  We present a model that generalizes the Bose-Fermi mapping for strongly
correlated 1D bosons in an optical lattice, to cases in which the average
number of atoms per site is larger than one. This model gives an accurate
account of equilibrium properties of such systems, in parameter regimes
relevant to current experiments. The application of this model to
non-equilibrium phenomena is explored by a study of the dynamics of an atom
cloud subject to a sudden displacement of the confining potential. Good
agreement is found with results of recent experiments. The simplicity and
intuitive appeal of this model make it attractive as a general tool for
understanding bosonic systems in the strongly correlated regime.
