  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 psi(t) that is assumed to
be an inverse power law. 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
when psi(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 non-stationary, thereby implying aging, while the Onsager
principle, is valid only in the case of fully aged systems. 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
the nature of the waiting time distribution. As a consequence of this procedure
we create a GME that it is a "bona fide" master equation, in spite of being
non-Markovian. 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 into the problem of how to
unravel non-Markovian master equations.
