Nonextensive Statistical Mechanics

144 4 Stochastic Dynamical Foundations of Nonextensive Statistical MechanicsFig. 4.23 Both panels represent probability density function P (Y ) vs. Y (properly scaled) in log–linear (top) and log–log (bottom) scales, where Y represents the sum of N independent variablesX each of them having a q-Gaussian distribution with q = 3/2 (< 5/3). Since the variables areindependent and the variance is finite, P (Y ) converges to a Gaussian as it is visible. It is alsovisible in the log–linear representation that, although the central part of the distribution approachesa Gaussian, the power-law decay subsists even for large N as depicted in log–log representation(from [252]).4.8 Generalizing the Langevin EquationThe standard Langevin equation is given by [302, 303]ẋ = f (x) + η(t) , (4.96)where x(t) is a stochastic variable, f (x) is an arbitrary function which representssome deterministic drift, and η(t) is a Gaussian-distributed zero-mean white noisesatisfying〈η(t) η(t ′ )〉=2 A δ(t − t ′ ) . (4.97)The noise amplitude A ≥ 0 stands for additive. The deterministic drift f (x) can beinterpreted either as a damping force (whenever x is a velocity-like quantity) or as anexternal force (when motion is overdamped and x represents a position coordinate).Other interpretations are possible as well, depending on the particular system we arefocusing on. This equation is known to lead to the standard Fokker–Planck equation(Fourier’s heat equation), whose basic solutions are Gaussians in the variable x/ √ t.