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Please help me integrate

  1. Dec 3, 2009 #1
    I am working on a problem (not homework) on diffusion and have landed up with an integral which I simply cannot integrate. No method seems to work.

    I shall first describe the problem a little for you to check if I have landed up on the correct route, and then present the integral.

    According to Fick's law, the flux of particles(Fl) crossing a point can be given by

    [tex]F_{l}[/tex] = [tex]- D \frac{\partial P(x,t)}{\partial x} [/tex]

    where D is some constant.

    My problem is concerned only with first passages, so I have been told to assume that there is an absorbing barrier at a certain length l away from the origin. The particle starts its motion from the origin and its motion is a diffusion process with equal probability of moving in either direction. Hence, a Guassian distribution is going to be used as the model.

    P(x,t) = [exp([tex]\frac{-x^{2}}{2Dt}[/tex]) - exp([tex]\frac{-l^{2}}{2Dt}[/tex])]/ [tex]\sqrt{2Dt}[/tex]

    On differentiating this term with respect to x only the first term survives.

    Now here comes my problem: I am supposed to find the mean time <T>, that the particle takes to hit the wall. Here is the integral:

    <T> = [tex]\frac{\int_{0}^{\infty} t F_{l}dt}{\int_{0}^{\infty}F_{l}dt}[/tex]

    = [tex]\frac{
    \int_{0}^{\infty} \frac{
    exp[-x^{2}/2Dt]}{\sqrt{t}
    }dt
    }

    {\int_{0}^{\infty} \frac{
    exp[-x^{2}/2Dt]dt}{t \sqrt{t}
    }}
    [/tex]

    I have a feeling this integral on the numerator might be unbounded. How do I integrate it? Following this I also have to do the case where the probability of moving towards any one side is biased. Any hints?
     
    Last edited: Dec 3, 2009
  2. jcsd
  3. Dec 3, 2009 #2
    Both integrals can be integrated using a substitition t = 1/z^2. Your numerator diverges. Check the expression for P, Wikipedia states that you should have erf instead of exp.
     
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