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Statistical physics reflection principle

  1. Jul 18, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider a random walker in one dimension, which can take right or left steps with equal probability. Assume that the walker starts at the location k>0 and there is an absorbing wall at point labelled as 0, that means if the walker reaches 0 the process stops and he stays there. Calculate the probability of reaching point m>0 in N steps. By taking continuum limit, find the solution to the diffusion in the presence of an absorbing wall at the origin.

    3. The attempt at a solution

    Actually this is an example of reflection principle (method of imagines boundary value problems.) However I couldnt figure out how its related. So if you give me hint to where should I start I'll be appreciated.
     
  2. jcsd
  3. Jul 18, 2013 #2
    I'm assuming you know what to do without the absorbing wall (i.e. how to derive the appropriate diffusion equation and show that the solution is Gaussian of some particular width in the continuum limit etc. etc. etc.) The diffusion equation is linear and has a unique solution given complete and consistent boundary conditions. So, given some set of boundary conditions, the solution will always be some linear combination of same-width Gaussians. For an absorbing wall at 0, the boundary condition is that the probably of reaching 0 in 0 steps is 0. Up to a sign (and only one sign will make sense here) there is one unique linear combination of same-width Gaussians that gives the value 0 at the point 0.
     
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