EXTREME DEVIATIONS AND APPLICATIONS

Stretched exponential probability density functions (pdf), having the form of the exponential of minus a fractional power of the argument, a re commonly found in turbulence and other areas. They can arise becaus e of an underlying random multiplicative process. For this, a theory o f 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 x(2) f ''(x) --> +infinity for \x\ --> infinity. Addi tional 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 simil ar to e(-nf(X/n)). This theory is then applied to products of independ ent random variables, such that their logarithms are in the above clas s, yielding usually stretched exponential tails. An application to fra gmentation is developed and compared to data from fault gouges. The pd f by mass is obtained as a weighted superposition of stretched exponen tials, reflecting the coexistence of different fragmentation generatio ns. For sizes near and above the peak size, the pdf is approximately l og-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 re sult together with a simple multiplicative model of local atom configu rations. Finally, we indicate the possible relevance to the distributi on of small-scale velocity increments in turbulent how.