By Didier Sornette
Concepts, equipment and methods of statistical physics within the learn of correlated, in addition to uncorrelated, phenomena are being utilized ever more and more within the normal sciences, biology and economics in an try and comprehend and version the big variability and dangers of phenomena. this is often the 1st textbook written via a widely known professional that gives a latest updated creation for staff open air statistical physics.
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Additional resources for Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools
Example text
Variables. 28), we obtain a relationship between the l-th cumulant of p(n+l)(x) and that of p(n)(x): cfn+l)(k) = 2cfn)(k). One cannot think of a simpler result! 71) 42 2. Sums of Random Variables Since a pdf is completely determined from the knowledge of all its cumulants (barring some special cases mentioned above), we can thus write P(x, N, c1, c2, c3, ... , cz, ... ) N = P(x, 2 , 2c1, 2c2, 2c3, ... , 2cz, ... ). 72) P(x, N, c 1, c 2, c 3, ... , cz, ... )denotes the pdf of the sum X, characterized by the cumulants c1, c2, c3, ...
This is justified for large N (leading to large x*) since x - x* is typically of order Ax and thus (x _ x*/~=x* ~ J~=x* f~=x* (f~=x* )2 * __ d [1l(df ldx)]l x=x . dx - (1. 70). We have shown that the last term in (1. 73) decays to zero for large x*, leading to the convergence of the distribution of the maximum to the Gumbel distribution. 71) of f(x). e. 74) where a= exp [ -~(x- x*) 2 J:=x*] -1 . (1. 75) Consider as an illustration the power-exponential pdf for which f (x) = xc with c > 0. We find x* "'(lnN) 1 fc, Ax"' (x*) 1 -c, leading to Axjx* "'(x*)-c and a~ -c(c-1)/2lnN.
This simply reflects the fact that the fractal object becomes more and more tenuous in comparison to the space of dimension d in which it is embedding, when its dimension dr is less than d. 23) where N(r) is the number of "elements" seen at the scaler, by the identification r --+ -Ji and N --+ ~ . To apply these notions to a random walk and derive its fractal dimension, one must define what one means by its "mass". A natural measure is its length (l) ~, for which one thinks of the step length as massive hard sticks.