Robson R E, Blumen A
Research School of Physical Sciences and Engineering, Australian National University, Canberra 2600, Australia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Jun;71(6 Pt 1):061104. doi: 10.1103/PhysRevE.71.061104. Epub 2005 Jun 15.
In this article we give a general prescription for incorporating memory effects in phase space kinetic equation, and consider in particular the generalized "fractional" relaxation time model equation. We solve this for small-signal charge carriers undergoing scattering, trapping, and detrapping in a time-of-flight experimental arrangement in two ways: (i) approximately via the Chapman-Enskog scheme for the weak gradient, hydrodynamic regime, from which the fractional form of Fick's law and diffusion equation follow; and (ii) exactly, without any limitations on gradients. The latter yields complete and exact expressions in terms of generalized Mittag-Lefler functions for experimentally observable quantities. These expressions enable us to examine in detail the transition from the nonhydrodynamic stage to the hydrodynamic regime, and thereby establish the limits of validity of Fick's law and the corresponding fractional diffusion equation.
在本文中，我们给出了在相空间动力学方程中纳入记忆效应的一般方法，并特别考虑了广义的“分数阶”弛豫时间模型方程。我们通过两种方式求解在飞行时间实验装置中经历散射、俘获和去俘获的小信号电荷载流子的该方程：（i）在弱梯度、流体动力学区域中，近似地通过查普曼 - 恩斯科格方法，由此可导出菲克定律和扩散方程的分数阶形式；（ii）精确求解，对梯度没有任何限制。后者以广义米塔格 - 莱夫勒函数给出了实验可观测量的完整且精确的表达式。这些表达式使我们能够详细研究从非流体动力学阶段到流体动力学区域的转变，从而确定菲克定律和相应分数阶扩散方程的有效性范围。
