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Homework Help: Particle distribution, Diffusion

  1. Nov 18, 2016 #1
    1. The problem statement, all variables and given/known data
    An initial particle distribution n(r, t) is distributed along an infinite line along the [itex]z[/itex]-axis in a coordinate system. The particle distribution is let go and spreads out from this line.
    [itex] a) [/itex] How likely is it to find a particle on a circle with distance [itex]r[/itex] from the [itex]z[/itex]-axis at the time [itex]t[/itex]?
    [itex]b)[/itex] What is the most likely distance [itex]r[/itex] from origo to find a particle at the time [itex]t[/itex]?

    2. Relevant equations

    The diffusion equation is given by
    [tex] \frac{\partial n}{\partial t} = D \nabla^2 n [/tex]
    where [itex] \nabla^2 [/itex] is the laplace-operator, [itex] D [/itex] is the diffusion constant and [itex] n [/itex] is the particle density.

    3. The attempt at a solution

    I take it by "line along the z-axis" they mean ON the z-axis(?).
    a) Im not sure how to go about this. Would it involve a fourier transform, or can it be done more easily? Any help on where/how to start would be appreciated.
    b) The most likely distance from the z-axis would be zero, because of symmetry(?). So the distance from origo would be z.
  2. jcsd
  3. Nov 18, 2016 #2


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    Staff: Mentor

    I interpreted it in the same way.

    a) What is the distribution of an initial point-like source? How can you generalize this to a 1-dimensional source?
    There is no symmetry you can use as distance cannot be negative and different distances have different differential volumes. The most likely point will be on the z-axis, but the most likely distance won't. The problem statement is confusing (is it translated?), as r seems to be the radial direction, but then it is the distance to the z-axis, not the distance to the origin (where the most likely value would be very messy to calculate).
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