# Emergent Properties-When is the Superposition Principle Inadequate

1. Apr 17, 2014

### leeone

So I understand the Superposition principle doesn't apply to non-linear systems. I want students to understand (in high school physics....which I will be teaching in about a year) that the superposition principle essentially says things add. So I wanted to come up with some examples when this isn't true (particularly in physics).

So what are some emergent properties applicable to physical systems/concepts?

I feel like there should be a lot of answers to this one.

2. Apr 17, 2014

### UltrafastPED

Right ... superposition applies only to linear systems. If you can use linear algebra as your mathematics, then superposition is a good word.

Forces are linear; they make up a linear vector space.

You can simply look at the differential equations; if they are linear, then superposition applies. For example, Maxwell's equations consist of four coupled linear partial differential equations in the electric and magnetic fields. Thus superposition applies for electric and magnetic fields ... But not necessarily within matter - thus we have non-linear optics, including the Kerr effect, frequency doubling, etc.

The Einstein field equations for general relativity consists of ten coupled non-linear partial differential equations. Thus the principle of superposition is not universally valid within general relativity, though it works well enough for Newton's version of gravitation.

Most undergraduate physics (and differential equations) are linear; we actually know how to solve all of these, and can find numerical solutions in all cases.

But non-linear systems are much more difficult, and only scattered systems have known solutions, and there are no general approaches for the solution of nonlinear systems of differential equations.

3. Apr 17, 2014

### leeone

All that is fine and dandy but I am thinking about concepts more applicable to high school level....like the binding of two atoms? Surely atoms bound to one another have emergent properties that weren't there beforehand.

4. Apr 18, 2014

### Q_Goest

Hi leeone,
What definition are you using for "emergent" (ie: strong versus weak emergence), or aren't you familiar with those terms? The distinction isn't always made in the literature. There are some questionable papers written suggesting that non-linear systems can support strong emergence but there's little support for them. Emergence, at least in classical physics, is always considered weakly emergent so there are no 'new properties' created at least at a classical scale where physical systems are separable. It's only when you look at quantum mechanical interactions that there is some legitimate talk about new properties being created.