Generating non-uniform distributions

<BODY LANG="EN"> <HR><center> <table border=1> <td><A NAME="tex2html63" HREF="node5.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html69" HREF="Lab5.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html71" HREF="node7.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node6.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="/hamlet-bin/qhandler?exit" target="results"><img height=36 width=36 alt="RESET" src="/UCESIcons/tools/stop.gif"></a></td> <td><a href="/UCES/hamlet/"><img height=36 width=36 alt="HAMLET" src="/UCESIcons/tools/hamlet.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces_info@krellinst.org"><img height=36 width=36 alt="FEEDBACK" src="/UCESIcons/tools/feedback.gif"></a></td> <td><a href="/UCES/hamlet/usage.html"><img height=36 width=36 alt="ABOUT" src="/UCESIcons/tools/about.gif"></a></td> </table> </center> <HR><H1><A NAME="SECTION00060000000000000000">Generating non-uniform distributions</A></H1> <P> The uniform random number generator can be used to obtain random numbers with non-uniform probability distributions. This can be accomplished by inverting the indefinite integral of the desired distribution (see lecture notes). <P> <I> As an example, generate energy values distributed according to the Boltzmann distribution, at a temperature of 298 <I>K</I>, and plot the distribution as a histogram. </I> <P> This can be done in the following way. If you have exited Mathematica since loading the graphics package, type <P> <table><td valign=top><a href="/hamlet-bin/qhandler/send9.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><code><<Graphics`Graphics`</code></td></table> <P> The histogram can be generated with the following commands (Note: here we need the inverse exponential function which is the natural logarithm, <B>Log[]</B>) <P> <table><td valign=top><a href="/hamlet-bin/qhandler/sende8.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><PRE>temp=298.0; energy := - 0.0083145 temp Log[Random[]]; npoints=5000; randoms=Table[energy,{n,1,npoints}]; nbins=20; scalefc=nbins/Max[randoms]; bins=Table[Round[randoms[[i]]*scalefc+0.5],{i,1,npoints}]; counts=Table[Count[bins,i],{i,1,nbins}]; BarChart[counts, PlotRange -> All];</PRE></td></table> <P> <I>Use predefined mathematica functions to calculate the mean, variance and standard deviation of your data.</I> First load the statistical package <P> <table><td valign=top><a href="/hamlet-bin/qhandler/send10.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><code><<Statistics`DescriptiveStatistics`</code></td></table> <P> Use <B>Mean[ ... ], Variance[...]</B> and <B>StandardDeviation[...]</B> commands to analyze the list of your data points. <P> <I>Show algebraically that this is the right algorithm for generating the Boltzmann distribution and verify that the generated data (energy values) are Boltzmann distributed.</I> <P> true cm <P> To generate velocity components, for example <IMG WIDTH=14 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline136" SRC="img9.gif" > , distributed according to Maxwell distribution, you need to invert the error function. This can be done in Mathematica by loading in a statistics package (the error function often shows up in statistical analysis because of its close relationship to the normal distribution). Be sure to first clear your variables. Also, start with a smaller number of points, for example npoints=500. For room temperature and mass of 1, the algorithm is <P> <table><td valign=top><a href="/hamlet-bin/qhandler/sende9.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><PRE><<Statistics`InverseStatisticalFunctions`; temp=298.0; mass=1.0; vx := (0.75 temp / mass ) * InverseErf[Random[Real, {-1.,1.}]]</PRE></td></table> <P> <I>Calculate the average and standard deviation for <IMG WIDTH=14 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline136" SRC="img9.gif" > both using your random number distribution and using the analytical solution for a couple of temperatures, say 300 K and 1000 K. What fraction of the atoms have Vx within <IMG WIDTH=18 HEIGHT=20 ALIGN=MIDDLE ALT="tex2html_wrap_inline140" SRC="img10.gif" > standard deviation from the mean?</I> <P> <HR><center> <table border=1> <td><A NAME="tex2html63" HREF="node5.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html69" HREF="Lab5.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html71" HREF="node7.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node6.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="/hamlet-bin/qhandler?exit" target="results"><img height=36 width=36 alt="RESET" src="/UCESIcons/tools/stop.gif"></a></td> <td><a href="/UCES/hamlet/"><img height=36 width=36 alt="HAMLET" src="/UCESIcons/tools/hamlet.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces_info@krellinst.org"><img height=36 width=36 alt="FEEDBACK" src="/UCESIcons/tools/feedback.gif"></a></td> <td><a href="/UCES/hamlet/usage.html"><img height=36 width=36 alt="ABOUT" src="/UCESIcons/tools/about.gif"></a></td> </table> </center> <HR><P><ADDRESS> <I>Hannes Jonsson<BR> Modified by Thomas L. Marchioro II<BR> and the <A NAME="tex2html1" HREF="/UCES/">Undergraduate Computational Engineering and Science project</A> </I> </ADDRESS> </BODY>