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Math Help - Brownian motioin in 1D binomial distribution

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    Brownian motioin in 1D binomial distribution

    It is well known that the probability density function of a point particle which undergows Brownian motion in (in 1D) is a binomial distribution Binomial Distribution -- from Wolfram MathWorld
    which at the continious limit turnes into normal distribution as it is shown inthe material above.
    Now assume a system of two point particles which are always at distance "a" from each other undergows Brownian motion (in 1D).
    Question: what is the probability density function for one of the point particles (does not matter which one since system is simmetric) at discrete and continious limit?

    Thanks.


    Diffusion Equation for the rod
    Last edited by Rafael; November 24th 2013 at 12:40 PM.
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    Re: Brownian motioin in 1D binomial distribution

    Hey Rafael.

    If you have two point particles that are always the same distance from each other and the random walk only lets you go up or down one unit for discrete (and is continuous when going to the limit), then the distribution for the other point is basically a shifted version which means you only have one real distribution.

    Since you already have the theory for this case, there is nothing else to do. Your random variable will always be Y = X + a.
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    Re: Brownian motioin in 1D binomial distribution

    Quote Originally Posted by chiro View Post
    Hey Rafael.

    If you have two point particles that are always the same distance from each other and the random walk only lets you go up or down one unit for discrete (and is continuous when going to the limit), then the distribution for the other point is basically a shifted version which means you only have one real distribution.

    Since you already have the theory for this case, there is nothing else to do. Your random variable will always be Y = X + a.
    Hey chiro,

    we are talking about 1D one dimention and both points are in same dimention i.e. poits are on the line and system can move itehr left or right on thet line.
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    Re: Brownian motioin in 1D binomial distribution

    So are they independent stochastic processes?
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    Re: Brownian motioin in 1D binomial distribution

    Quote Originally Posted by chiro View Post
    So are they independent stochastic processes?
    We have a stochastic process but the distance between point particles is always "a".
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    Re: Brownian motioin in 1D binomial distribution

    Quote Originally Posted by Rafael View Post
    we are talking about 1D one dimention and both points are in same dimention i.e. poits are on the line and system can move itehr left or right on thet line.
    I think you are getting confused by chiro's X and Y notation. X and Y are not the spacial dimensions. X is a random variable which is the position of the first particle. Y is a random variable which is the position of the second particle. Since the position of the second particle directly depends on the position of the first particle we can relate these random variables by Y=X+a. Then the probability that position Y is equal to some figure "d" is equal to the probability that position X is equal to d-a
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    Re: Brownian motioin in 1D binomial distribution

    Quote Originally Posted by Shakarri View Post
    I think you are getting confused by chiro's X and Y notation. X and Y are not the spacial dimensions. X is a random variable which is the position of the first particle. Y is a random variable which is the position of the second particle. Since the position of the second particle directly depends on the position of the first particle we can relate these random variables by Y=X+a. Then the probability that position Y is equal to some figure "d" is equal to the probability that position X is equal to d-a
    Dear Shakarri,

    I agree that if the probability that position Y is equal to some figure "d" then the position X is equal to d-a, but what is the distribution of position Y? In my opinion this is no longer same distribution as for free point particle. I think one should build analogical to Pascals triangle scheme.
    First we assume that both particles can move stochastically right or left. Now if both particles move to right then our system moves to the right, if both particles move to the left our system moves to the left, if one of the particles moves to the right and the other to the left system stays at the same place. And then next etc untill one can understand the pattern and find it's analytical expression and then take the continuous limit.
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