gtspring2009:spieker_blog:fokkerplanck

Fokker-Planck Equation

08/28/09 I believe that quite a bit of the material we are covering in the Stat. Mech. II class that I am taking could be put to use for channelflow and I have decided that I will put a section on each of these topics on my blog and what I think they could possibly be used for. I would appreciate feedback as to the worthiness/feasibility of these ideas. The second I had after the auto-correlation would be putting the Fokker-Planck Equation to use. Again, we have only derived it for the Langevin equation and multiplicative noise, but for those dynamical systems:

$\partial_t p(z,t)= -(p(z,t)A)' + \frac{1}{2}(p(z,t)B)''$

where

$A(z) = \int dy (y-z)p(z \rightarrow y; \Delta t)$

and

$B(z) = \int dy (y-z)^2 p(z \rightarrow y; \Delta t)$

In this notation, $p(x \rightarrow y;t+\Delta t)$ is the probability for the system to go from x to y in a time $t+\Delta t$ . I need to learn more about the range of applicability of this equation from Kurt Wiesenfeld, but, I think we could use it to not only verify that $||\mathbf{u}|| \xrightarrow[t \rightarrow \infty]{} 0$ , but also show other patterns of the probability distribution of velocity vectors within plane couette. The exists (and I will put it up if someone wants to see it) a multivariable Fokker-Planck equation so that we could break Navier-Stokes in components and try it that way. Again, I don't know for sure if this new found tool is of any use to us, just putting it out there for now.

09/16/09 Predrag: Domenico's thesis topic is (in the long run) put such ideas into our work on hydrodynamics. Briefly: deterministic dynamics allows for arbitrarily fine resolution of the state space, for example exponentially many periodic orbits of arbitrarily long periods, but any noise in real systems limits the possible resolution. Domenico's goal is to develop the method that tells you what the finest attainable resolution of states space is for a given system and a given noise, i.e. at what point to stop searching for longer orbits. Ask him to give you a lecture about it.