Hit and miss in higher dimensions:

<BODY LANG="EN"> <HR><center> <table border=1> <td><A NAME="tex2html53" HREF="node4.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html59" HREF="Lab5.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html61" HREF="node6.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node5.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="SECTION00050000000000000000">Hit and miss in higher dimensions:</A></H1> <P> The dart experiment can be generalized to three dimensions by generating triplets of random numbers and calculating how many triplets land inside a sphere that is inscribed in the cube of volume 1. In a similar way, the code can be generalized to any dimension. Multidimensional integrals are very common, for example in statistical mechanics. For each dimension the probability that the random number coordinate lands within the target is less than one, <I>p</I><1, and the joint probability that a set of <I>D</I> coordinates all fall within the target is <IMG WIDTH=17 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline124" SRC="img8.gif" > which becomes smaller and smaller as <I>D</I> is increased. <P> A module for running the dart experiment in arbitrary dimensions is in the file <code>!hannes/Chem465/dartsinDdim</code>. You can load it by typing <P> <table><td valign=top><a href="/hamlet-bin/qhandler/send7.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><code><<dartsinDdim</code></td></table> <P> and you can run it for <I>N</I> darts in <I>D</I> dimensions by typing <P> <table><td valign=top><a href="/hamlet-bin/qhandler/send8.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><code>dartsinDdim[N,D]</code></td></table> <P> <I>Try a few runs, starting with the one dimensional case, <I>D</I>=1, and work your way up until the probability of hitting the target with the number of darts you are throwing becomes negligible. </I> This illustrates the problem of doing high dimensional integrals (remember, the configurational integral of a 100 atom liquid has 300 dimensions!). <P> <HR><center> <table border=1> <td><A NAME="tex2html53" HREF="node4.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html59" HREF="Lab5.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html61" HREF="node6.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node5.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>