Assume a Spherical Cow

Over at bento-box, there's a nice response to my recent post about simulations. He makes the very good point that the Sandia press release in question could sensibly be read as referring to the fact that recent computer technology requires fewer simplifying approximations:

Well, it isn't really until quite recently that computers have gotten fast enough that many of these approximations can be toned-down. Simulations are starting to match up on a more than qualitative level with experiment on more than simple and uninteresting systems. But computers have been around for a long time and there is a long history of simulating chemical systems. There is a long history of "bad" simulations, lots and lots of computer generated results that were either totally wrong or only qualitatively good at best. That is to say, there is a long history of the simulation being wrong, but the theory being "correct". So to look at simulations as a test of the theory, at least in chemical simulations, one had to wait till fairly recently to find this a reasonable prospect.

I'm still highly dubious about the idea of using simulations as a "test" of theory, no matter how few approximations you make. I'll probably have more to say about this later, but I've got to get to my day job, so I'm going to cop out a bit, and re-post some thoughts on the question of simplifying approximations from back in 2003, below the fold and after the obligatory DonorsChoose icon:

There's an old joke in physics circles about a dairy farmer who, in a fit of desperation over the fact that his cows won't give enough milk, consults a theoretical physicist about the problem. The physicist listens to him, asks a few questions, and then says he'll take the assignment. A few weeks later, he calls up the farmer, and says "I've got the answer." They arrange for him to give a presentation of his solution to the milk shortage.

When the day for the presentation arrives, he begins his talk by saying, "First, we assume a spherical cow..."

OK, it's not a great joke, and it's the sort of joke that's only funny to physics majors. Because this is a common feature of physics lectures-- you begin considering the problem using simplifying assumptions which are often bizarrely unrealistic. Any object you need to model is first considered as a sphere, if not a point. Strings are assumed to be massless, surfaces frictionless, and mirrors and lenses loss-less.

(This is sometimes taken to absurd extremes-- a former colleague claimed to have had great success modeling the growth of his baby daughter as a sphere accreting cells at a fixed rate...)

None of these assumptions are remotely realistic, but in their own way, they're absolutely essential. Students are often baffled and bored by the simplified cases-- one of the best students from last term's introductory mechanics class remarked at the end of the term that "we neglected friction and air resistance, and all the interesting stuff." But if you don't make those assumptions, you can never discover the underlying laws and general principles that make physics such a successful science.

The ability to abstract away the "interesting stuff" and get down to the basic principles is one of the qualifications of the Great Names in physics. In a way, this is what Einstein did, by demolishing the idea of simultaneity (though another way of looking at it would be to say that he removed a simplifying assumption that people didn't know they were making). And going back even farther, it was the specific failure to consider the "interesting stuff" my student commented on that let Newton and Galileo get the whole field of physics started.

Newton's Laws have been around for long enough that they've sort of become part of the atmosphere surrounding us. Phrases like "an object at rest tends to remain at rest" or "for every action there is an equal and opposite reaction" have attained the same sort of unconscious quote status as various Shakespeare references ("there's a method to his madness" and the like)-- say the phrase to a randomly chosen person, and it will at least sound familiar, even if they don't understand what it means.

Of course, if you stop to think about it, the full statement of Newton's first law-- "an object at rest tends to remain at rest, an object in motion tends to continue in motion in a straight line at constant speed, unless acted on by an external force"-- is a little tricky to really see in action. Half of it is trivial-- stationary objects rarely start into spontaneous motion-- but the other half is actually not that obvious. Pick an object near you, and start it moving-- odds are, it will stop moving in fairly short order. Thrown objects will be hauled down to the ground by gravity, while sliding objects are subject to friction. It takes a day like today, when there's a quarter-inch layer of ice on everything to really make you believe in Newton's first law.

Plenty of people tried to develop general theories of motion before Newton came around, and all of them got tripped up by their inability to see past the inescapable forces of friction and gravity. Aristotle is probably the most famous of these failures, and while Galileo did better, he didn't get all the way there. Newton is justly famous for being the first to realize that gravity was also an external force, and get to the laws that bear his name.

