I ended up in a brief discussion on Twitter about models, in which the other party was suggesting that models can be manipulated to produce a desired result. Their background was finance, and they linked to some kind of Financial Times report to support their view. I, quite innocently, pointed out that this was more difficult with physical models, and the discussion went downhill from there. I do think, however, that my point is defensible, which I shall try to do here.
What I mean by physical models is those that are meant to represent the physical world, as opposed to – for example – financial, or economic models. The crucial point about physical models (which does not – as far as I’m aware – apply to other types of models) is that they’re typically founded on fundamental conservation laws; the conservation of mass, momentum, and energy. This has two major consequences; you are restricted as to how you can develop your model, and others can model the same physical system without needing to know the details of your model.
A set of fundamental equations that is often used to model physical systems is the Navier Stokes equations. These are essentially equations that describe the evolution of a gas/fluid in the presence of dissipation. The first of these equations represents mass conservation
which we can expand to:
What this equation is essentially saying is that the density, , in a particular volume cannot change unless there is a net flux, , into that volume.
We can also write the equivalent equation for momentum
which we can again write out (but which I’ll only do for the component) as
The left-hand-side is similar to that for the equation for mass conservation; the momentum in a volume can change if there is a net flux of momentum into that volume. There are, however, now terms on the right-hand-side. These are forces. The first is simply the pressure force; if there is a pressure gradient across this volume, then it means that there is net force on the volume, and the momentum will change. The final term is the viscous force. Microscopically you can think of this as representing a change of momentum due to the exchange of gas particles between neighbouring volumes. If there is no viscosity, then there will be no net change in momentum. If there is a viscosity, then the momentum gained will be different to the momentum lost, and there will be a net change in momentum. This can then be expressed macroscopically as a force.
You can also write out a similar equation for energy, that can also include the change in energy due to work done by neighbouring volumes, energy changes due to viscous dissipation, and – potentially – heat conduction. I won’t write this one out, as it just gets more and more complicated and harder and harder to explain. The point is, though, that these equations describe the evolution of a gas/fluid and are used extensively across all of the physical sciencies; from studying star and planet formation, through to atmospheric dynamics. They conserve mass, momentum, and energy. There are certain parameters (viscosity, heat conduction, …) that can be adjusted, but these are typically constrained both by physical arguments and by observations. For example, one could set the ocean heat diffusion to be so small that the surface warmed incredibly fast. It would, however, be fairly obvious that this was wrong given that it would neither match the observed surface warming, nor the warming of the deeper parts of the oceans.
Now, I don’t have any specific expertise in modelling outside of the physical sciences, so maybe there are equivalent conservation-type laws that one can apply to other types of modelling. It does seem, however, that when a physicist applies some form of conservation law to economics, they get told that they don’t really understand economics terminology, nor the basics of economic growth. The latter may well be true, but it does still seem consistent with the basic idea that such modelling isn’t constrained by the type of conservation laws that constrain most physical systems.
To be clear, I’m not suggesting that physical models are somehow better than other types of models, or that physiscists are somehow better than other types of researchers; I’m simply pointing out that the existence of fundamental conservation laws makes it quite difficult to produce some kind of desired result using a physical model. I’m also not saying that it isn’t possible, simply that it’s harder when compared to models that don’t have underlying conservation laws. It’s also easier to pick up on such issues, given that you don’t need to know the details of the other model in order to model the same system independently. Essentially, that it may be regarded as easy to engineer a desired result with some models, does not necessarily make it the case for all types of models. Of course, I’m sure there are subtleties that I haven’t considered; this isn’t meant to be a definitive argument. Also, if someone can convince me that I’m wrong, feel free to try and do so.