  We study the quantum dynamics of filling an empty lattice of size $L$, by
connecting it locally with an equilibrium thermal bath that injects
non-interacting bosons or fermions. We adopt four different approaches, namely
(i) direct exact numerics, (ii) Redfield equation, (iii) Lindblad equation, and
(iv) quantum Langevin equation -- which are unique in their ways for solving
the time dynamics and the steady-state. Our setup offers a simplistic platform
to understand fundamental aspects of dynamics and approach to thermalization.
The quantities of interest that we consider are the spatial density profile and
the total number of bosons/fermions in the lattice. The spatial spread is
ballistic in nature and the local occupation eventually settles down owing to
equilibration. The ballistic spread of local density admits a universal scaling
form. We show that this universality is only seen when the condition of
detailed balance is satisfied by the baths. The difference between bosons and
fermions shows up in the early time growth rate and the saturation values of
the profile. The techniques developed here are applicable to systems in
arbitrary dimensions and for arbitrary geometries.
