By Andrea Gabrielli, F. Sylos Labini, Michael Joyce, Luciano Pietronero
The physics of scale-invariant and complicated platforms is a singular interdisciplinary box. Its rules let us examine normal phenomena in a considerably new and unique means, ultimately resulting in unifying ideas self sustaining of the specific constitution of the structures. the target is the learn of complicated, scale-invariant, and extra normal stochastic buildings that seem either in house and time in an unlimited number of typical phenomena, which express new different types of collective behaviors, and the fostering in their knowing. This ebook has been conceived as a methodological monograph during which the most equipment of contemporary statistical physics for cosmological constructions and density fields (galaxies, Cosmic Microwave heritage Radiation, etc.) are offered intimately. the most objective is to offer sincerely, to a attainable point, those tools, with a undeniable mathematical accuracy, delivering additionally a few paradigmatic examples of functions. this could bring about a brand new and extra normal framework for the statistical research of the various new facts about the varied cosmic constructions which represent the big scale Universe and for his or her theoretical interpretation and modeling.
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Additional info for Statistical Physics for Cosmic Structures (Lecture Notes in Physics)
More precisely, it comes from the Feynman graphs expansion of complete correlation functions in ﬁeld theory . In this context one can see that the cumulants are given by only the sum of connected graphs. 8). First of all, we see that C2 (r) ˜ in the case of a mass distribution with ρ0 > 0) measures (or equivalently ξ(r) the spatial memory of mass density ﬂuctuations on the scale r. In fact, if C2 (r) is diﬀerent from zero, it means that the knowledge of the mass density at a generic point of the space inﬂuences the conditional probability density of the mass density at another point at distance r from the ﬁrst one.
16) i=1 N where p˜i (ki ) is the FT of pi (xi ). 6-bis) If Y = i=1 Xi , where the variables Xi are mutually independent, then the characteristic function φy (k) of the stochastic variable Y is simply given by N φy (k) = p˜i (k) . 17) 38 2 Uniform and Correlated Mass Density Fields We can now introduce the cumulant generating function ψ(k) which is deﬁned by: ψ(k) = log ˜(k) . 18) Let us assume that all the moments of p(x) exist and are ﬁnite, so that ˜(k), and therefore ψ(k), can be expanded in Taylor series.
KN }. The fact that p(x) is a probability density function, and the general theorems of the Fourier transform (FT) imply, among others, the following simple properties of the characteristic function ˜(k): 1) ˜(0) = 1. p(k)| ≤ 1. 2) |˜ 3) ˜(k) is a uniformly continuous function of its argument for all real k. N i=1 4) If the moment Xili exists, it is given by: Xili = n n i=1 i i=1 ∂ ∂ki li p˜(k) . 13) are also called the complete correlation coefﬁcients of the stochastic vector X, and they are the discrete analogues of ρ(r2 ) .
Statistical Physics for Cosmic Structures (Lecture Notes in Physics) by Andrea Gabrielli, F. Sylos Labini, Michael Joyce, Luciano Pietronero