Nonlinear Physical Systems

[Q_Goest]

The Wikipedia article on Nonlinearity states:

This article describes the use of the term nonlinearity in mathematics. For other meanings, see nonlinearity (disambiguation).

In mathematics, nonlinear systems represent systems whose behavior is not expressible as a sum of the behaviors of its descriptors. In particular, the behavior of nonlinear systems is not subject to the principle of superposition, as linear systems are. Crudely, a nonlinear system is one whose behavior is not simply the sum of its parts or their multiples.

(emphasis theirs) Ref: http://en.wikipedia.org/wiki/Nonlinearity

Interestingly, the page regarding http://en.wikipedia.org/wiki/Nonlinearity_%28disambiguation%29" [Broken] states:

Nonlinear generally refers to a situation that has a disproportionate cause and effect.

(Not sure what "disproportionate" truly means here either.)

Although the Nonlinearity article specifically says that it is describing the use of the term in mathematics, the article clearly implies the use of the term to physical systems, as does the disambiguation article. Examples given at the end of the page include the Navier-Stokes equations for fluid flow.

What's your take on this? Is a physical system, such as the flow of a fluid as described by the Navier-Stokes equations, a system "whose behavior is not expressible as a sum of the behaviors of its descriptors"? By making this statement, is the intent only that one can not describe the flow of fluid mathematically, or is there something else about the flow of fluid which is inherently (strongly) emergent in the sense that it is irreducible to what happens within each volume, element or portion of the fluid field?

Also, how exactly do you define "descriptors" in this example? I would assume what is meant would be the fluid's properties such as velocity or momentum for example. What would be a complete list of "descriptors" for a fluid system as described by the NS equations?

 

[AlephZero]

My 2 cents based on my work experience developing computer methods, mostly for mechanics (statics and dynamics) and heat transfer:

I think the key concept is the principle of superposition.

Beyond that strict "mathematical" definition of linearity, there is a practical distinction between

Mild nonlinearity: the governing equations and the "interesting" solutions are similar to linear ones, except that some parameters are smooth functions of others (e.g. the dynamics of a mechanical system with temperature dependent material properties, where the "answers" might include frequencies which are smooth functions of temperature, etc)

Strong nonlinearity: the physical nature of the solutions is entirely dependent on the presence of the nonlinearity, e.g. a supersonic fluid flow including shock waves.

Putting it a different way, mildly nonlinear systems "nearly" satisfy the superposition principle, strongly nonlinear systems don't satisfy it at all.

Mildy nonlinear systems can often by solved numerically by linear methods, plus a quickly convergent iteration procedure. For strongly nonlinear systems that approach usually fails.

 

[Crosson]

I think that AlephZero did a very good job at describing how systems are classified as nonlinear. I will take a stab at the more philosophical topic of nonlinearity in nature.

I like wikipedia, but

In mathematics, nonlinear systems represent systems whose behavior is not expressible as a sum of the behaviors of its descriptors.

I would change to read

"In physics, a model is nonlinear if it applies to a system whose behavior is not attributable to subsystems described by the same model".

The important point is that nonlinearity is a mathematical phenomenon, and as such it applies to physical models, not physical systems.

For example, consider the following linear theory of cows:

2 cows = 1 cow + 1 cow

Clearly the positive integers are not the best model for cows, because they leave out too much detail. But remember that every model leaves out details; if it did not we would be dealing with a duplicate of the phenomenon itself. The point is not to call cows a "nonlinear system" based on the fact that there is a nonlinear biological model of cows. New advances in math can allow us to replace nonlinear models with linear ones.

 

[Q_Goest]

Thanks Aleph.

I think the key concept is the principle of superposition.
. . .
Putting it a different way, mildly nonlinear systems "nearly" satisfy the superposition principle, strongly nonlinear systems don't satisfy it at all.

Also from Wikipedia:

For linear physical quantities, this implies that the net result at a given place and time caused by two or more independent phenomena is the sum of the results which would have been caused by each phenomenon individually.

Ref: http://en.wikipedia.org/wiki/Superposition_principle

From my limited understanding of this, a nonlinear physical system would therefore react to 2 or more independent phenomena, at a given place and time, in a way which is not the simple sum of the effect of those independent phenomena.

Is that what you're saying? Can you provide an example of two (or more) such phenomena and how they interact at some point such that the net result is more than the simple sum of the effect of these phenomena individually?

