Nonextensive Statistical Mechanics

148 4 Stochastic Dynamical Foundations of Nonextensive Statistical Mechanicswhich is the value of x to be used in g(x). This is the well-known Itô–Stratonovichcontroversy [303,307]. In the Itô definition, the value before the pulse must be used,whereas in the Stratonovich definition, the values before and after the pulse contributein a symmetric way. In the particular instance when noise is purely additive,both definitions agree. In what follows, we shall adopt the Stratonovich definition(the Itô definition leads in fact to very similar results). By using standard procedures[303, 304], Eq. (4.98) leads toP(u, t)twhere the current is defined as follows:withj(u, t) ≡ J(u)P(u, t) − j(u, t)=− , (4.100)u[D(u)P(u, t)]u, (4.101)J(u) ≡ f (u) + Mg(u)g ′ (u) , (4.102)D(u) ≡ A + M[g(u)] 2 . (4.103)Equation (4.100) can be rewritten as a Fokker–Planck equation, namelyP(x, t)t[ f (x)P(x, t)]=− + M (g(x)x x[g(x)P(x, t)])+ A 2 P(x, t).xx 2 (4.104)In some processes, the deterministic drift derives from a potential-like functionV (x) = (τ/2)[g(x)] 2 , where τ is some nonnegative proportionality constant. Therefore,using f (x) =−dV/dx, we obtain the conditionf (x) =−τg(x)g ′ (x) . (4.105)Let us note that the particular case g(x) ∝ f (x) ∝ x, which is a natural firstchoice for a physical system, verifies this condition. However, since no extra calculationaldifficulties emerge, we will discuss here the more general case of Eq.(4.105). Notice that, in the absence of deterministic forcing, condition (4.105) issatisfied for any g(x) by setting τ = 0.We shall restrict here to the stationary solutions corresponding to no flux boundaryconditions (i.e., j(−∞) = j(∞) = j(u) = 0), although more general conditionscould in principle also be considered. If Eq. (4.105) is satisfied, the stationarysolution P(u, ∞) isoftheq-exponential form, namelyP(u, ∞) ∝ e −β[g(u)]2q , (4.106)