Sean at Cosmic Variance has an interesting article on decoherence called Quantum Hyperion. It describes a paper by Zurek and Paz that calculates that if Saturn’s moon Hyperion were an isolated system, then within twenty years it would evolve into a non-localized quantum state. It is only the interaction with the outside world that keeps Hyperion looking like a moon.
I’ve been thinking about Lévy processes, a topic that I mentioned once before. A Lévy process is a generalization of both Brownian motion and a Poisson process. Brownian motion and Poisson processes are both continuous-time stochastic processes but have very different behavior. A Brownian motion follows a very jagged path that is almost always continuous. A Poisson process stays at one place for a long time, and then suddenly jumps to a new place. What they have in common is that changes over two disjoint time intervals are independent of each other, and if the two time intervals are the same the changes have the exact same distribution.
Lévy processes include generalizations such as various combinations of Brownian motions and a Poisson process, but they also include more exotic possibilities. The sample path of a combination of a Brownian motion and a Poisson process will almost always have only finite number of discontinuities. In general, Lévy processes can generate sample paths with infinitely many jump discontinuities in almost every interval. Over a finite time horizon, this can give rise to fait-tailed distributions such as the Cauchy distribution.
In lieu of actually finishing any of the half-finished posts, I thought I’d see if anyone has any requests for posts on particular topics. Anything on your mind?
Sorry for the light posting. I have fifty million half-finished posts that I haven’t been able to find the energy to finish, mainly because I spend all my time hitting “refresh” on financial news sites. I normally use mathematics to take my mind off the news, but this time it isn’t working. The last time I was this distracted from math was in the immediate aftermath of September 11th, and before that my own birth.