Model number

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


   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 
   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=, 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.


RandomWalk1D pic1

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

Ten1DRandomWalks pic2

Ten 1D random walks.



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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Key terms
Random walk
one dimension
Transport Physiology

Please cite in any publication for which this software is used and send one reprint to the address given below:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

Model development and archiving support at 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.