The step of ignoring friction and gravity is the absolutely critical moment in early physics. It's what lets us get from an incomplete and unsatisfying description of the everyday world to a broad and elegant description of everything. If you believe that the natural state of inanimate objects is to be motionless on the ground, you won't do too badly at describing the sort of objects Aristotle had to work with in ancient Greece, but you'll have a terrible time trying to describe the motion of planets, stars, and galaxies, or the behavior of atoms and molecules. To get rules that apply to everything, you need the imagination to start with the ideal case, and work back to the more complicated reality.

This extends well beyond mechanics, of course. In dealing with electricity and magnetism, you start with the assumption of perfect conductors and insulators, and work back to real materials. In quantum mechanics, you start with impenetrable potential barriers, and work back to finite potential values. In atomic physics, everything starts with two-level atoms, despite the fact that, as my former boss was once (mis)quoted, "There are no two-level atoms, and sodium is not one of them."

You need to work the simple, idealized problems first, to get the general rules, and then apply those rules to whatever more realistic situation you're interested in. Daft as it may seem, you need to solve the spherical cow problem before you can understand the problems of a cow shaped like, well, a cow. And sometimes, genius lies in knowing how to look at a cow and see a sphere.

(Of course, hand in hand with knowing how to make the "spherical cow" approximation to render a problem soluble goes the knowledge of when not to make those kind of approximations. Excessive zeal in making simplifying assumptions is the bane of more than a few physicists, and whole fields of social science. But that's a topic for another day...)


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It takes a day like today, when there's a quarter-inch layer of ice on everything to really make you believe in Newton's first law.

So, did you write part of this post at another time, or are you in the Southern Hemisphere (or the far Northern?)

So, did you write part of this post at another time, or are you in the Southern Hemisphere (or the far Northern?)

It was originally posted in April of 2003. I hadn't noticed the ice reference on a quick re-read, but it does sound sort of odd, given that it's currently about 80 and sunny in Schenectady.

I thought we were to assume a spherical cow of uniform density. It goes all wonky* otherwise.

It's frictionless, too. It's very improtant that the cow be frictionless.

"I thought we were to assume a spherical cow of uniform density. It goes all wonky* otherwise."

It's frictionless, too. It's very improtant that the cow be frictionless.

Well, unless surface area or volume is important, you might as well model the cow as a point mass.

BTW, recently I used the "spherical cow" as shorthand for "unrealistic simplification" in a discussion with a bunch of fairly well educated CS folks at work -- I was a bit surprised that none of them got the reference. I'm Math/CS myself, but I worked in the Physics dept and had almost enough credits to have minored in Physics as well (I would have stayed an extra semester for it if my Uni had allowed a double major with a minor. It wasn't even an honest double major; CS is just a branch of mathematics anyway.) I should have known. They hadn't heard the joke about the relative fire-extinguishing techniques of the Engineer, the Physicist and the Mathematician either. Perhaps I have over-estimated their education.

Count on a physicist to screw up a joke. A funnier and more understandable version of the joke asks the physicist to predict horse races. In that context the punchline "Begin by assuming a spherical racehorse in a vacuum" actually makes a modicum of sense.

I'm still highly dubious about the idea of using simulations as a "test" of theory

Simulations are a way of exploring theory, and extracting highly non-trivial consequences of the theory. In that sense they are absolutely essential for tests of theory, because you need them to calculate the predictions in highly complicated situations.

And in some cases they *can* be tests in and of themselves, kindof -- if simulations robustly show that the theory in some case makes a prediction which is clearly wrong (in a case relevant to my own work, if it makes a `supernovae' which doesn't explode) then you don't need to compare very carefully to experiment -- you've seen that the theory doesn't work. That's a bit of a cop-out, of course, because you're implicitly comparing to an observation, if a fairly simple one.

I think the confusion here comes from the fact that in some fields of observational science simulation is often used to solve what's essentially an inverse problem -- find a mechanism which produces a particular phenomenon.

