  We investigate thermalization process of boosted charged AdS black holes in
the Einstein-Maxwell system in the presence of an ionic lattice formed by
spatially varying chemical potential. We calculate perturbations of the black
holes by the lattice and investigate how the momentum relaxation occurs through
umklapp scattering. In the WKB approximation, both of the momentum relaxation
rate and entropy production rate are analytically obtained and the first law of
black holes is derived in the irreversible process. Interestingly, both the
analytical and numerical calculations show that the momentum relaxation rate or
the entropy production rate does not approach zero in the zero temperature
limit unless the velocity of the black hole is zero. In the dual field theory
side, this indicates that persistent current does not exist even in the zero
temperature limit, implying that the "ionic lattice" does not behave as a
perfect lattice in a strongly coupled dual field theory.
