Introductory Remarks on Large Deviation Statistics
نویسنده
چکیده
The physical concept of entropy as it is used in thermodynamics is related to the mathematical formulation of a Shannon entropy. Usually only the Shannon entropy of equilibrium distributions such as a canonical distribution is considered. Large deviations statistics goes beyond that framework. Entropies are considered for arbitrary distributions or physical states, and they describe, e.g., \how fast" nonequilibrium distributions and states \die out" with increasing number of degrees of freedom or increasing number of particles. Hence the concept of an entropy acquires a new meaning, referring to the statistical uctuations in collectives of empirical events. In the particular case of experiments with independent and identically distributed (i.i.d.) events, Shannon entropy can be shown to play its usual role (Sanov's theorem). Jaynes' maximum entropy principle, important in statistical physics, is a consequence of Sanov's theorem and thereby obtains a precise interpretation. In the general case of non-i.i.d. events, all sorts of (even non-convex) entropies can arise. As illustrative examples, large deviation statistics of phase transitions and multifractals are addressed.
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