The Large Deviation Approach to Statistical Mechanics - 2009

Title : The Large Deviation Approach to Statistical Mechanics Author(s): Touchette, Hugo Link(s) : http://arxiv.org/abs/0804.0327

First mathematical example: Given a sequence \(b=b_1b_2\cdots b_n\) where \(b_i\in \{0,1\}\) is a coin flip, define \(R_n = \frac{1}{n}\sum_{i=1}^{n}b_i\), i.e. the fraction of 1s in \(b\). We want to know the probability that \(R_n\) is one of the rational values \(0,\frac{1}{n},\frac{2}{n},\cdots,\frac{n}{n}\). Since all bit strings occur with probability \(2^{-n}\) we need to count how many bit strings are there such that \(R_n(b)=r\).