Here is a curious symmetry. To achieve randomness in science or technology, our best strategy is exponentials. You can toss a coin, but the outcome isn’t so random because it is sensitive only algebraically to the details of the throw. For truer randomness you need a chaotic system with exponential sensitivities, like a pinball machine or the Lorenz equations. Run such a system for a moment and your randomness might be 99%. If that’s not enough, run it a little longer to get 99.99%. The point is that with each new step, your knowledge about the system shrinks by a constant factor, soon reaching zero for practical purposes.
And to achieve certainty, our best strategy is exponentials again! At the level of fundamental physics, anything can happen because of quantum tunnelling. But some things “never” happen in practice, such as the radioactive decay of an iron-56 atom. Why? Because the frequency of quantum events shrinks exponentially with the width of a potential barrier. Thickening up that barrier in a physics experiment is like adding another level of error correction in an electronic circuit or taking another step of a random algorithm or making another compressed sensing measurement. With each new step, your uncertainty about the system shrinks by a constant factor, soon reaching zero for practical purposes.
[1 September 2011]