I had a little look at two well-understood phenomena on a recent trip: small amplitude “water” waves (really surface waves of any fluid at large enough wavelengths where viscosity can be ignored) and probability distribution evolution (I might want to relate this to barrier crossing in classical systems — what is the probability of particle crossing the barrier at a given moment of time). Hopefully, I can send the article dump this weekend.
Small amplitude waves
I’ve always liked watching bodies of water, clouds, and other natural phenomena (water waves around Kingston / Kingston Mills (the latter a detour)). I thought the simplest might be what I see .
While there are more extensive analyses on the internet (even wikipedia), say looking at the shallow water equations, it was satifisfying to look at this on my own.
A few observations:
- 3 coupled equations (horizontal velocity, pressure and surface height in integrated over depth Navier-Stokes — ignoring air — (2 equation) and continuity equation)
- advection can be dropped to study linear equations (small amplitude)
- As wave length decreases (or frequency increases), the latter (frequency or wavelength tends to a constant value — I’d need to check my notes, though because we are dealling). That I observed constant wavelength patterns (differing depending on the puddle/slope of “rain stream”), I thought that the observeation was consistent with the phenomena (wave-like properties of surface relatively insensitve to disturbance)
Probability evolution
The second issue had two parts:
- “Deriving” the evolution of a probability distribution.
- Non-gaussian noise (e.g., Cauchy noise)
For the first, the result should look like the Fokker-Planck equation in many cases (mainly a continuity of probability density equation; change of prob density = – divergence of flux). Also check out Ito calculus. For the second issue, I think the divergence of moments needs to be handled carefuully. While I had some thoughts on based on the divergence of (make sure that a differential in random variables, position (if not a random variable), and time all scale appropriately).
jonathanstolle 