PHYSICS 897M-01:Monte Carlo Methods

This course offers a systematic introduction to one of the most powerful and versatile numerical technique for large complex systems. The Monte Carlo method can be applied to solve virtually any problem where one is interested in finding some system property averaged (with the appropriate weight) over a multitude of possible system states. Important applications can be found in natural sciences, engineering, behavioral studies, and finance, to name a few.   

All material in the course is covered by written lecture notes (see website). The course starts with a short introduction to the probability theory and algorithms for generating random numbers – indeed, there is nothing random or probabilistic about computers following a deterministic set of code instructions. It is followed by the theory of Monte Carlo sampling techniques used in simulations of classical models; these include cluster, invaded cluster, histogram, multiple histogram, entropic, tempering, parallel tempering, and kinetic equations. The recommended text for this part of the course is the Monte Carlo Methods in Statistical Physics book by Newman and Barkema. The book is easy to read and it offers a good coverage of classical methods developed prior to 1999 though the actual order of presenting the course material is different. Also, less time is spent on the discussion of programming tricks used in writing codes. Instead, more recent state-of-the-art algorithmic developments and such as Worm Algorithm and Wang-Landau version of the entropic sampling are covered.

The last part of the course deals with quantum Monte Carlo methods.   Since there are no textbooks yet which cover significant recent developments in the field, students are supposed to study lecture notes and research papers to understand (i) when and how quantum mechanical problems can be mapped to classical ones (ii) what are the simulation objects and their states which emerge from the mapping, and (iii) which methods are best suited for the efficient sampling of the resulting configuration space. The learning curve starts with the Variational and Diffusion Monte Carlo methods used for simulations of the ground state (zero temperature) properties. Feynman path-integral representation of quantum mechanics and diagrammatic expansions are introduced next. The course concludes with the discussion of the most efficient modern methods used to work with path-integrals and diagrams.