Determinism, Indeterminism, and the nature of the laws of physics…

The laws of physics are causal, but this fact does not imply that they can be used to determine each and everything that you feel should be determinable using them, in each and every context in which they apply. What matters is the nature of the laws themselves. The laws of physics are not literally boundless; nothing in the universe is. They are logically bounded by the kind of abstractions they are.


Let’s take a concrete example.

Take a bottle, pour a little water and detergent in it, shake well, and have fun watching the Technicolor wonder which results. Bubbles form; they show resplendent colors. Then, some of them shrink, others grow, one or two of them eventually collapse, and the rest of the network of the similar bubbles adjusts itself. The process continues.

Looking at it in an idle way can be fun: those colorful tendrils of water sliding over those thin little surfaces, those fascinating hues and geometric patterns… That dynamics which unfolds at such a leisurely pace. … Just watching it all can make for a neat time-sink—at least for a while.

But merely having fun watching bubbles collapse is not physics. Physics proper begins with a lawful description of the many different aspects of the visually evident spectacle—be it the explanation as to how those unreal-looking colors come about, or be it an explanation of the mechanisms involved in their shrinkage or growth, and eventual collapse, … Or, a prediction of exactly which bubble is going to collapse next.


For now, consider the problem of determining, given a configuration of some bubbles at a certain time t_0, predicting exactly which bubble is going to collapse next, and why… To solve this problem, we have to study many different processes involved in the bubbles dynamics…


Theories do exist to predict various aspects of the bubble collapse process taken individually. Further it should also be possible to combine them together. The explanation involves such theories as: the Navier-Stokes equations, which govern the flow of soap water in the thin films, and of the motion of the air entrapped within each bubble; the phenomenon of film-breakage, which can involves either the particles-based approaches to modeling of fluids, or, if you insist on a continuum theory, then theories of crack initiatiation and growth in thin lamella/shells; the propagation of a film-breakage, and the propagation of the stress-strain waves associated with the process; and also, theories concerning how the collapse process gets preferentially localized to only one (or at most few) bubbles, which involves again, nonlinear theories from mechanics of materials, and material science.

All these are causal theories. It should also be possible to “throw them together” in a multi-physics simulation.

But even then, they still are not very useful in predicting which bubble in your particular setup is going to collapse next, and when, because not the combination of these theories, but even each theory involved is too complex.

The fact of the matter is, we cannot in practice predict precisely which bubble is going to collapse next.


The reason for our inability to predict, in this context, does not have to do just with the precision of the initial conditions. It’s also their vastness.

And, the known, causal, physical laws which tell us how a sensitive dependence on the smallest changes in the initial conditions deterministically leads to such huge changes in the outcomes, that using these laws to actually make a prediction squarely lies outside of our capacity to calculate.

Even simple (first- or second-order) variations to the initial conditions specified over a very small part of the network can have repercussions for the entire evolution, which is ultimately responsible for predicting which bubble is going to collapse next.


I mention this situation because it is amply illustrative of a special kind of problems which we encounter in physics today. The laws governing the system evolution are known. Yet, in practice, they cannot be applied for performing calculations in every given situation which falls under their purview. The reason for this circumstance is that the very paradigm of formulating physical laws falls short. Let me explain what I mean very briefly here.


All physical laws are essentially quantitative in nature, and can be thought of as “functions,” i.e., as mappings from a specific set of inputs to a specific set of outputs. Since the universe is lawful, given a certain set of values for the inputs, and the specific function (the law) which does the mapping, the output is  uniquely determined. Such a nature of the physical laws has come to be known as determinism. (At least that’s what the working physicist understands by the term “determinism.”) The initial conditions together with the governing equation completely determine the final outcome.

However, there are situations in which even if the laws themselves are deterministic, they still cannot practically be put to use in order to determine the outcomes. One such a situation is what we discussed above: the problem of predicting the next bubble which will collapse.

Where is the catch? It is in here:

When you say that a physical law performs a mapping from a set of input to the set of outputs, this description is actually vastly more general than what appears on the first sight.

Consider another example, the law of Newtonian gravity.

