  We investigate the use of a logarithmic density variable in estimating the
Lagrangian displacement field, motivated by the success of a logarithmic
transformation in restoring information to the matter power spectrum. The
logarithmic relation is an extension of the linear relation, motivated by the
continuity equation, in which the density field is assumed to be proportional
to the divergence of the displacement field; we compare the linear and
logarithmic relations by measuring both of these fields directly in a
cosmological N-body simulation. The relative success of the logarithmic and
linear relations depends on the scale at which the density field is smoothed.
Thus we explore several ways of measuring the density field, including
Cloud-In-Cell smoothing, adaptive smoothing, and the (scale-independent)
Delaunay tessellation, and we use both a Fourier space and a geometrical
tessellation approach to measuring the divergence. We find that the relation
between the divergence of the displacement field and the density is
significantly tighter with a logarithmic density variable, especially at low
redshifts and for very small (~2 Mpc/h) smoothing scales. We find that the
grid-based methods are more reliable than the tessellation-based method of
calculating both the density and the divergence fields, though in both cases
the logarithmic relation works better in the appropriate regime, which
corresponds to nonlinear scales for the grid-based methods and low densities
for the tessellation-based method.
