Distributions in Direction-Biased Random Walk

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


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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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