Where Do Simplified Models Stop Being Useful in Physics?

[cedricuk]

Physics often uses simplified models to explain real-world behaviour. These models can make complicated systems easier to understand, but they also rely on assumptions that may not hold in every situation.

For example, frictionless surfaces, point masses and ideal gases are useful starting points, even though real systems rarely behave exactly that way.

I’m interested in how people decide when a simplified model is still useful and when its assumptions become too restrictive. Is it mainly a matter of comparing predictions with experimental results, or are there other ways to recognise when a model needs to be refined?

I’d appreciate examples from different areas of physics where a simple model provides a good explanation in some conditions but becomes misleading in others.

 

[berkeman]

I’d appreciate examples from different areas of physics where a simple model provides a good explanation in some conditions but becomes misleading in others.

Projectile motion problems are often simplified by assuming no air resistance. That works fine for basic problems with low-velocity projectiles where an answer that is "close enough" to the real world motion is sufficient. But in ballistics, including air resistance in the calculations is usually important if you want to hit the target. That includes even more complications from cross-winds, etc.

 

[pines-demon]

This post is very vague but the idea is that we always know (unless we are talking about quantum gravity) what the deeper theory is.

For example for projectiles. We know we have to include a bunch of factors, Coriolis force, air resistance, or even relativistic calculations. However we also know from the problem when these factors are less important and we can choose to ignore them to see how well it fits the problem or to what level of perturbation theory we need to have a good enough fit.

Most problems in condensed matter could be solved by turning to your atomistic simulation guy to make very heavy calculations in order to validate their experiment/phenomena. However most of the times the calculation is so long or expensive (too many atoms) that this is not the usual approach. Theoretical CM physicists have become experts in taking a very big Hamiltonian that includes everything and then grinding it with approximations to obtain an effective equation that takes into account the features that are absolutely key for the experiment.

 

[Borek]

It is experimental: when predictions calculated using simplified model stop predicting right, you need to switch to the less simplified model.

 

[cedricuk]

Projectile motion problems are often simplified by assuming no air resistance. That works fine for basic problems with low-velocity projectiles where an answer that is "close enough" to the real world motion is sufficient. But in ballistics, including air resistance in the calculations is usually important if you want to hit the target. That includes even more complications from cross-winds, etc.

That’s a good example of how the purpose of a model affects how much simplification is acceptable. Ignoring air resistance works well for learning the basic principles, but when accuracy matters, factors like drag and crosswinds can make a significant difference. It really comes down to whether the assumptions are reasonable for the situation being modelled.

 

[cedricuk]

This post is very vague but the idea is that we always know (unless we are talking about quantum gravity) what the deeper theory is.

For example for projectiles. We know we have to include a bunch of factors, Coriolis force, air resistance, or even relativistic calculations. However we also know from the problem when these factors are less important and we can choose to ignore them to see how well it fits the problem or to what level of perturbation theory we need to have a good enough fit.

Most problems in condensed matter could be solved by turning to your atomistic simulation guy to make very heavy calculations in order to validate their experiment/phenomena. However most of the times the calculation is so long or expensive (too many atoms) that this is not the usual approach. Theoretical CM physicists have become experts in taking a very big Hamiltonian that includes everything and then grinding it with approximations to obtain an effective equation that takes into account the features that are absolutely key for the experiment.

That’s an interesting point about choosing approximations based on the problem rather than trying to include every possible factor. The condensed matter example highlights this especially well: even when a more complete calculation is possible in principle, its computational cost can make an effective model much more practical. The challenge seems to be identifying which details can be simplified without losing the behaviour you’re trying to explain.

 

[cedricuk]

It is experimental: when predictions calculated using simplified model stop predicting right, you need to switch to the less simplified model.

That makes sense. Comparing a model’s predictions with experimental results gives a practical way to judge whether its assumptions are still adequate. If the results consistently diverge, it’s a sign that some neglected factor may need to be included or the model refined. Thanks for sharing your perspective.