Distributions in Direction-Biased Random Walk

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This Demonstration simulates a random walk that steps up with probability and down with probability (), but cannot step to negative values.

Contributed by: Simon Mochrie (March 2011)
Open content licensed under CC BY-NC-SA


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This simulation is a model for a particle undergoing Brownian motion subject to a constant force directed downward, but with an impenetrable barrier at zero. Force and probability are related via , where is the step size, is Boltzmann's constant, and is the absolute temperature.

The result might represent the height of a colloidal particle (i.e., a micron- or submicron-sized) in a beaker of water. It could also represent the height of a gas molecule in the Earth's atmosphere.

For a sufficiently long simulation, the distribution of particle positions (sufficiently far from zero) may be seen to be a negative exponential function of the height with a characteristic decay length of .



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