There’s a paradox of white noise: in its standard idealization it has equal energy at all frequencies, hence infinite energy. Curiously this paradox is linked to two of Einstein’s great papers from his annus mirabilis.
The ultraviolet catastrophe. The first paradox was a big problem implicit in 19th century physics. Statistical mechanics predicted that a cavity should support electromagnetic waves of all frequencies, each with an equal amount of energy. So at each instant a cavity should radiate infinite energy! Planck’s quantization, it was later realized, resolved this difficulty by positing that in fact, higher frequencies have less energy in them. Einstein investigated more deeply the implications of quantization for the behavior of light photons, for which he won the Nobel Prize.
Brownian motion. Meanwhile another work of Einstein’s that year eventually led to today’s standard idealization of Brownian motion, which has become a central subject in mathematics. The idealization is this: Brownian motion is the indefinite integral of white noise. But white noise has infinite variance! This time the physicists are not concerned, since bouncing molecules don’t go down to the infinitesimal scale, but the paradox still needs a mathematical resolution if we are to work with these ideas rigorously. In effect mathematicians take the view that white noise itself is never measured, only integrals of it, and these are finite because of sign cancellations. The higher the frequency, the greater the cancellation. But the details make stochastic analysis technical and forbidding.
Did Einstein notice that blackbody radiation and Brownian motion are linked in this way?
[6 May 2017]