National supercomputer facilities are political not scientific exercises. NASA built "Columbia" with Intel Itanium processors at astounding cost. As everybody but perhaps NASA knew well, the Itanium was crap on all fronts. Columbia was rebuilt with Itanium-2s. (Was there a trade-in allowance?) Had it been built with AMD Opterons it would not only be faster, the energy saved over its lifetime vs. Itaniums would have paid for the iron. Opterons are native 64-bit chips compatible all the way back to 80x86. Itaniums are custom hardware. Before you write Itanium code you must figure out what the Hell is going on in the hardware, from scratch, and why the new compilers bite.

It was a a spherical homogeneous isotropic cow. "8^>)

A well-written simulation is internally self-consistent and perfect. As with economics, the proper question is "does it correspond to anything in the real world?" Economics' reply is "heteroskedasticity." Sandia asks for a bigger hot fusion rig so turbulence can average out. Seems to Uncle Al that the physical model has magnetohydrodynamic instabilities adding to sunspots rather than averaging to zero.

I would propose that one uses the tool(s) best suited to the task at hand. There are some things that lend themselves to analytical representation and others that lend themselves to simulation. For example, the analytical representation of a harmonic oscillator is straightforward but a simulation of it is rather difficult, if not misleading, because of the rapid accumulation of error. On the other hand, if you want to look at low energy laser beam broadening using radiative transfer, an analytical solution is rather daunting but a Monte Carlo simulation is straightforward, short, and relatively rapid using contemporary processors.

By Simple Country… (not verified) on 22 Jun 2006 #permalink

And in some cases they *can* be tests in and of themselves, kindof -- if simulations robustly show that the theory in some case makes a prediction which is clearly wrong (in a case relevant to my own work, if it makes a `supernovae' which doesn't explode) then you don't need to compare very carefully to experiment -- you've seen that the theory doesn't work.

Last I checked (which was talking to Alan Calder at a conference a couple of years ago), core-collapse supernova models were still only 2d or "2-and-a-half d". Given that we know that *some* core-collapse supernovae have huge asymmetries (GRBs), it's reasonable to suppose that the lacking extra dimension is the reason that the core-collapse picture wasn't working when calculated out.... Unless we've got full 3d models that show it's not working, I wouldn't say that this is a case where you've shown that the theory doesn't work. Even if so, it probably means that while the basic idea is correct, there is still some wrinkle missing. With core-collapse supernovae, there are other reasons to believe the basic picture is right -- the back-of-the-envelope energetics work out, we see pulsars at the center of supernova remnants, the precursor to SN1987A was a supergiant, etc.

Also, in the last year or so somebody (again, I think it was Calder's group) *did* get a Type Ia supernova explosion to work, by starting the ignitition slightly off of the center of the white dwarf.

In any event : my original point, and the one that Chad was echoing, was the whole "ace card" thing, which suggests that simulation has become a "trump card" and that the positions of simulation and experiment have been reversed. There have long been cases where we didn't have the computer power to really simulate properly the implications of a theory, and as computers get more powerful that's more possible... but there remain cases where it's tough to do that. (I was just talking to a student the other day about simplifying assumptions needed to make QCD calucations of glueballs. I didn't understand much of it, but it is clear that assumptions were needed to be made to make the calculations tractable.) Ultimately, though, it comes back to the data....


Sheesh! You guys can't let a guy make a joke without piling on him right after the punch line telling him he made it wrong! (Which he didn't, even if there are other ways to say it.)


Ultimately, whether a simulation or class of simulations are valid or not is context driven. In at least some cases, the reasons they're called "computer experiments" are at least acceptable, in some cases they're not. It's just like any other method, the ultimate test is reality.

I do have to point out, this argument is nothing new. It's been going on since before the first digital computer came out. After all, there are stories floating around about when the student's job was to kick the "monte carlo" box full of balls in the office every time she/he went by, and record the positions on the tablet supplied...

My favorite one liner about the whole thing is "Simulation doesn't tell you anything you don't already know." If you're careful, it might tell you the implications of what you do know.