Would you agree or disagree that this phenomena is strictly a limitation of the mathematical model? (Note - I'm only referring to phenomena that can be described at a classical mechanical level such as fluid flow as described by NS equations, not quantum mechanical phenomena.) That is, I would claim that nature 'knows' how things should react, so what happens may not be calculable (even in principal <gasp!>) but they are caused by local effects such that at least at a classical mechanical level, a physical system can be viewed as a simple summation of it's constituent parts and what happens at any point is not dependant on the system as a whole but only on local causal actions.


Thanks Crosson,

The important point is that nonlinearity is a mathematical phenomenon, and as such it applies to physical models, not physical systems.

I'd agree with this, it seems the wording in Wikipedia is not the best.

But remember that every model leaves out details; if it did not we would be dealing with a duplicate of the phenomenon itself.

I like that one.

 

[Claude Bile]

Is that what you're saying? Can you provide an example of two (or more) such phenomena and how they interact at some point such that the net result is more than the simple sum of the effect of these phenomena individually?

I can provide one - ultra-short laser pulses are measured by taking two pulses and firing them at one another. The resultant second harmonic generation (which varies as E^2) is used to measure the pulse duration. On their own, each pulse does not generate significant amounts of second harmonic radiation, only when the pulses overlap can an appreciable signal be measured.

Would you agree or disagree that this phenomena is strictly a limitation of the mathematical model? (Note - I'm only referring to phenomena that can be described at a classical mechanical level such as fluid flow as described by NS equations, not quantum mechanical phenomena.) That is, I would claim that nature 'knows' how things should react, so what happens may not be calculable (even in principal <gasp!>) but they are caused by local effects such that at least at a classical mechanical level, a physical system can be viewed as a simple summation of it's constituent parts and what happens at any point is not dependent on the system as a whole but only on local causal actions.

So you are proposing that on a fundamental level, we can regard everything as being linear?

I would say the opposite. Take my above example. This nonlinear behaviour manifests because atoms do not have a linear response to an applied electric field. In fact, when you delve into most aspects of atomic behaviour, linearity is the exception rather than the rule. No system is fundamentally linear, but can be approximated as linear under certain conditions.

Claude.

 

[Q_Goest]

Hi Claude,
Thanks for the example.

So you are proposing that on a fundamental level, we can regard everything as being linear?

No, far from it. I'd agree that many phenomena can be viewed as nonlinear. I'm trying to understand why Wiki would describe a nonlinear system as "not simply the sum of its parts or their multiples". What does that really mean?

That phrase generally implies one can not break up a physical system (example: the flow of fluid as described by the NS equations) into its constituent parts without loosing something. As if there is something which happens, that can not happen, unless the entire fluid system is actually (physically) interacting. But I don't think that's what is meant.

From what I hear Aleph saying and from the entry for superposition, I think what is meant is that the affect at some location is more than the superposition of 2 or more independent phenomena in a way which is not the simple sum of the effect of those independent phenomena.

I'd still like to know what that really means for the example I gave regarding the NS equations and fluid flow.

Also, I guess I've concluded that the Wikipedia article is simply poorly worded, but I'd also like a better understanding of what is meant by a nonlinear system being more than the sum of its parts.

 

[Claude Bile]

No, far from it. I'd agree that many phenomena can be viewed as nonlinear. I'm trying to understand why Wiki would describe a nonlinear system as "not simply the sum of its parts or their multiples". What does that really mean?

Sounds like a badly worded way of describing the fact that nonlinear systems do not obey superposition. Wiki should always be taken with a grain of salt because it's articles are not peer-reviewed in the normal sense, as poor wording such as this tends to pervade these types of articles.

That phrase generally implies one can not break up a physical system (example: the flow of fluid as described by the NS equations) into its constituent parts without loosing something. As if there is something which happens, that can not happen, unless the entire fluid system is actually (physically) interacting. But I don't think that's what is meant.

No, I think you are walking down the wrong path with this trail of thought. It is a mere statement of the fact that, in a nonlinear system, f(x+y) does not equal f(x) + f(y), where f(x) is the response of the "stimulus", x. In other words, the response of the system to the two stimuli added together is not the same as the sum of the responses to individual stimuli. I don't think you are so much "losing" anything as simply not describing the system correctly anymore when you attempt to break up a nonlinear system as you would a linear one.

