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WiFO215
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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
F_{l} = - D \frac{\partial P(x,t)}{\partial x}
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(\frac{-x^{2}}{2Dt}) - exp(\frac{-l^{2}}{2Dt})]/ \sqrt{2Dt}
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> = \frac{\int_{0}^{\infty} t F_{l}dt}{\int_{0}^{\infty}F_{l}dt}
= \frac{<br /> \int_{0}^{\infty} \frac{<br /> exp[-x^{2}/2Dt]}{\sqrt{t}<br /> }dt<br /> }<br /> <br /> {\int_{0}^{\infty} \frac{<br /> exp[-x^{2}/2Dt]dt}{t \sqrt{t}<br /> }}<br />
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 anyone side is biased. Any hints?
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
F_{l} = - D \frac{\partial P(x,t)}{\partial x}
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(\frac{-x^{2}}{2Dt}) - exp(\frac{-l^{2}}{2Dt})]/ \sqrt{2Dt}
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> = \frac{\int_{0}^{\infty} t F_{l}dt}{\int_{0}^{\infty}F_{l}dt}
= \frac{<br /> \int_{0}^{\infty} \frac{<br /> exp[-x^{2}/2Dt]}{\sqrt{t}<br /> }dt<br /> }<br /> <br /> {\int_{0}^{\infty} \frac{<br /> exp[-x^{2}/2Dt]dt}{t \sqrt{t}<br /> }}<br />
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 anyone side is biased. Any hints?
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