科研之友
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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 density 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 nonMarkovian SLE, which describes the effect of CTRW- like noise on the relaxation of dynamic and stochastic systems, is generalized to take into account the influence of relaxing systems on the statistical properties of noise. Some applications of the generalized non- Markovian SLE are discussed. In particular, it is applied to study two modifications of the CTRW approach. One of them considers cascaded CTRWs in which the controlling process is actually a CTRW- like one controlled by another CTRW process, controlled in turn by a third one, etc. Within the MR a simple expression for the PDF W (t ) of the 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 the long- time tailed W (t ). The cascaded CTRWs can model the effect of the complexity of a system on the 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 the correlations of PDFs of consecutive renewals. The analysis shows that fluctuations of W (t ) can strongly affect the kinetics of the process. Possible manifestations of this effect are discussed.
- 出版日期2008-3
