dos Santos B Coutinho, Tsallis C
Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro, RJ 22290-180, Brazil.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Dec;82(6 Pt 1):061119. doi: 10.1103/PhysRevE.82.061119. Epub 2010 Dec 13.
We consider a class of single-particle one-dimensional stochastic equations which include external field, additive, and multiplicative noises. We use a parameter θ ∊ [0,1] which enables the unification of the traditional Itô and Stratonovich approaches, now recovered, respectively, as the θ=0 and θ=1/2 particular cases to derive the associated Fokker-Planck equation (FPE). These FPE is a linear one, and its stationary state is given by a q-Gaussian distribution with q=(τ+2M(2-θ))/(τ+2M(1-θ)<3), where τ ≥ 0 characterizes the strength of the confining external field and M ≥ 0 is the (normalized) amplitude of the multiplicative noise. We also calculate the standard kurtosis κ(₁) and the q-generalized kurtosis κ(q) (i.e., the standard kurtosis but using the escort distribution instead of the direct one). Through these two quantities we numerically follow the time evolution of the distributions. Finally, we exhibit how these quantities can be used as convenient calibrations for determining the index q from numerical data obtained through experiments, observations, or numerical computations.
我们考虑一类包含外场、加性噪声和乘性噪声的单粒子一维随机方程。我们使用一个参数θ∈[0,1]，它能统一传统的伊藤方法和斯特拉托诺维奇方法，现在分别作为θ = 0和θ = 1/2的特殊情况恢复，以推导相关的福克 - 普朗克方程（FPE）。这些FPE是线性的，其稳态由q - 高斯分布给出，其中q = (τ + 2M(2 - θ))/(τ + 2M(1 - θ) < 3)，这里τ≥0表征限制外场的强度，M≥0是乘性噪声的（归一化）幅度。我们还计算了标准峰度κ(₁)和q - 广义峰度κ(q)（即使用伴护分布而非直接分布的标准峰度）。通过这两个量，我们数值跟踪分布的时间演化。最后，我们展示如何将这些量用作方便的校准，以便从通过实验、观测或数值计算获得的数值数据中确定指数q。
