Nonextensive Statistical Mechanics

110 4 Stochastic Dynamical Foundations of Nonextensive Statistical Mechanicsrecovers the Fokker–Planck equation). Finally, at a larger scale, we enter into thethermodynamical description, i.e., the level referred to as the macroscopic one. Statisticalmechanics bridges from the microscopic level up to the macroscopic one.For pedagogical reasons, we first discuss the Fokker–Planck-like equations(Sections 4.2, 4.3, and 4.4), and then the Langevin-like equations (Section 4.5). It isonly in the next chapter that we focus on the microscopic level, with its deterministicequations.4.2 Normal DiffusionThe basic equation of normal diffusion is the so-called heat equation, firstintroducedby Fourier. It is given, for d = 1, byp(x, t)t= D 2 p(x, t)x 2 (D > 0) , (4.1)where D is the diffusion coefficient. Let us assume the simplest initial condition,namelyp(x, 0) = δ(x) , (4.2)where δ(x) is Dirac’ s delta distribution. The corresponding solution is given byp(x, t) =1√2π Dte −x2 /2Dt(t ≥ 0) . (4.3)We can verify that∫ ∞−∞dx p(x, t) = 1 (t ≥ 0) , (4.4)and that〈x 2 〉≡∫ ∞−∞dx x 2 p(x, t) = Dt . (4.5)This corresponds to what is normally referred to as normal diffusion. Many typesof functions 〈x 2 〉(t) exist (see, for instance, [265, 266]). But a very frequent one is〈x 2 〉∝t μ (μ ≥ 0) , (4.6)where x can be a d-dimensional quantity (and not necessarily the simple d = 1case that we are focusing here). Diffusion is said normal or anomalous for μ = 1or μ ≠ 1, respectively. If μ