  The crust of a neutron star is known to melt at a temperature that increases
with increasing matter density, up to about $10^{10}$ K. At such high
temperatures and beyond, the crustal ions are put into collective motion and
the associated entropy contribution can affect both the thermodynamic
properties and the composition of matter. We studied the importance of this
effect in different thermodynamic conditions relevant to the inner crust of the
proto-neutron star, both at beta equilibrium and in the fixed-proton-fraction
regime. To this aim, we solved the hydrodynamic equations for an ion moving in
an incompressible, irrotational, and non-viscous fluid, with different boundary
conditions, thus leading to different prescriptions for the ion effective mass.
We then employed a compressible liquid-drop approach in the one-component
plasma approximation, including the renormalisation of the ion mass to account
for the influence of the surrounding medium. We show that the cluster size is
determined by the competition between the ion centre-of-mass motion and the
interface properties, namely the Coulomb, surface, and curvature energies. In
particular, including the translational free energy in the minimisation
procedure can significantly reduce the optimal number of nucleons in the
clusters and lead to an early dissolution of clusters in dense
beta-equilibrated matter. On the other hand, we find that the impact of
translational motion is reduced in scenarios where the proton fraction is
assumed constant and is almost negligible on the inner-crust equation of state.
Our results show that the translational degrees of freedom affect the
equilibrium composition of beta-equilibrated matter and the density and
pressure of the crust-core transition in a non-negligible way, highlighting the
importance of its inclusion when modelling the finite-temperature inner crust
of the (proto-)neutron star.
