Physics 329: Monte Carlo Techniques

• General Reference
  • Numerical Recipes, Chapter 7, Sections 7.0 to 7.3
• Source code covered in class can be found on the phy329 account on einstein in ~phy329/rand/ex1 and ~phy329/rand/ex2
• Handout (PS) and slides (PS).
• Lecture 1 (Dec 2)
  • tsrand.f: Demonstration program showing use of srand() to set seed for uniform random number generator rand(). Plot of first 50 random numbers generated using tsrand with seeds 0, 1, 2 and 3. Note that there may be excess correlation between sequences generated with seeds separated by some constant amount, so seeds themselves should generally be chosen randomly. As pointed out in Numerical Recipes, it is healthy to always view system-supplied random number generators with a certain amount of suspicion, particularly if you are going to generate a lot of random numbers. However, I feel that the SGI supplied routine will suffice for our purposes.
  • nurand.f: Fortran routine for generating non-uniform deviates given user-supplied probablility-distribution function (PDF). Sample PDFs, pdfs.f for generating uniform and gaussian-distributed deviates. tnurand.f: Driver program for nurand.f. Plot showing normalized counts (1000 bins) for gaussian-distributed deviates---blue: n = 100 000 random numbers, red: n = 10 000 000 random numbers, dashed black line: peak of analytic PDF. Doit: Shell script which performs runs then invokes sm with command file sm_nurand to create plot.
• Lecture 2 (Dec 4)
  • dla.f: Program which performs Diffusion Limited Aggregation (DLA) simulation in 2D with variable non-stochastic central force (as discussed in class). This central "bias" speeds cluster formation but alters the grown cluster's "fractal properties" (for example, the cluster's radius grows more slowly with particle number for any non-zero bias than for zero bias). Sample usage on SGIs.
  • Plot of final state in pure DLA simulation with 10 000 particles. Supermongo input file which produced the plot. Detail of above simulation. Same simulation after an additional 50 000 particles have been added. Here, the effect of launching all particles from a fixed, finite radius is apparent. Bias = 0.0625 simulation after 10 000 steps. Note how structure is already slightly less "distributed" than the pure DLA case. Bias = 0.1 and Bias = 0.9 simulations, also after 10 000 steps. It is left as an exercise to the reader to investigate and/or predict run time as a function of bias.
  • A nice DLA specimen from the Laboratory of Computer Science at MIT.
  • An "embedded" specimen from Eric Weeks' (UT grad student) collection of algorithmically-generated pictures.