All dynamics are ultimately nonlinear. Linear dynamics are simply those nonlinear dynamics that can be safely compressed into a linear mathematical form…
When you trade financial markets you quickly learn that 10% up and 10% down are not the same thing!
If we were to grow $100 in 1000 incremental steps of 0.0095315% we will arrive at $110. And if we subsequently unwind/decline from this $110 in 1000 incremental steps of 0.0095315% we will arrive back at $100. If however we were to grow the same $100 to $110 in one incremental step of 10%, and then decline from $110 in one incremental step of 10% we will not arrive back at $100 but at $99 instead.
So why is this? The reason is simple. Because we are rounding out very tiny errors on each step smooth incremental change will synchronize up and down movements, but because large errors are too big to be rounded out coarse incremental change will not.
Now believe it or not, this simple idea is the cornerstone of all mathematical science. This simple idea is the basis for something called “linear approximation” or “Linearization”.
Linear Approximation is used in virtually every area of science; from physics to engineering, from economics to ecology. By eliminating insignificant deviations we can “approximate” the behaviour of a “nonlinear” system as being “linear” – which allows us to “compress” this behaviour into neat “solvable” mathematical equations.
Without linearization most of the equations of science would effectively be unsolvable, and consequently “without linearization almost nothing would be predictable”…
The science of physics specialises in linearization; that is to say that physics “approximates” the behaviour of all systems. In general the basic idea behind linear approximation is to ignore “insignificant deviations”. The two primary tools of linearization are
- [Suppression of Positive Feedback] — Pure Continuity: Incremental space and time are made infinitely small, and so the behaviour of a body in motion can be approximated as “linear dynamics”.
- [Dominance of Negative Feedback] — The Law of Large Number (LLN): The number of elements within a system are assumed to be infinitely large, and so and the behaviour of very large systems (of elements) can be approximated by the behaviour of the “average” element.
[Note: You might have noticed that the concept of “infinity” plays a major role in the process of linearization.]
All classical mechanics and electromagnetism uses the concepts of continual time and continual space in order to deploy (differential) calculus. Thermodynamics uses the LLN to explain macro phenomenon such as temperature and pressure. Even quantum mechanics uses linear mathematics, although it is an interesting case because it employs both tools of linearization — the Schrödinger Wave Equation is an differential equation of motion, that only works thanks to the LLN…
We live in a universe of nonlinear dynamics, it is just that some dynamics are less nonlinear than others. Physics is, in a sense, the science of “linear dynamics”, that is to say it is the science of the nonlinear stuff that can be “compressed into a linear mathematical form”.
All the mathematical laws of physics are linear (approximations). In fact the reason that these “Laws of Physics” are actually considered to be “Universal Laws” is precisely because the dynamics behind these laws can be linearized (which means we can repeat experiments over and over again, with the same initial conditions, and always get the same predictable result).
In truth however, even this is a little bit disingenuous because nothing is infinitely predictable, but the predictable things are considered to be predictable because it takes a long time for the errors to surface (so far into the future in fact, that we have plenty of time to make the appropriate adjustments).
The reality is however that nothing happens in the absence of anything else, but to-date physics has pretty much only focused on those things that happen in the presence of minimal disturbance. Minimal disturbance means linear dynamics; and linear dynamics are predictable because
“linear (approximated) behaviour” is really just another way of saying that a system’s “behaviour is not affected (very much) by incompressible feedback”…
Absence of Feedback
Physics in general doesn’t like to deal with “un-linearizable feedback” because the dynamics of such “coarse feedback” are messy and incompressible, and physics likes to isolated itself from such messy and incompressible dynamics.
In the realms of astronomy, mechanics, thermodynamics, electromagnetism, or even quantum mechanics; things are easy to linearize because things tend to be under the influence of “constant forces” (and even when we do have to deal with some feedback in the form of “friction” we are still usually dealing with a “constant” frictional force). It is this absence of constantly changing feedback that make these linearizable systems predictable…
We live in a universe of nonlinear dynamics, some of it linearizable, some of it not. For the last 350 years we have codified the linearizable stuff and not ventured much beyond. As things stand today in the early part of the 21st century, the furthest our mathematics has managed to reach into the true bowels of nonlinearity is probably the Navier-Stokes equations of turbulence. In these equation we have attempted to tame turbulence; but turbulence is not easily tamed because we cannot mathematically compress the excessive nonlinear dynamics generated by excessive amounts of internal feedback.
In the simplest possible terms turbulence occurs when the amount of energy in the system overwhelms the system’s natural ability to neutralize internal diversity and thereby achieve macro-stability. In a way we could say that excessive energy can hold a system’s “information entropy” too far away from (approximated) linearity, and consequently beyond the reach of mathematical compression.
The End of Physics?
For the last 350 years rigorous science has rigorously followed “the scientific method”. The scientific method basically says that a theory is useless unless it is “falsifiable”. This means that a hypothesis must offer the possibility that experiment could prove it to be false — or in other words the theory must make predictions that can be tested by experiment. But as we have just discussed “only systems that are virtually immune to feedback are predictable”; which effectively means that every other type of system is unpredictable and consequently not bound by, or amenable to, the scientific method.
So does this mean we have reached the end of physics? Well no, it simply means that are in the process of transitioning from a realm of pure “quantitative” to a realm that is considerably more “qualitative”. In the past we restricted ourselves to investigating and codifying only simple linear systems and simple linear dynamics, but in the future we will need to expand our focus to embrace the mathematically incompressible physics of “complex system and complexity dynamics”.
So no, we have absolutely not reached the end of physics, but we have probably pretty much reached “the end of the beginning”…