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摘要翻译：
在马尔可夫表示(MR)和基于连续时间随机游动(CTRW)的非马尔可夫随机Liouville方程(SLE)中，详细分析了连续时间随机游动(CTRW)方法的一些具体特征和扩展。在MR中，CTRW过程由多维马尔可夫过程表示。在此表示中，波动更新的概率分布函数W(t)与某一马尔可夫控制过程在某一跳变状态下的再发生概率分布函数联系起来。在MR中，将描述类CTRW噪声对动态和随机系统松弛影响的非马尔可夫SLE推广到考虑松弛系统对噪声统计性质的影响。应用广义非马尔可夫SLE研究了CTRW方法的两种改进。其中一种是考虑控制过程实际上是类CTRW过程的级联CTRW，由另一个CTRW过程控制，再由第三个CTRW过程控制等。在MR范围内，根据级联中控制PDFs的马尔可夫变体，得到了总控制过程的PDF W(t)的简单表达式。对于由长时间尾W(t)确定的异常过程，该表达式特别简单和有指导意义。级联CTRWs可以模拟系统的复杂性对弛豫动力学的影响（在玻璃、分形、分支介质、超参数结构等中）。另一个CTRW修正描述了由波动W(t)控制的过程动力学。在MR中，该问题以一般形式进行分析，没有对连续更新的PDFs相关性的限制性假设。分析表明，W(t)能强烈地影响该过程的动力学。讨论了这种效应的可能表现形式。
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英文标题：
《Non-Markovian Stochastic Liouville equation and its Markovian
representation. Extensions of the continuous time random walk approach》
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作者：
A. I. Shushin
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最新提交年份：
2007
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分类信息：
一级分类：Physics 物理学
二级分类：Statistical Mechanics 统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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一级分类：Physics 物理学
二级分类：Soft Condensed Matter 软凝聚态物质
分类描述：Membranes, polymers, liquid crystals, glasses, colloids, granular matter
膜，聚合物，液晶，玻璃，胶体，颗粒物质
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英文摘要：
Some specific features and extensions of the continuous time random walk (CTRW) approach are analyzed in detail within the Markovian representation (MR) and CTRW-based non-Markovian stochastic Liouville equation (SLE). In the MR CTRW processes are represented by multidimensional Markovian ones. In this representation the probability distribution function (PDF) W(t) of fluctuation renewals is associated with that of reoccurrences in a certain jump state of some Markovian controlling process. Within the MR the non-Markovian SLE, which describes the effect of CTRW-like noise on relaxation of dynamic and stochastic systems, is generalized to take into account the influence of relaxing systems on statistical properties of noise. The generalized non-Markovian SLE is applied to study two modifications of the CTRW approach. One of them considers the cascaded CTRWs in which the controlling process is actually CTRW-like one controlled by another CTRW process, controlled in turn by the third one, etc. Within the MR simple expression for the PDF W(t) of total controlling process is obtained in terms of Markovian variants of controlling PDFs in the cascade. The expression is shown to be especially simple and instructive in the case of anomalous processes determined by long time tailed W(t). The cascaded CTRWs can model the effect of complexity of a system on relaxation kinetics (in glasses, fractals, branching media, ultrametric structures, etc.). Another CTRW-modification describes the kinetics of processes governed by fluctuating W(t). Within the MR the problem is analyzed in a general form without restrictive assumptions on correlations of PDFs of consecutive renewals. The analysis shows that W(t) can strongly affect the kinetics of the process. Possible manifestations of this effect are discussed.
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PDF链接：
https://arxiv.org/pdf/705.2358
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