# RandomWalk2D

Random walk diffusion in two dimensions with choice of fixed step size, random step size, random direction and step size.

Model number: 0372

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## Description

This model simulates the simplest of the diffusion problems, that of one particle moving in two dimensions with equivalent directional diffusivities. This simulation utilizes a random number generator to calculate the x and y movement of any given step in the random walk process. In the visualization above a set of nstep moves are made which are shown in green. At the end of every nstep moves, an endpoint is reached and a mark is made which is indicated by the red star. After nmark sets of nstep moves the simulation ends. For each step, two random numbers are chosen from a uniform distribution between 0 and 1. If the user has chosen fixed steps, the x and y steps are plus or minus 1 dependent on whether the random numbers are greater than a half. If the user has chosen variable steps, the random numbers are scaled to be between -1 and +1. If the user has chosen angle and step random, the step is the first random number and the direction multiplies the second random number by 2*PI.

Random walk with every 80th point marked in red.

## 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.

## Related Models

- Diffusion Tutorial,
- 1-D Diffusion modeled as a partial differential equation,
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- 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

Get Model history in CVS.Posted by: RCJ

## Acknowledgements

Please cite **www.physiome.org** in any publication for which this software is used and send an email
with the citation and, if possible, a PDF file of the paper to: staff@physiome.org.

Or send a copy to:

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

[This page was last modified 14Mar18, 3:17 pm.]

**Model development and archiving support at
physiome.org provided by the following grants:** NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration,
JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ,
4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation,
8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer
Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior
support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass
Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973
JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.