By Didier Sornette
Innovations, tools and methods of statistical physics within the learn of correlated, in addition to uncorrelated, phenomena are being utilized ever more and more within the ordinary sciences, biology and economics in an try and comprehend and version the big variability and dangers of phenomena. this can be the 1st textbook written via a widely known professional that gives a contemporary up to date advent for employees outdoor statistical physics. The emphasis of the e-book is on a transparent figuring out of strategies and techniques, whereas it additionally presents the instruments that may be of instant use in purposes. even if this ebook developed out of a direction for graduate scholars, it is going to be of significant curiosity to researchers and engineers, in addition to to post-docs in geophysics and meteorology.
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Extra info for Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools
The location of this transition is at the inﬂection point deﬁned by d2 G< (x)/dx2 = 0. For simplicity, we consider unimodal pdf’s such that f (x) is monotonically increasing. 78) noting f = df /dx and f = d2 f /dx2 . 8 Maximum of Random Variables and Extreme Value Theory 27 large x. In contrast, for power law tails, this term approaches a constant for large x. 79) ∆x dx leading to ∆x ∼ 1 . e. the tail of the distribution decays faster than any power law. 81) f (x) = f (x∗ ) + (x − x∗ )fx=x∗ + (x − x∗ )2 fx=x∗ + .
It is thus important to determine the statistical properties of such rare but extreme events, with applications to the calculation of the largest risks that insurance companies and governments must face, to the construction of wave and tsunami barriers, to the deﬁnition of engineering building codes and so on. The theory of extreme events is called extreme value theory (EVT). The two classic references are Gumbel  and Galambos . A more recent book  provides a synthetis of recent results on EVT, with application to ﬁnance and insurance.
Consider two objects of dimensions d1 and d2 imbedded in a space of dimension d. It is a well-known result that the intersection of the two objects has dimension d1 + d2 − d with probability one. For instance, two planes in space intersect generically along a line (2 + 2 − 3 = 1). A plane and a line in space intersect generically at a point (2 + 1 − 3 = 0). e. for the set of crossing to become almost vanishing. In other words, in a space of four or more dimensions, a random walk has very little chance to cross itself and this explains why four dimensions plays a special role in theories of interacting ﬁelds, such as spin models that we will study later on.
Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools by Didier Sornette