If you have only two bodies interacting gravitationally, i.e., if all other bodies in the universe can be ignored (because their influence on the two bodies is negligibly small in the problem as posed), then the set of the required input data is indeed very small. The system itself is simple because there is only one interaction going on—that between two bodies. The simplicity of the problem design lends a certain simplicity to the system behaviour: If you vary the set of input conditions slightly, then the output changes proportionately. In other words, the change in the output is proportionately small. The system configuration itself is simple enough to ensure that such a linear relation exists between the variations in the input, and the variations in the output. Therefore, in practice, even if you specify the input conditions somewhat loosely, your prediction does err, but not too much. Its error too remains bounded well enough that we can say that the description is deterministic. In other words, we can say that the system is deterministic, only because the input–output mapping is robust under minor changes to the input.

However, if you consider the N-body problem in all its generality, then the very size of the input set itself becomes big. Any two bodies from the N-bodies form a simple interacting pair. But the number of pairs is large, and worse, they all are coupled to each other through the positions of the bodies. Further, the nonlinearities involved in such a problem statement work to take away the robustness in the solution procedure. Not only is the size of the input set big, the end-solution too varies wildly with even a small variation in the input set. If you failed to specify even a single part of the input set to an adequate precision, then the predicted end-state can deterministically become very wildly different. The input–output mapping is deterministic—but it is not robust under minor changes to the input. A small change in the initial angle can lead to an object ending up either on this side of the Sun or that. Small changes produce big variations in predictions.

So, even if the mapping is known and is known to work (deterministically), you still cannot use this “knowledge” to actually perform the mapping from the input to the output, because the mapping is not robust to small variations in the input.

Ditto, for the soap bubbles collapse problem. If you change the initial configuration ever so slightly—e.g., if there was just a small air current in one setup and a more perfect stillness in another setup, it can lead to wildly different predictions as to which bubble will collapse next.

What holds for the N-body problem also holds for the bubble collapse process. The similarity is that these are complex systems. Their parts may be simple, and the physical laws governing such simple parts may be completely deterministic. Yet, there are a great many parts, and they all are coupled together such that a small change in one part—one interaction—gets multiplied and felt in all other parts, making the overall system fragile to small changes in the input specifications.

Let me add: What holds for the N-body problem or the bubble-collapse problems also holds for quantum-mechanical measurement processes. The latter too involves a large number of parts that are nonlinearly coupled to each other, and hence, forms a complex system. It is as futile to expect that you would be able to predict the exact time of the next atomic decay as it is to expect that you will be able to predict which bubble collapses next.

But all the above still does not mean that the laws themselves are indeterministic, or that, therefore, physical theories must be regarded as indeterministic. The complex systems may not be robust. But they still are composed from deterministically operating parts. It’s just that the configuration of these parts is far too complex.


It would be far too naive to think that it should be possible to make exact (non-probabilistic) predictions even in the context of systems that are nonlinear, and whose parts are coupled together in complex manner. It smacks of harboring irresponsible attitudes to take this naive expectation as the standard by which to judge physical theories, and since they don’t come up to your expectations, to jump to the conclusion that physical theories are indeterministic in nature. That’s what has happened to QM.

It should have been clear to the critic of the science that the truth-hood of an assertion (or a law, or a theory) is not subject to whether every complex manner in which it can be recombined with other theoretical elements leads to robust formulations or not. The truth-hood of an assertion is subject only to whether it by itself and in its own context corresponds to reality or not.

The error involved here is similar, in many ways, to expecting that if a substance is good for your health in a certain quantity, then it must be good in every quantity, or that if two medicines are without side-effects when taken individually, they must remain without any harmful effects even when taken in any combination—that there should be no interaction effects. It’s the same error, albeit couched in physicists’ and philosopher’s terms, that’s all.

… Too much emphasis on “math,” and too little an appreciation of the qualitative features, only helps in compounding the error.


A preliminary version of this post appeared as a comment on Roger Schlafly’s blog, here [^]. Schlafly has often wondered about the determinism vs. indeterminism issue on his blog, and often, seems to have taken positions similar to what I expressed here in this post.

The posting of this entry was motivated out of noticing certain remarks in Lee Smolin’s response to The Edge Question, 2013 edition [^], which I recently mentioned at my own blog, here [^].


A song I like:
(Marathi) “kaa re duraavaa, kaa re abolaa…”
Singer: Asha Bhosale
Music: Sudhir Phadke
Lyrics: Ga. Di. Madgulkar


[In the interests of providing better clarity, this post shall undergo further unannounced changes/updates over the due course of time.

Revision history:
2019.04.24 23:05: First published
2019.04.25 14:41: Posted a fully revised and enlarged version.
]

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Where the Mind Stops—Not!

The way people use language, changes.

