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Thread: 3D Random Walk

  1. #1

    3D Random Walk




    Okay, so I've got this random walk problem here:

    (a) Using an appropriate Dirac-delta function, find the probability density w(s) for displacements of uniform length l, but in any random direction of three-dimensional space. (Hint: Remember that the function w(s) must be such that Integral[w(s)ds] =1 when integrated over all space.)

    (b) Use the result of part (a) to calculate Q(k) [the moment generating function]. (Perform the integration in spherical coordinates.)


    -- What I've done --

    Okay so for part (a) I took w(s)=Delta(r-l)/(4 pi l^2). I think this is the appropriate density since I take this question to have the essential meaning that the walker is equally likely to step from the origin to anywhere on the surface of a sphere of radius l. Maybe this is wrong? Either way, I used this in part b.

    Then, for part (b) I attempted to calculate the moment generating function, which I previously derived as Q(k)=Int[ds e^(ik.s)] over all space. (I got the dot product result from the derivation in 3 space using cartesian coordinates, but I don't think this should be affected by switching to spherical coordinates. k=<k1,k2,k3>) Anyway, I go ahead and plug in s = <r,theta,phi> and just get a disaster of a dot product up in the exponential with all sorts of trig functions. No idea how to integrate this.

    If anyone can help and either point out where I went wrong in the density or point me in the right direction for computing this integral, I'd really appreciate it.

  2. #2
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    I don't intend to bump my own question, but nobody has responded and I've made 'some' progress. The integral I'm stuck on is pretty much stuck at is e^(cosx + sinx) dx evaluated from 0 to 2pi. If anyone can help me do that, I might be able to finish up the problem. If I manage a result, I'm more than happy to post.

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