Linear Models of Reality

In engineering, great use is made of linear mathematical models. If you read popular books on chaos theory or new-age physics, you learn that linear models are oversimplified and bad, and are equated with uncreative straight-line thinking. I'd like to both touch on the element of validity in the criticism and also explain why linear models are valuable and often the right choice.

### What is a Model?

A mathematical model is a representation of how something is going to behave, most often how things are going to change over time. Mathematical models are always abstractions that ignore some details. A mechanical engineer can model the strength of steel bar using only a few numbers read from a handbook. Metallurgists who create new metal alloys need a more fine-grained understanding: what aspects of the crystalline structure create aspects of stiffness or susceptiblity to cracking. They have their own mathematical models. Various kinds of physicists work on even smaller-scale understandings of matter with their own models.

In some sense, the finest scale models such as quantum physics are better models because they can accurately predict the behavior of matter across a wider range of conditions. However, there are several good reasons why civil engineers don't design bridges with quantum physics:

• Much simpler models get the job done better. Simpler models are easier to use, and perhaps more important, can lead to an intuitive understanding of interactions in the design.
• The concerns at the different levels are so different that quantum physics doesn't have much to say about the best shape of metal beams.
• A simple model will have a limited range of situations in which it is acceptably accurate, but we may know (or be able to show) that simplifying assumptions are valid. When the steel bar melts, its mechanical properties change greatly. However, the engineer is not concerned with how the beam will behave once it melts, because at that point it is useless. More relevant is the the decision of whether to add insulation to prevent the beam from melting in a fire.

### What is a Linear Model?

First of all, a linear model does not say that everything is a straight line. Linear refers to the form of the differential equations, not the predicted behavior of the system. Linear models can exhibit quite complex behavior when they have multiple inputs and outputs, and even the simplest models predict wave-like oscillating motion and exponential increasing or fading away, and combinations of these.

What linear does mean is that if the system is stable, then the output is proportional to the input. If you increase the input 50%, the output will increase 50%. If you ask somone a harmless question, and they scream at you, then it is a reasonable metaphor to say they "went nonlinear" because your input created a disproportionate output.

A linear system may also be unstable, which means that even in the absence of input, the output rapidly explodes toward infinity. When someone blows their top, you could as well attribute this to unstable (but linear) behavior. In real unstable systems, some nonlinearity will intervene preventing the output from growing forever. However, if your plan is to keep the output from blowing up by carefully controlling the input, then you can use a linear model and ignore the nonlinearity that would have resulted without your intervention.

### Why are Linear Models so Popular?

No system is actually linear, and linear models make up an infinitesimal fraction of all possible mathematical models, so why do we use them? Fifty years ago, a large part of the reason was that linear models are much easier to analyze with pencil and paper. Because the work was manageable, people concentrated on linear models and linear effects, and ignored phenomena that didn't fit the tool. This is a bit like the story of the man who is found looking for his keys under the streetlight, and when asked where he lost them, he says "over by the bushes, but the light is better here."

Then computers cast some light in the nonlinear bushes. With computers, we could look at the behavior of nonlinear systems, and found that even simple nonlinear systems showed interestingly complex behavior. Systems with no randomness built into them produce seemingly random fluctuating outputs. This unpredictable behavior of nonlinear systems was called chaos, and people worked on trying to understand the behavior of these systems.

From this work, we now understand more about how nonlinear systems (like the earth's weather) can exhibit unpredictable behavior. We have also learned more about the role of nonlinearity in living and self-organizing systems. It turns out that a wide range of stable systems from cells in the body to entire societies operate on the edge of chaos. There's enough unpredictable behavior to keep things from getting stuck in a rut, but on average things are pretty linear.

Although in many areas of science linear models do a good job of predicting the things that are predictable, there is a special reason for the relevance in engineering. Engineers don't want to make things that are unpredictable. If we can constrain the problem definition so that everything is linear, then we are happy. If something goes nonlinear, that is a problem to be fixed by keeping well away from there.

Last update 10 February 2004

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