From what I hear Aleph saying and from the entry for superposition, I think what is meant is that the affect at some location is more than the superposition of 2 or more independent phenomena in a way which is not the simple sum of the effect of those independent phenomena.

Yes, that is closer to the mark.

I'd still like to know what that really means for the example I gave regarding the NS equations and fluid flow.

At this level, I would suggest investigating mathematical definitions of linearity and how the apply to the equations you are interested in.

Also, I guess I've concluded that the Wikipedia article is simply poorly worded, but I'd also like a better understanding of what is meant by a nonlinear system being more than the sum of its parts.

I guess it is simply a convoluted way of saying that you can't break a nonlinear system down into parts - essentially when you are analysing the behaviour of one part, you cannot ignore the effect of the other parts. For my above example, I cannot ignore the presence of the second laser pulse when analysing the effect of the 1st. (I hope I haven't added to your confusion).

Claude.

 

[polar]

From my limited understanding of this, a nonlinear physical system would therefore react to 2 or more independent phenomena, at a given place and time, in a way which is not the simple sum of the effect of those independent phenomena.

Is that what you're saying? Can you provide an example of two (or more) such phenomena and how they interact at some point such that the net result is more than the simple sum of the effect of these phenomena individually?

Imagine a water-wheel turning a mill. The water going into the thing is constant. The frictional load is linear, depending on the speed of the wheel. Now, create some same-sized holes in all of the buckets to allow just a portion of the water to escape as the thing turns.

It might seem that if you let this thing turn long enough, it would eventually settle into a constant speed of rotation, and also it might seem that mathematically you could have enough information to try and determine precisely what this ultimate speed would be. Each of the functions involved is linear and can be accurately defined.

What the thing actually does, though, if you were to really build such a device, is that it continually hunts for a steady speed but it never achieves it. You will have created a non-linear machine, and its speed will vary around two different steady-stated points, but it will never settle in.

This was the example that was provided to me when I asked this same question. I don’t have the skills to be able to determine if it is true or not, but I can accept the concept.

 

[AlephZero]

From my limited understanding of this, a nonlinear physical system would therefore react to 2 or more independent phenomena, at a given place and time, in a way which is not the simple sum of the effect of those independent phenomena.

Is that what you're saying? Can you provide an example of two (or more) such phenomena and how they interact at some point such that the net result is more than the simple sum of the effect of these phenomena individually?

That's one aspect of it. Actually the way I would have thought of it was that twice as much "cause" (whether twice as much of one phenomenon, or the sum of two different ones) doesn't produce twice as much "effect".

A simple example of that would be a con-di nozzle. Once the flow is choked, increasing the inlet pressure doesn't change the flow rate.

Classifying systems as linear or nonlinear may be a bit more subtle than just that fact though. For example consider a block of materal at uniform temperature, then cooled according to Newton's law of cooling - heat flux Q = h(T-T_0) where T is the body temp, T_0 is the external temperature, and h is the heat transfer coefficient.

In one sense this is a linear system. Double the temperature difference gives double the heat flow. Double the heat transfer coefficient also gives double the heat flow. However if you ask the naive question "does the block take twice as long to cool twice as many degrees in temperature"? The answer will be "no", so you might decide the system was nonlinear based on that observation.

Would you agree or disagree that this phenomena is strictly a limitation of the mathematical model?...

I wouldn't call it a "limitation" - but the map is not the country. Any mathmatical discussion about linear/nonlinear systems is a discussion about the map. The "real world" is whatever it is - but at the practical working level, I prefer not to ask that sort of philosophical question!

Re the "water wheel" system, something very similar has been found in a real aerodynamic situation. When autopilots came into use in commercial flying, there were complaints about motion sickness in some aircraft designs, mostly from passengers at the rear of the plane.

It turned out that some (but not all) commercial passenger planes can't actually fly "straight" in a stable manner. They have two stable modes of flight, yawed 1 or 2 degrees to the left or to the right. Nobody had noticed this when flying manually, but the autopilot was trying to maintain the unstable straight flight and the plane was randomly flipping from one stable condition to the other, and making the guys in the back seats ill.

Re nonlinear systems in general (and the difference between the map and the country), I saw this joke recently:

Q: Why did the neanderthals become extinct?
A: They died of starvation, after discovering vector algebra. When a hunting party saw a mammoth northwest of them, half of them shot their arrows north and the the other half shot them west...

 

[Q_Goest]

Thanks again everyone for all your input. Much appreciated!