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Volume 13: Number 4: Article 4
Introductory Remarks on Large Deviation Statistics
Anton Amann, Universitätsklinik für Anästhesie und Allgemeine Intensivmedizin,
Leopold-Franzens-Universität Innsbruck, Anichstr. 35, A-6020 Innsbruck,
Austria
Harald Atmanspacher, Institut für Grenzgebiete der Psychologie,
Wilhelmstr. 3a, D-79098 Freiburg, Germany
and Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstrasse,
D-85740 Garching, Germany
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" non-equilibrium
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 fluctuations
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.
Keywords: large deviations, Shannon entrophy, Jaynes' maximum entropy
principle, multifractals
FULL TEXT:
Introductory Remarks on Large
Deviation Statistics
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