There are many paths that a quantum can take. But there are no gears, ratchets and wheels. It’s all abstract. The 19th century physicists thought with all those mechanical gears and wheels and nails and collisions. But Maxwell got it right. He realized that there are no gears or nails. Maxwell was a smart guy. Also Pascal. Pascal also was a very, very smart guy. He was a mathematician. Pascal’s mathematical triangle is the abstract scheme which quanta somehow follow. There are many paths between different nodes of the Pascal triangle. Let us label the one node in the first row of the Pascal triangle as A, the two nodes in the second row as B1 and B2, those in the third row as C1, C2, C3, and so on and so forth. There are many paths and you have to sum up the quantum’s motion along each of them. For example, suppose there are only three rows. So, there are only a 3-factorial number of nodes: i.e., six in all. And you can connect these six nodes via all these tiny little arrows. And, so, in case there are only three rows to the triangle, you end up with these paths:
A -> B1 -> C1
A -> B1 -> C2
A -> B2 -> C2
A -> B2 -> C3
Of course, as the number of rows increases, the number of paths increases too. The factorial function is like that. It blows up. We spend seven years teaching our graduate students the necessary math [sic] so that they can calculate how these little quanta behave. But the essentials of that abstract mathematical process are very, very simple. I am sure my friend Smriti [/Kiran/Shazia/Shaina/…] can understand it. I thank her for inviting me here. Now, assuming that the path-lengths between the adjacent nodes in those paths are constant, then, the probability that the quantum will arrive at a detector, say, C2, can be calculated by taking the number of paths that have C2 as the final letter (2 here), and dividing it by the total number of paths (4 here). So, the probability in this case is 50.00…% You can calculate the probability to as much precision as you like: just keep on adding the recurring 0! Yes, you can do that. That is a neat trick which I learnt from my high-school teacher.

But no one understands quantum mechanics. Yes, a quantum is a particle. But it is nothing like a classical particle. It is quantum particle. No one understands what it means. No one can understand what it means. What this quantum particle actually does in that triangular box is, it goes over all those paths, before it is detected at any of the detectors. And so, you have to sum over all the paths. That is the way nature has chosen to do her book-keeping. Even if there is only a single quantum, you still have to take all the paths in your calculations. All the paths obtained by joining all those tiny little arrows. So, a single quantum simultaneously goes over the first path, the second path, the third path, etc. How it manages to be every where at the same time? That is something we don’t understand. No one understands. No one can understand. It’s not a classical particle. A classical particle follows only one path at a time. But a quantum particle goes over all the paths at the same time. This is called superposition. But it’s not an ordinary superposition. It is the quantum superposition. And you can calculate the probabilities with it…

And you can build a quantum machine. There is a lot of room at the bottom—in fact, the room goes on becoming bigger and bigger as you go down and further down the Pascal triangle. But, no one understands how this triangular box “really” works. No one ever can.

* * * * *   * * * * *   * * * * *

The Many Worlds Interpretation:

The essential confusion is about the measurement problem or the field-function collapse, and the probabilistic nature of the detection events.

Therefore, the only valid answer can be that when you conduct a quantum experiment and detect a quantum at a detector, say at C2, this detection event happens in our world. However, there also are other worlds. The mathematical Hilbert space is big enough to contain many worlds! It contains our physical world, as well as every other possible physical world. Let us be polite to all these worlds. In the above example of a Pascal’s triangle of 3 rows, the Hilbert space contains six worlds. As Feynman ingeniously pointed out, as the number of rows increases, the number of physical worlds contained in the mathematical Hilbert space goes up dramatically.

Suppose a quantum goes from row A to B to C following the path: A -> B1 -> C2. But in the process of the quantum going from A to B1 rather than B2, the entire universe branches into a second world. The quantum has gone from A to B1, but this occurrence has happened only in our world. But there is another world in which it actually has gone from A to B2. Even though we cannot observe it, ever. It exists. Hilbert space can be proved to contain it. And similarly, for every branching occasion and every branched out world.

And, let us all be polite: please don’t tell me that there can be only one world. I acknowledge and in fact in my work I encourage the idea that you might have a philosophically interesting idea there. But there are many worlds. And, this idea sounds very plausible even if it may not be immediately compelling, because there are no hidden variables in this theory, and yet everything is deterministic. So, there have to be many worlds. At least, many physicists take very favourably to this idea.

After all, physics is the most fundamental and most abstract science. Computer scientists may think they are the only ones to do the abstract thinking. But they are wrong. When they model the searching and sorting algorithms, they may construct what they call an abstract tree. They may show all the branches and the leaves of this tree data structure at the same time. But, their theories still are not sufficiently abstract. They still insist on telling you that the actual computer actually traverses the tree via only a single pathway at a time—depth-first, or breadth-first, or whatever-first. So, in that sense, they do make a distinction between what is only potentially traversed and what is actually traversed. And, it is this distinction that compels them to have this entire tree only in one world. If they were to think more abstractly, if they were to use the insights of quantum mechanics, they would realize that all the various branches of the tree are actually traversed quite at the same time, but in different worlds.

We the physicists think about the most fundamental principles. We therefore have to be most abstract. And, mathematical. Mathematics is fundamental to physics. Therefore, the Hilbert space is more fundamental than the physical world; it contains all the possible physical worlds. We thus are in logic forced to insist that all the branches and leaves of the tree are physically traversed at the same time. That’s quantum mechanics for you. But simultaneous traversals require many different worlds.

Ergo, there are many worlds. Just the way computer scientists use an entire tree even if only one pathway would be traversed, similarly, we use the entire multiplicity of the physical worlds hidden in the Hilbert space, even if the events occurring only in our world would be observed. This is another reason why we like the MWI: it helps simplify our calculations—apart from, of course, fully satisfactorily solving the measurement problem and the probabilistic nature of quantum phenomena. So what if it takes many worlds! How does that pose a problem?

* * * * *   * * * * *   * * * * *

A note on a more serious note: The above-discussed analogy is entirely classical, even though it does help pin-point the quantum idiocy to such an astounding extent. In case you don’t know QM, do not let yourself think that the above analogy is what QM is really like. In particular, the system evolution here occurs via the classical Newtonian gravity and momentum exchange, not according to Schrodinger’s equation, and there are no phases here—there are no interference effects. Similarly, in the Feynman interpretation, for a quantum system, depending on the context, the accounting might have to include the additional two paths: A + B1 + C3 and A -> B2 -> C1 paths. So, the analogy as given above remains entirely classical. Even if it helps bring out the quantum idiocy—I mean, not the idiocy of science popularizers, but that of physicists themselves—to this recognizable an extent.

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A Song I Like:
(Hindi) “mila hai kisi kaa jhoomka…”
Music: Salil Choudhary
Singer: Lata Mangeshkar
Lyrics: Shailendra

[Guess I will not bother with this post much further, though, as usual, a chance exists that I might come back and streamline things a bit. The world is quantum.]

[E&OE]