In the mid- and late-1990s, when the Internet was new, when blogs had yet to become widespread, when people would often use their own Web sites (or the feedback forms and “guestbooks” at others’ Web sites) to express their own personal thoughts, opinions and feelings—in short, when it still was Web 1.0—one would often run into expressions of the title sort. For example: XYZ is a very great course—NOT! XYZ university has a very great student housing—NOT! XYZ is a very cute product—NOT! … You get the idea—you really do! (NO not!!)… That’s the sense in which the title of this post is to be taken.

For quite some time, I had been thinking of a problem, a deceptively simple problem, from engineering sciences and mechanics. Actually, it’s not a problem, but a way of modeling problems.

Consider a body or a physical object, say a piece of chalk. Break it into two pieces. Easy to do so physically? … Fine. Now, consider how you would represent this scenario mathematically. That is the problem under consideration. … Let me explain further.

The problem would be a mere idle curiosity but for the fact that it has huge economic consequences. I shall illustrate it with just two examples.

Example 1: Consider hot molten metal being poured in sand molds, during casting. Though “thick,” the liquid metal does not necessarily flow very smoothly as it runs everywhere inside the mold cavity. It brushes against mold-walls, splashes, and forms droplets. These flying droplets are more effective than the main body of the molten metal in abrading (“scrubbing”) the mold-walls, and thereby dislodging sand particles off the mold walls. Further, the droplets themselves both oxidize fast, and cool down fast. Both the oxidized and solidified droplets, and the sand particles abraded or dislodged by the droplets, fall into the cooling liquid metal. Due to oxidized layer the solidified droplets (or due to the high melting point of silicates, the sand particles) do not easily remelt once they fall into the main molten metal. The particles remain separate, and thus get embedded into the casting, leading to defective castings. (Second-phase particles like oxidized droplets and sand particles adversely affect the mechanical load-carrying capacity of the casting, and also lead to easier corrosion.) We need the flow here to be smooth, not so much because laminar flow by itself is a wonderful to have (and mathematically easier to handle). We need it to remain smooth mainly in order to prevent splashing and to reduce wall-abrasion. The splashing part involves separation of a contiguous volume of liquid into several bodies (the main body of liquid, and all the splashed droplets). If we can accurately, i.e. mathematically, model how droplets separate out from a liquid, we would be better equipped to handle the task of designing the flow inside a mold cavity.

Example 2: Way back in mid-1980s, when I was doing my MTech at IIT Madras, I had already run into some report which had said that the economic losses due to unintended catastrophic fractures occurring in the US alone were estimated to be some $5 billion annually. … I quote the figure purely from my not-so-reliable memory. However, even today, I do think that the quoted figure seems reasonable. Just consider just one category of fractures: the loss of buildings and human life due to fractures occurring during earthquakes. Fracture mechanics has been an important field of research for more than half a century by now. The process of fracture, if allowed to continue unchecked, results in a component or an object fragmenting into many pieces.

It might surprise many of you (in fact, almost anyone who has not studied fluid mechanics or fracture mechanics) that there simply does not exist any good way to mathematically represent this crucial aspect of droplets formation or fracture: namely, the fact of one body becoming several bodies. More accurately, no one so far (at least to my knowledge) has ever proposed a neat mathematical way to represent such a simple physical fact. Not in any way that could even potentially prove useful in building a better mechanics of fluids or fracture.

Not very surprising. After all, right since Newton’s time, the ruling paradigm of building mathematical models has been: differential equations. Differential equations necessarily assume the existence of a continuum. The region over which a given differential equation is to be integrated, may itself contain holes. Now, sometimes, the existence of holes in a region of space by itself leads to some troubles in some areas of mechanics; e.g., consider how the compatibility criteria of elasticity lose simplicity once you let a body carry holes. Yet, these difficulties are nothing once you theoretically allow the original single body to split apart into two or more fragments. The main difficulty is the following:

A differential equation is nothing but an equation defined in terms of differentials. (That is some insight!) In the sense of its usage in physics/engineering, a differential equation is an equation defined over a differential element. A differential element (or an infinitesimal) is a mathematical abstraction. It begins with a mathematically demarcated finite piece of a continuum, and systematically takes its size towards zero. A “demarcated finite piece” here essentially means that it has boundaries. For example, for a 1D continuum, there would be two separate points serving as the end-points of the finite piece. Such a piece is, then, subjected to the mathematical limiting process, so as to yield a differential element. To be useful, the differential equation has to be integrated over the entire region, taking into consideration the boundary and initial values. (The region must be primarily finite, and it usually is so. However, sometimes, through certain secondary mathematical considerations and tricks involving certain specific kinds of boundary conditions, we can let the region to be indefinitely large in extent as well.)

