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
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.
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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The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome 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.