Srokowski Tomasz
Institute of Nuclear Physics, Polish Academy of Sciences, PL 31-342 Kraków, Poland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Jul;92(1):012125. doi: 10.1103/PhysRevE.92.012125. Epub 2015 Jul 20.
The non-Markovian stochastic dynamics involving Lévy flights and a potential in the form of a harmonic and nonlinear oscillator is discussed. The subordination technique is applied and the memory effects, which are nonhomogeneous, are taken into account by a position-dependent subordinator. In the nonlinear case, the asymptotic stationary states are found. The relaxation pattern to the stationary state is derived for the quadratic potential: the density decays like a linear combination of the Mittag-Leffler functions. It is demonstrated that in the latter case the density distribution satisfies a fractional Fokker-Planck equation. The densities for the nonlinear oscillator reveal a complex picture, qualitatively dependent on the potential strength, and the relaxation pattern is exponential at large time.
讨论了涉及 Lévy 飞行以及谐波和非线性振荡器形式势的非马尔可夫随机动力学。应用了从属技术，并通过位置依赖的从属过程考虑了非均匀的记忆效应。在非线性情况下，找到了渐近稳态。推导了二次势到稳态的弛豫模式：密度像 Mittag-Leffler 函数的线性组合一样衰减。结果表明，在后一种情况下，密度分布满足分数阶福克 - 普朗克方程。非线性振荡器的密度呈现出复杂的情况，定性地取决于势的强度，并且在长时间时弛豫模式是指数型的。
