Integrating Diffusion: Problem Solving and Fick's Law Explained

[WiFO215]

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(F_l) 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> = $$\frac{\int_{0}^{\infty} t F_{l}dt}{\int_{0}^{\infty}F_{l}dt}$$

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

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?

 

[hamster143]

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.