J. Phys. I France

**7**(1997) 1155-1171

## Extreme Deviations and Applications

**U. Frisch**

^{1}and D. Sornette^{2, 3}^{1}Observatoire de la Côte d'Azur CNRS UMR 6529, BP 4229, 06304 Nice Cedex 4, France

^{2}Department of Earth and Space Sciences and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90095-1567, USA

^{3}Laboratoire de Physique de la Matière Condensée CNRS URA 190, Université de Nice-Sophia Antipolis, Parc Valrose, BP 91, 06108 Nice Cedex 2, France

(Received 16 October 1996, revised 6 March 1997, accepted 15 May 1997)

** Abstract **

Stretched exponential probability density functions (pdf), having the form of the exponential of minus a fractional power
of the argument, are commonly found in turbulence and other areas. They can arise because of an underlying random multiplicative
process. For this, a theory of *extreme* deviations is developed, devoted to the far tail of the pdf of the sum
*X* of a finite number
*n* of independent random variables with a common pdf e
^{-f(x)}. The function
*f*(*x*) is chosen (i) such that the pdf is normalized and (ii) with a strong convexity condition that
*f*''(*x*)>0 and that
for
. additional technical conditions ensure the control of the variations of
*f*''(*x*). The tail behavior of the sum comes then mostly from individual variables in the sum all close to
*X*/*n* and the tail of the pdf is
. This theory is then applied to products of independent random variables, such that their logarithms are in the above class,
yielding usually stretched exponential tails. An application to fragmentation is developed and compared to data from fault
gouges. The pdf by mass is obtained as a weighted superposition of stretched exponentials, reflecting the coexistence of different
fragmentation generations. For sizes near and above the peak size, the pdf is approximately log-normal, while it is a power
law for the smaller fragments, with an exponent which is a decreasing function of the peak fragment size. The anomalous relaxation
of glasses can also be rationalized using our result together with a simple multiplicative model of local atom configurations.
Finally, we indicate the possible relevance to the distribution of small-scale velocity increments in turbulent flow.

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*Les Editions de Physique 1997*