Emergent Properties-When is the Superposition Principle Inadequate

[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.

 

[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.

 

[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.

 

[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.

 

[sardar]

The Superposition Principle is a fundamental concept in physics that states that the total response of a system is equal to the sum of its individual responses. This principle is applicable to many linear systems, where the output is directly proportional to the input. However, there are certain cases where the Superposition Principle is inadequate and does not accurately describe the behavior of a system.

One example of when the Superposition Principle is inadequate is in non-linear systems. In these systems, the output is not directly proportional to the input, and the superposition of individual responses does not accurately predict the total response of the system. This is because non-linear systems have emergent properties that arise from complex interactions between their components, making it difficult to predict the behavior of the system based on its individual parts.

Another example of when the Superposition Principle is inadequate is in chaotic systems. Chaotic systems are highly sensitive to initial conditions, meaning that even small changes in the starting conditions can lead to vastly different outcomes. In these systems, the superposition of individual responses may not accurately predict the total response, as the system's behavior is highly unpredictable and may exhibit emergent properties that are not present in its individual components.

Other examples of emergent properties in physical systems include phase transitions, where a system undergoes a sudden change in its physical properties due to small changes in external conditions, such as temperature or pressure. Another example is self-organization, where complex patterns and structures emerge from simple interactions between individual components.

In summary, the Superposition Principle is inadequate in systems with non-linear behavior, chaotic systems, and systems with emergent properties. These emergent properties are important to consider in physics as they can lead to unexpected and complex behaviors, highlighting the limitations of reductionist approaches in understanding the world around us.