Since the basic definition of the differential element itself refers to a continuum, i.e. to a continuous region of space, this entire paradigm requires that cuts or holes not existing initially in the region cannot at all be later introduced. A hole is, as I said above, mostly acceptable in mathematical physics. However, the hole cannot grow so as to actually severe a single contiguous region of space into two (or more) separate regions. A cut cannot be allowed to run all the way through. The reason is: (i) either the differential element spanning the two sides of the cut must be taken out of the model—which cannot be done under the differential equations paradigm, (ii) or the entire model must be rejected as being invalid.

Thus, no cut—no boundary—can be introduced within a differential element. A differential element may be taken to end on a boundary, in a sense. However, it can never be cut apart. (This, incidentally, is the reason why people fall silent when you ask them the question of one of my previous posts: can an infinitesimal carry parts?)

You can look at it as a simple logical consistency requirement. If you model anything with differential elements (i.e. using the differential equations paradigm), then, by the logic of the way this kind of mathematics has been built and works, you are not allowed to introduce a cut into a continuum and make fragments out of it, later on.  In case you are wondering about a logically symmetrical scenario: no, you can also not join two continua into one—the differential equation paradigm does not allow you to do that either. And, no, topology does not lead to any actual progress with this problem either. Topology only helps define some aspects of the problem in mathematically precise terms. But it does not even address the problem I am mentioning here.

Such a nature of continuum modeling is indeed was what I had once hinted at, in one of my comments at iMechanica [^]. I had said (and none contradicted me at that forum for it) that:

As an aside, I think in classical mathematics there is no solution to this issue, and there cannot be—you simply cannot model a situation like “one thing becomes two things” or “two infinitesimally close points become separated by a finite distance” within any continuum theory at all…

In other words, this is a situation where, if one wishes to think about it in mathematical terms, one’s mind stops.

Or does it?

Today, I happened to idly go over these thoughts once again. And then, a dim possibility of appending a NOT appeared.

The reason I say it’s a dim possibility is because: (i) I haven’t yet carefully thought it through; (ii) and so, I am not sure if it really does not carry philosophic inconsistencies (philosophy, here, is to be rather taken in the sense of philosophy of science, of physics and mathematics); (iii) I already know enough to know that this possibility would not in any way help at least that basic fracture mechanical problem which I have mentioned above; and (iv) I think an application simpler than the basic problem of fracture mechanics, should be possible—with some careful provisos in place. May be, just may be. (The reason I am being so tentative is that the idea struck me only this afternoon.)

I still need to go over the matter, and so, I will not provide any more details about that dim possibility, right here, right today. However, I think I have already provided a sufficiently detailed description of the problem (and the supposed difficulty about it) that, probably, anyone else (trained in basic engineering/physics and mathematics) could easily get it.

So, in the meanwhile, if you can think of any solution—or even a solution approach—that could take care of this problem, drop me a line or add a comment.  … If you are looking for a succinct statement of the problem out of this (as usual) verbose blog-post, then take the above-mentioned quote from my iMechanica comment, as the problem statement. … For years (two+ decades) I thought no solution/approach to that problem was possible, and even at iMechanica, it didn’t elicit any response indicating otherwise. … But, now, I think there could perhaps be a way out—if I am consistent by basic philosophic considerations, that is. It’s a simple thing, really speaking, a very obvious one too, and not at all a big deal… However, the point is, now the (or my) mind no longer comes to a complete halt when it comes to that problem…

Enough for the time being. I will consider posting about this issue at iMechanica after a little while. … And, BTW, if you are in a mode to think very deeply about it, also see something somewhat related to this problem, viz., the 2011 FQXi Essay Contest (and what its winners had to say about that problem): [^]. Though related, the two questions are a bit different. For the purpose of this post, the main problem is the one I mentioned above. Think about it, and have fun! And if you have something to say about it, do drop me a line! Bye for now!!

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A Song I Like
(Hindi) “nahin nahin koee tumasaa hanseen…”
Singers: Kishore Kumar, Asha Bhosale
Music: Rajesh Roshan
Lyrics: Anand Bakshi

[PS: Perhaps, a revision to fix simple errors, and possibly to add a bit of content here and there, is still due.]
[E&OE]