  In this paper, we analyze the dynamics of the Coulomb Glass lattice model in
three dimensions near a local equilibrium state by using mean-field
approximations. We specifically focus on understanding the role of localization
length ($\xi$) and the temperature ($T$) in the regime where the system is not
far from equilibrium. We use the eigenvalue distribution of the dynamical
matrix to characterize relaxation laws as a function of localization length at
low temperatures. The variation of the minimum eigenvalue of the dynamical
matrix with temperature and localization length is discussed numerically and
analytically. Our results demonstrate the dominant role played by the
localization length on the relaxation laws. For very small localization lengths
we find a crossover from exponential relaxation at long times to a logarithmic
decay at intermediate times. No logarithmic decay at the intermediate times is
observed for large localization lengths.
