  We study a microscopic mean-field model for the dynamics of the electron
glass, near a local equilibrium state. Phonon-induced tunneling processes are
responsible for generating transitions between localized electronic sites,
which eventually lead to the thermalization of the system. We find that the
decay of an excited state to a locally stable state is far from being
exponential in time, and does not have a characteristic time scale. Working in
a mean-field approximation, we write rate equations for the average occupation
numbers, and describe the return to the locally stable state using the
eigenvalues of a rate matrix, A, describing the linearized time-evolution of
the occupation numbers. Analyzing the probability distribution of the
eigenvalues of A we find that, under certain physically reasonable assumptions,
it takes the form $P(\lambda) \sim \frac{1}{|\lambda|}$, leading naturally to a
logarithmic decay in time. While our derivation of the matrix A is specific for
the chosen model, we expect that other glassy systems, with different
microscopic characteristics, will be described by random rate matrices
belonging to the same universality class of A. Namely, the rate matrix has
elements with a very broad distribution, i.e., exponentials of a variable with
nearly uniform distribution.
