The Value of Idealized Models

Has an attempt to actually create the famed spherical cow gone badly wrong? Credit: olkacf/

Every physicist has to know the joke about the dairy farmer. Seriously, if you don’t know it, you can’t call yourself a physicist. It really should be added to the IOP’s requirements for the accreditation of physics degrees. If you have such a degree, and none of your lecturers ever told you the joke, please write a letter of complaint to your alma mater immediately. In case you find yourself in that unhappy situation, here it is:

A dairy farmer, struggling to make a profit, asked the academic staff of his local university to help improve the milk-yield of his herd. Realising it was an interdisciplinary problem, he approached a theoretical physicist, an engineer and a biologist. After due consideration, the engineer contacted him. “Good news!” she said. “My new design of milking machine will reduce wastage, increasing your yield by 5%.” The farmer thanked her, but explained that nothing short of a 100% increase could save the farm from financial ruin. The biologist came up with a better plan: genetically-modified cows would produce 50% more milk. But it was still not enough.

At last the theoretical physicist called, sounding very excited. “I can help!” he said. “I’ve worked out how to increase your milk yield by six hundred percent.”

“Fantastic!” said the farmer. “What do I have to do?”

“It’s quite straightforward,” explained the physicist. “You just have to consider a spherical cow in a vacuum…”

I shouldn’t break the cardinal rule never to explain a joke, but… the gag works because you recognize the theoretical physicist’s habit of simplifying and idealizing real-world problems. At least, I hope you recognise it, although I wonder if it’s a dying art. With the availability of vast computer-processing power and fantastically detailed experimental data in many fields, there is an increasing trend to construct hugely complex and comprehensive theoretical models, and number-crunch them into submission. Peta-scale computers can accurately simulate the trajectories of vast numbers of atoms interacting in complex biological fluids, and can even model the non-equilibrium thermodynamics of the atmosphere realistically enough to fluke an accurate weather forecast occasionally.

Superficially, it might seem like a good thing if our theoretical models can match real-world data. But is it? If I succeed in making a computer spit out accurate numbers from a model that is too complex for my meagre mortal mind to disentangle, can I claim to have learnt anything about the world?

In terms of improving our understanding and ability to develop new ideas and innovations, making a computer produce the same data as an experiment has little value. Imagine I construct a computer model of an amoeba, that includes the dynamics of every molecule and every electron in it. I can be confident that the output of this model will perfectly match the behaviour of the amoeba. So there is no point in wasting computer-time simulating that model; I already know what the results will be, and it will teach me precisely nothing about the amoeba.

If I want to learn how an amoeba (or anything) works, by theoretical modelling, I need to leave things out of the model. Only then will I discover whether those features were important and, if so, for what.

When I was a physics undergraduate, I remember once explaining Galileo’s famous experiment to a classicist friend; the possibly apocryphal one where he dropped large and small stones from the leaning tower of Pisa, to demonstrate that gravity applies the same acceleration to all bodies. “But a stone falls faster than a feather,” she protested. I said that was just because of the air resistance, so the demonstration would work perfectly if you could take the air away. “But you can’t,” she pointed out. “The theory’s pointless if it doesn’t apply to the real world. So Galileo was wrong.” I have a strong suspicion that she was just trying to wind me up – and succeeding. The point, which she probably appreciated really, is that the idealized scenario teaches us about gravity, and we can’t hope to understand the effects of gravity-plus-air before we understand gravity alone.

Similarly, if Newton had acknowledged that no object has ever found itself perfectly free of any unbalanced force, he would never have formulated his first law of motion. If Schrödinger had fretted that an electron and proton cannot be fully isolated from all external influences, he would have failed to solve the structure of the hydrogen atom and establish the fundamentals of quantum mechanics. The simplicity of the laws of nature can only be investigated by idealized models (like the one-dimensional “fluid” below), before adding the bells and whistles of more realistic scenarios.

Representation of a one-dimensional fluid under flow. Credit: Mike Evans

With increasing research emphasis on throwing massive experimental and computational power at chemically complex biophysical and nanotechnological systems, and in the face of financial pressure to follow applications-led research, it would be easy to forget the importance of developing idealized models, elegant enough to deduce general principles that transcend any one specific application. So let’s adiabatically raise a semi-infinite glass of (let’s assume) milk, and drink to the health of the spherical cow.

Mike Evans

Mike Evans

Dr Mike Evans is a lecturer in Applied Mathematics at Leeds University. His research is on non-equilibrium statistical mechanics, soft matter and complex systems
Mike Evans

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