Entropy, Large Deviations, and Statistical Mechanics: 271 (Die Grundlehren der Mathematischen Wissenschaften) - Hardcover

Ellis, R. S.

Hardcover

ISBN 10: 038796052X ISBN 13: 9780387960524

Publisher: Springer-Verlag New York Inc., 1985

Synopsis

This book has two main topics: large deviations and equilibrium statistical mechanics. I hope to convince the reader that these topics have many points of contact and that in being treated together, they enrich each other. Entropy, in its various guises, is their common core. The large deviation theory which is developed in this book focuses upon convergence properties of certain stochastic systems. An elementary example is the weak law of large numbers. For each positive e, P{ISn/nl 2: e} con- verges to zero as n --+ 00, where Sn is the nth partial sum of indepen- dent identically distributed random variables with zero mean. Large deviation theory shows that if the random variables are exponentially bounded, then the probabilities converge to zero exponentially fast as n --+ 00. The exponen- tial decay allows one to prove the stronger property of almost sure conver- gence (Sn/n --+ 0 a.s.). This example will be generalized extensively in the book. We will treat a large class of stochastic systems which involve both indepen- dent and dependent random variables and which have the following features: probabilities converge to zero exponentially fast as the size of the system increases; the exponential decay leads to strong convergence properties of the system. The most fascinating aspect of the theory is that the exponential decay rates are computable in terms of entropy functions. This identification between entropy and decay rates of large deviation probabilities enhances the theory significantly.

About this title

About the Author

Richard S. Ellis received his B.A. degree in mathematics and German literature from Harvard University in 1969 and his Ph.D. degree in mathematics from New York University in 1972. After spending three years at Northwestern University, he moved to the University of Massachusetts, Amherst, where he is a Professor in the Department of Mathematics and Statistics and Adjunct Professor in the Department of Judaic and Near Eastern Studies. His research interests in mathematics focus on the theory of large deviations and on applications to statistical mechanics and other areas. Information on his interests outside mathematics is available at http://www.math.umass.edu/~rsellis. He is Alison’s husband and Melissa’s and Michael’s father.

Edition details

• Publisher: Springer-Verlag New York Inc.
• Publication date: 1985
• Language: English
• ISBN 10: 038796052X
• ISBN 13: 9780387960524
• Binding: Hardcover
• Number of pages: 379