As well known, the generalized Langevin equation with a memory kernel decreasing at large times as an inverse power law of time describes the motion of an anomalously diffusing particle. Here, we focus attention on some new aspects of the dynamics, successively considering the memory kernel, the particle's mean velocity, and the scattering function. All these quantities are studied from a unique angle, namely, the discussion of the possible existence of a distribution of relaxation times characterizing their time decay. Although a very popular concept, a relaxation time distribution cannot be associated with any time-decreasing quantity (from a mathematical point of view, the decay has to be described by a completely monotonic function). Technically, we use a memory kernel decaying as a Mittag-Leffler function (the Mittag-Leffler functions interpolate between stretched or compressed exponential behaviour at short times and inverse power law behaviour at large times). We show that, in the case of a subdiffusive motion, relaxation time distributions can be defined for the memory kernel and for the scattering function, but not for the particle's mean velocity. The situation is opposite in the superdiffusive case.
Subdiffusion equation with fractional Caputo time derivative with respect to another function in modeling transition from ordinary subdiffusion to superdiffusion
The Dance of Odd-Diffusive Particles: A Fourier Approach
University of Augsburg（奥格斯堡大学）；Leibniz-Institute for Polymer Research（莱布尼茨高分子研究所）；University of Potsdam（波茨坦大学）；Asia Pacific Centre for Theoretical Physics（亚太理论物理中心）
Anomalous Polymer Dynamics Is Non-Markovian: Memory Effects and The Generalized Langevin Equation Formulation
任何关于聚合物物理学的第一门课程都教导聚合物的标记单体的动力学是有害的亚扩散，即，对于某些$α<1$，标记单体的均方位移随着$t^α$增加，直到聚合物的末端弛豫时间$τ$。超过时间$τ$，标记单体的运动变为扩散。聚合物物理学中异常动力学的经典例子是单一聚合物系统，如幻影劳斯（Phantom Rouse）、自回避劳斯（Self-avoiding Rouse）、自回避齐姆（Self-avoiding Zimm）、爬行（reptation）、通过膜中窄孔的易位（translocation through a narrow pore in a membrane）和多聚合物系统，如聚合物熔体。在这篇教学论文中，我报告说，所有这些情况下的聚合物系统的异常动力学的特点是强大的幂律记忆内核内的{ \it统一}广义朗之万方程（GLE）计划，因此，是非马尔可夫。幂律记忆核的指数与聚合物对局部应变的弛豫响应有关，并且来自聚合物的平衡统计物理。在这些系统中的聚合物的标记的单体的异常动力学，然后通过波动耗散定理（FDT）从幂律记忆内核的GLE再现。使用这个GLE制定我进一步表明，这些聚合物系统上的（弱）施加的字段所造成的漂移的特性也得到了相应的内存内核。
