Non-Markovian Stochastic Liouville equation and its Markovian representation. Extensions of the continuous time random walk approach

TL;DR

This paper extends the continuous time random walk approach by representing non-Markovian processes through Markovian models, enabling analysis of complex relaxation kinetics and noise effects in stochastic systems.

Contribution

It introduces a Markovian representation for non-Markovian CTRW processes and generalizes the stochastic Liouville equation to account for system influences on noise.

Findings

Derived simple expressions for cascade CTRWs in anomalous processes

Showed how W(t) influences relaxation kinetics

Analyzed non-restrictive correlations in renewal PDFs

Abstract

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.