torsdag 10 september 2009
Wigner without Computer is Unreasonable
Computational solution of the Navier-Stokes equations.
In his 1963 Nobel Lecture discussed in the previous post: Events, Laws of Nature and Invariance Principles, Eugene Wigner expresses the physicist dream of a Theory of Everything TOE as some fundamental invariance principle or conservation law in the form of a differential equation, to which the World would be the solution.
If we narrow down the World to fluid mechanics, which is a reasonable a approximation as concerns macroscopic phenomena, then we already have a TOE of fluid mechanics
in the form of the Navier-Stokes equations expressing conservation of mass, momentum and energy. From Wigner's point of view this would close the scientific field of fluid mechanics since everything there is to know, is known: The Navier-Stokes equations!
This TOE would seem to represent extremely effective knowledge, since the NS equations can written down in two lines and can be taught to most people in less than an hour. It would be like a very compact two-line genetic code of fluid mechanics.
But thus is too simple, you say, right? Fluid mechanics is more than just jotting down the NS equations! Yes, you are right! The NS equations also have to be solved to tell us anything, and that turns out to be impossible by analytical mathematics: Only very simple analytical solutions are known which tell you very little about fluid mechanics. So the NS equations alone is not a TOE.
But computing solutions of the NS eqautions numerically using computers is today possible, because computers are now powerful enough, which in a sense gives you a TOE for fluid mechanics. However, there is hook: You have to compute solutions one by one with different data and you cannot get all solutions in one shot.
Today you have a wonderful laboratory in your laptop allowing you to explore the rich field of fluid mechanics successively by computing solutions of the NS equatiopns, studying their properties and hopefully discovering regularities or even laws supporting understanding. Some of what you can learn from this laboratory is presented on my knols on Fluid Mechanics.
Wigner's vision of a TOE represents a pre-computer classical approach to physics, which is beyond reach because even if the basic equation is a simple analytical equation like NS, the World as the solution to the equation is not simple and cannot be described by analytical mathematics, at least not a priori. This does not say that an a posteriori description by analytical mathematics is also impossible. Once solutions have been computed one can start to look for regularities and maybe find some which can be expressed by analytical mathematics, but only a posteriori. Wigner without computer is unreasonable.