Introduction to Large Deviations

One of the common themes of probability theory and statistical mechanics is the discovery of regularity in the midst of chaos. The laws of probability theory, which include laws of large numbers and central limit theorems, summarize the behavior of a stochastic system in terms of a few parameters (e.g., mean and variance). In statistical mechanics, one derives macroscopic properties of a substance from a probability distribution that describes the complicated interactions among the individual constituent particles. A central concept linking the two fields is entropy.1 The term was introduced into thermodynamics by Clausius in 1865 after many years of intensive work by him and others on the second law of thermodynamics. An early important step in its development and enrichment was the discovery by Boltzmann of a statistical interpretaton of entropy. Boltzmann’s discovery, which was published in 1877, has three parts. We have augmented part (c) to include the possibility of phase transitions.