By Piero Olla (auth.)

This textbook deals a sophisticated undergraduate or preliminary graduate point creation to subject matters equivalent to kinetic idea, equilibrium statistical mechanics and the idea of fluctuations from a contemporary standpoint. the purpose is to supply the reader with the required instruments of chance concept and thermodynamics (especially the thermodynamic potentials) to permit next examine at complicated graduate point. whilst, the booklet bargains a bird's eye view on arguments which are frequently left out primarily curriculum classes. additional good points contain a spotlight at the interdisciplinary nature of the topic and in-depth dialogue of other interpretations of the idea that of entropy. whereas a few familiarity with uncomplicated thoughts of thermodynamics and likelihood idea is thought, this doesn't expand past what's usually acquired in simple undergraduate curriculum courses.

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**Sample text**

Suppose that the statistics is spatially homogeneous, so that ψ(x)ψ(x ) = C(x − x ) In this case, the following relation will hold: ψk ψk = = dd x dd x ψ(x)ψ(x ) exp[−i(k · x + k · x )] dd y C(y) exp(−ik · y) dd x exp[−i(k + k ) · x ], We notice that the integral (2π )−d dd x exp[−i(k + k ) · x ] is just a representation of the Dirac delta δ(k + k ). 8 The Wiener-Khinchin Theorem 33 the Fourier transform of the correlation function. This relation between correlations of Fourier components, ψk , and the Fourier transform of the correlation, Ck , is the content of the Wiener-Khinchin theorem.

P. Bouchaud, A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127 (1990) More on this also in: • D. ) in the form N XN = xk . k=1 A fundamental question is the existence of limit forms at large N for the PDF ρ(XN ). Such limit distributions indeed exist, and their form depends solely on the behavior of the PDF ρ(x) at large values of the argument: the so-called tails of the distribution. In particular, the existence of the first moments of the distribution is crucial in the determination of the limit form for ρ(XN ).

The most severe choice, of course, is k = 1, that is equivalent to assuming that fN fully factorizes in products of f1 ’s (equivalently, that ρN factorizes in products of ρ1 ’s). In particular: f2 (x, z; t) f1 (x, t)f1 (z, t). 9) This assumption goes under the name of mean field approximation, and corresponds in fact to disregarding the fluctuations f˜1 − f1 in the dynamics. Utilizing the mean field approximation Eq. 9) in Eq. 8), we obtain the nonlinear diffusion equation ∂2 ∂f1 (x, t) = 2 ∂t ∂x and, if f1 varies at scale dz q(x − z)f1 (x, t)f1 (z, t), λ: ∂ 2 f 2 (x, t) ∂f1 (x, t) =α 1 2 , ∂t ∂x +∞ α= q(x)dx.