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What probability density is used in Brownian motion?

  1. Dec 4, 2015 #1
    1. The problem statement, all variables and given/known data

    I have a free Brownian particle and its coordinate is given as a function of time:

    upload_2015-12-4_13-58-5.png

    And its first moment, or mean, is given as

    upload_2015-12-4_13-58-40.png

    But what kind of probability density was used to calculate this first moment?

    2. Relevant equations

    I know that the first moment is calculated by using probability density

    <x> = ∫x*W(x)dx from -infinity to +infinity.

    X itself is dependent on t, so is it W(x) or W(x,t) or W(t)? I'm little lost on this one.

    3. The attempt at a solution

    I did find some kind of density function in regards to Brownian motion

    upload_2015-12-4_14-6-0.png

    But I can't see how this would yield the correct result. And still, must I integrate over x or t?
     

    Attached Files:

  2. jcsd
  3. Dec 4, 2015 #2

    Orodruin

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    Just take the expectation value of your first expression (expectation values are linear) keeping in mind that the only actually random variable in it is the driving force ##\zeta(t')##.
     
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