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Generalized master equation via aging continuous-time random walks.

Paolo Allegrini1, Gerardo Aquino, Paolo Grigolini

  • 1Istituto di Linguistica Computazionale del Consiglio Nazionale delle Ricerche, Area della Ricerca di Pisa, Via Moruzzi 1, 56124 Pisa, Italy.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 20, 2003
PubMed
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We establish the equivalence between continuous-time random walks (CTRW) and generalized master equations (GME) under stationary conditions. This allows for a bona fide master equation, even for non-Markov systems, offering insights into quantum master equations.

Area of Science:

  • Statistical Physics
  • Quantum Mechanics
  • Complex Systems

Background:

  • Continuous-time random walks (CTRW) model particle movement with variable waiting times.
  • Generalized master equations (GME) describe system dynamics but can be complex.
  • The Onsager principle relates to equilibrium and time-reversal symmetry in physical systems.

Purpose of the Study:

  • To investigate the equivalence between CTRW and GME.
  • To determine the conditions under which the Onsager principle holds for CTRW and GME.
  • To develop a non-Markovian GME that functions as a bona fide master equation.

Main Methods:

  • Assumed fulfillment of the Onsager principle.
  • Established equivalence between GME and Montroll-Weiss CTRW.

Related Experiment Videos

  • Analyzed sojourn time distributions (inverse power law, exponential, Poisson).
  • Considered dichotomous fluctuations and stationary conditions.

    • Main Results:

    • Equivalence between GME and Montroll-Weiss CTRW is limited to exponential sojourn times due to non-stationarity (aging) in CTRW.
    • The Onsager principle is strictly valid only for fully aged systems or those without aging (Poisson distribution).
    • Stationary conditions on both CTRW and GME lead to total equivalence, irrespective of waiting-time distribution.
    • A non-Markovian GME equivalent to a bona fide master equation was derived.

    Conclusions:

    • Stationary conditions are key to unifying CTRW and GME, enabling a master equation for non-Markov processes.
    • The memory kernel of the GME reveals system-bath interactions; Poisson distributions imply fast bath fluctuations.
    • Deviations from Poisson distributions lead to infinite memory, potentially clarifying the unraveling of non-Markov quantum master equations.