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Diffusion Length

  1. Aug 9, 2010 #1
    I am trying to do simulations of a random walk, I get out a normal distribution in 1D how do I get the "diffusion length" from the gaussian fit?
     
  2. jcsd
  3. Aug 9, 2010 #2

    mathman

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    From wikipedia

    Gaussian random walk

    A random walk having a step size that varies according to a normal distribution is used as a model for real-world time series data such as financial markets. The Black-Scholes formula for modeling equity option prices, for example, uses a gaussian random walk as an underlying assumption.

    Here, the step size is the inverse cumulative normal distribution Φ − 1(z,μ,σ) where 0 ≤ z ≤ 1 is a uniformly distributed random number, and μ and σ are the mean and standard deviations of the normal distribution, respectively.

    For steps distributed according to any distribution with a finite variance (not necessarily just a normal distribution), the root mean squared expected translation distance after n steps is

    E|S_n| = σ√n.
     
  4. Aug 9, 2010 #3
    So, if I am looking for the diffusion length of an exciton with lifetime [tex] \tau [/tex], where [tex] l_{D}=\sqrt{D_{X}\tau} [/tex], and I want to find out what the equivalent diffusion length in my simulation is where I am using random steps of length dx, I can fit the gaussian and find the E mentioned above?
     
  5. Aug 10, 2010 #4

    mathman

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    Your original question and your comment are confusing me. Are you talking about a random walk with steps of fixed length (random direction) or are the step lengths distributed normally? Also, how many dimensions is your walk? I am not familiar with the physics notion (exciton) and the diffusion length (?) formula.
     
  6. Aug 18, 2010 #5
    I think I figured it out.
    In general (1D) you can solve for:

    [itex] \frac{\partial n_{x}}{\partial t} = D_{x}\frac{\partial^{2} n_{x}}{\partial x}-\frac{n_{x}}{\tau} + I(x,t) [/itex]

    This can be solved with a Gaussian and [itex]\sigma^{2} = 4D_{x}t[/itex]. What I was trying to do was using a random step matlab simulation with a time step, lifetime, and spatial step figure out what the equivalent diffusion length was.
     
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