Reddit reviews Statistical Mechanics: Algorithms and Computations (Oxford Master Series in Physics)

http://www.amazon.com/Statistical-Mechanics-Computations-Computational-Theoretical/dp/0198515367

The prof. did a course on coursera; fun, but, more useful to teach a physicist python than the other way around. The book is unfortunately a little too dense for me. [hmm, no, I'm not a python. CS geek.]
hmm ... in retrospect, less the libraries than the use of python. Still, quite interesting.

this book provides a very gentle introduction:
http://www.amazon.com/Statistical-Mechanics-Computations-Computational-Theoretical/dp/0198515367




The first example in that book is the following monte carlo algorithm for finding an approximate value for the area of a unit circle, which you probably know about already:

Generate 10000 uniformly random numbers in (0,1). This gives 5000 uniformly random pairs (x_n,y_n) in (0,1)x(0,1), i.e. the unit square. Now count how many of these pairs satisfy x_n^2 +y_n^2 < 1. Divide this count by 5000. This is your estimate for the area.


Of course you can generalize this to finding the volume of higher dimensional balls in R^d. If you didn't know how to evaluate the corresponding integral exactly then it would be a plausible way to estimate the volume numerically. Indeed one of the main uses of MC(MC) methods is in high dimensional integration.

Edit: Here's another example. Choosing a uniformly random shuffled deck of cards is easy - at least on a computer - choose a random card from the deck; that will be the top card, then choose one of the remaining 51 cards to be the second card, etc....


However, humans don't have a built in function for choosing one card uniformly at random from a deck. The general idea to get around this limitation is to implement some sort of Markov Chain on the space of card configurations, which, when run for a sufficiently long amount of steps, results in a sufficiently shuffled deck:
http://www.youtube.com/watch?v=adNeBLkP8ZM

A lot of recent research in this area has been focused on getting good asymptotics for how many steps you need the Markov chain for in order to get a sufficiently 'shuffled deck'.

I liked Krauth's Statistical Mechanics and Computations. You might actually be extremely interested in it since it goes over density matrix computation and path integrals. https://www.amazon.com/Statistical-Mechanics-Algorithms-Computations-Physics/dp/0198515367 People also recommended Sethna's book but I haven't read it.

As for complexity there's a lot of stuff out there. I actually had the pleasure of sitting in Stanley's class, and he makes this chapter available for free http://polymer.bu.edu/hes/ssbook-chapter5.pdf, and I would also recommend his paper "Scaling, Universality, and Renormalization". Those are some pretty good general starts to go over power laws, but I find a lot of those scale free systems are phenomenological and have very specific studies and papers instead of anything generic. You might be interested in Barabasi's Network Science book if you want something more applied and less mathematical, but again it's focused on a specific application.