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Fokker-Planck Equation November 8, 2008

Posted by keithkchan in Statistial Physics.
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You probably have heard of Fokker-Planck equation many times. I have heard of it in two different occasions this week. So I decided to blog a little bit about it.

The simplest stochastic process is the random process. A simple example is that a drunkard who is so drunk that he has no memory which direction that he has walked, so that each of his step is independent of the previous steps. Tossing a coin falls into this category. Now suppose that the drunkard still has a little bit memory that he remembers he previous only the previous step, so that the previous step influences his next step. This type of process is called Markov process. This kind of process proves to be relatively tractable yet practical. Brownian motion is well modeled by the Markov process. Another example is the transition probability in quantum mechanics calculated by the Fermi golden rule. Fokker-Plancker equation is a Markov process.

One of the usually way is to start with a little bit strange integral, and then interchange the order of integration, and Taylor expanding up to second order, blah blah (you can find it good text book, e.g. Huang 1992). Here is the product \frac{\partial P}{\partial t}= - \frac{ \partial}{\partial x} (AP) + \frac{1}{2}\frac{\partial^2}{\partial x^2}(BP)

Here P(x,t) is the probability distribution. Both A(x) and B(x) are functions of x. We can regard Fokker-Planck equation as a generalization of the diffusion equation. Now the diffusion constant is generalized to the functions A and B.

In general, we can only it numerically. Today some guy from Courant Institute gave a talk on a solver that solve a particular from of A and B. His speech was so fast that I did not get much, so I can’t really blog about it.

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Comments»

1. mystery reader - November 9, 2008

yay!

Now maybe I can request a post, about something I have wanted to ask you for a while.

I have been curious about your views on the: Tensor interaction constraints from beta decay recoil spin asymmetry of trapped atoms and its utility in negative trion injection using accelarators

thank you.
i cant wait for an insightful post on this exciting topic

2. einsteins ghost - November 9, 2008

I declare keith, my heir.

3. keithkchan - November 9, 2008

Thanks for your comments. I didn’t tell your identity is revealed to administrator when you make a post. I know who you guys are. Haha.

4. Ahmes - November 10, 2008

The only thing Einstein could have ever been was a complete skylarking idiot. His “theories” are so boring and incomplete that I fall to sleep with laughter at any sight or sound of that German fraud.

5. keithkchan - November 10, 2008

Ahmes
Seems to have real ghost in this site now.
Why don’t you look into the sky and come up with another theory? You will be the next Einstein.
In fact, many smart people have tried to modify gravity. None of them really succeed, including Mr D, G and P. First you need to come up with a consistent theory, secondly and more difficultly, it needs to pass all the tests Mr Einstein have passed. So far Mr Einstein have passed all the tests we have with flying colours. So we haven’t really detected any “German fraud” (although I am exactly what this term means) so far.


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