# RandomWalk1D

20,000 1-D random walks are taken, the summation of Gaussian distributed steps with mean of 0 and variance = 1 or uniformly distributed steps from -1 to 1. The positions at a specific step number are

Model number: 0184

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## Description

1 DIMENSIONAL DIFFUSION, RANDOM DIRECTION, RANDOM STEP SIZE N walks (nwalks) are taken consisting of N steps (nsteps). Each walk starts at x=0. The steps are either left or right depending on a uniform random number(0 to 1) being greater than 1/2; the step lengths are taken from a uniform random distribution from 0 to 1. The final positions are gridded into bins of width xL.delta. The distribution and the cumulative distributions are plotted along the expected Gaussian distribution and the expected cumulative distribution. The Gaussian distribution is given by Gaussian distribution = (A/(sigma*sqrt(2*PI))*exp(-(xL-xmean)^2/(2*sigma^2))); where A= xL.delta, tmean=0, and sigma = sqrt(ntime/3) for steps drawn from the uniform distribution from -1 to 1, or sigma = sqrt(ntime) for steps drawn from the Gaussian distribution with mean = 0 and variance = 1.

A Gaussian model fitting the end positions of a 1-D Random Walk.

Ten 1D random walks.

## Equations

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.

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## References

None.name="Related Models" id="Related Models">Related Models

- Diffusion Tutorial,
- 1-D Diffusion modeled as a partial differential equation,
- 1-D Diffusion with asymmetrical Consumption modeled as a partial differential equation,
- 1-D diffusion-advection equation with Robin boundary condition
- Random Walks of multiple particles in 1 dimension
- Random Walk of single particle in 2 dimensions
- Fractional Brownian Motion Walk in 2 dimensions
- Diffusion in a uniform slab
- Two Slab diffusion: Different diffusion coeffs in adjacent slabs require special boundary conditions
- Heat equation in two dimensions with Dirichlet boundary conditions
- Safford 1977 Dead end pore model for Calcium diffusion in muscle
- Safford 1978 Water diffusion in heart
- Suenson 1974 Diffusion in heart tissue, sucrose and water
- Facilitated diffusion through 2 regions
- Barrer Diffusion: Diffusion through 1-D slab with recipient chamber on right

## Key Terms

## Model History

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## Acknowledgements

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The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

[This page was last modified 02Nov16, 2:41 pm.]

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