Brownian Motion

1D Uniform • Consider a 1D random walk where a particle has a displacement of ∆???????????????? in time interval of ∆????????. After ???????? steps, the time elapsed is ????????=????????∆????????, and the average variance of the displacement would be 〈∆????????2〉= ????????〈∆????????????????2〉. If one would look at the displacement after time 1 2∆????????, after time ????????, there would be a total number of 2???????? steps. In order to keep the same variance, i.e., 〈∆????????2〉=????????〈∆????????????????2〉, the step size would need to be adjusted to 1 √2????????????????????????. 〈∆????????2〉=(2????????)〈�1 √2∆????????????????�2〉=????????〈∆????????????????2〉 • Create a 1D random walk where ∆????????=±1 for time ∆????????, and simulate the displacement after total time ????????=????????∆????????. However, introduce an integer parameter ???????? which reduces the time for each step to 1 ????????∆????????. The total number of steps is ????????????????. • Plot 〈∆????????2〉 ????????????????.???????? for different values of ????????. Is it still linear? 1 D Normal • Repeat the above for ∆???????????????? being a random normally distributed step. 3D Brownian Motion • Repeat the above (1D Uniform and Normal) for in three-dimensional space.

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