Multiplicative noise: A mechanism leading to nonextensive statistical mechanics

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Abstract
A large variety of microscopic or mesoscopic models lead to generic results that accommodate naturally within Boltzmann–Gibbs statistical mechanics [based on S1≡−k∫du p(u)ln p(u)]. Similarly, other classes of models point toward nonextensive statistical mechanics [based on Sq≡k[1−∫du[p(u)]q]/[q−1], where the value of the entropic index q∈R depends on the specific model]. We show here a family of models, with multiplicative noise, which belongs to the nonextensive class. More specifically, we consider Langevin equations of the type u̇=f(u)+g(u)ξ(t)+η(t), where ξ(t) and η(t) are independent zero-mean Gaussian white noises with respective amplitudes M and A. This leads to the Fokker–Planck equation ∂tP(u,t)=−∂u[f(u)P(u,t)]+M∂u{g(u)∂u[g(u)P(u,t)]}+A∂uuP(u,t). Whenever the deterministic drift is proportional to the noise induced one, i.e., f(u)=−τg(u)g′(u), the stationary solution is shown to be P(u,∞)∝{1−(1−q)β[g(u)]2}1/(1−q) [with q≡(τ+3M)/(τ+M) and β=(τ+M/2A)]. This distribution is precisely the one optimizing Sq with the constraint 〈[g(u)]2〉q≡{∫du [g(u)]2[P(u)]q}/{∫du [P(u)]q}=const. We also introduce and discuss various characterizations of the width of the distributions.