Generalized Master Equation Via Aging Continuous-Time Random Walks

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This article discusses the problem of the equivalence between continuous-time random walk (CTRW) and generalized master equation (GME).

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11 p.

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Allegrini, Paolo; Aquino, Gerardo; Grigolini, Paolo; Palatella, Luigi & Rosa, Angelo 2003.

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This article discusses the problem of the equivalence between continuous-time random walk (CTRW) and generalized master equation (GME).

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11 p.

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Copyright 2003 American Physical Society. The following article appeared in Physical Review E, 68:5; http://pre.aps.org/abstract/PRE/v68/i5/e056123

Abstract: We discuss the problem of the equivalence between continuous-time random walk (CTRW) and generalized master equation (GME). The walker, making instantaneous jumps from one site of the lattice to another, resides in each site for extended times. The sojourn times have a distribution density ψ(t) that is assumed to be an inverse power law with the power index μ. We assume that the Onsager principle is fulfilled, and we use this assumption to establish a complete equivalence between GME and the Montroll-Weiss CTRW. We prove that this equivalence is confined to the case where ψ(t) is an exponential. We argue that is so because the Montroll-Weiss CTRW, as recently proved by Barkai [E. Barkai, Phys. Rev. Lett. 90, 104101 (2003)], is nonstationary, thereby implying aging, while the Onsager principle is valid only in the case of fully aged systems. The case of a Poisson distribution of sojourn times is the only one with no aging associated to it, and consequently with no need to establish special initial conditions to fulfill the Onsager principle. We consider the case of a dichotomous fluctuation, and we prove that the Onsager principle is fulfilled for any form of regression to equilibrium provided that the stationary condition holds true. We set the stationary condition on both the CTRW and the GME, thereby creating a condition of total equivalence, regardless of the nature of the waiting-time distribution. As a consequence of this procedure the authors create a GME that is a bona fide master equation, in spite of being non-Markov. We note that the memory kernel of the GME affords information on the interaction between system of interest and its bath. The Poisson case yields a bath with infinitely fast fluctuations. We argue that departing from the Poisson form has the effect of creating a condition of infinite memory and that these results might be useful to shed light on the problem of how to unravel non-Markov quantum master equations.

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  • Physical Review E, 2003, College Park: American Physical Society

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  • Publication Title: Physical Review E
  • Volume: 68
  • Issue: 5
  • Peer Reviewed: Yes

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  • 2003

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  • Nov. 24, 2011, 12:20 a.m.

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  • July 22, 2013, 12:10 p.m.

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Allegrini, Paolo; Aquino, Gerardo; Grigolini, Paolo; Palatella, Luigi & Rosa, Angelo. Generalized Master Equation Via Aging Continuous-Time Random Walks, article, 2003; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc67635/: accessed July 21, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.