  We address the equilibrium and out-of-equilibrium behavior of the particle
density in many-body systems undergoing quantum transitions driven by the
chemical potential $\mu$. They originate from a nontrivial interplay between
noncritical short-range and critical long-range quantum fluctuations. As a
paradigmatic model we consider the one-dimensional fermionic Kitaev chain, for
which very accurate numerical studies can be performed, up to $O(10^4)$ sites.
The search for dynamic scaling behaviors of the particle density is complicated
by the fact that its equilibrium (ground state) behavior is dominated by
short-range fluctuations, giving rise to regular background terms and peculiar
logarithmic terms from resonances between renormalization-group perturbations
associated with the energy and identity operator families within the conformal
field theory. To study these issues, we focus on two dynamic protocols, either
instantaneous quenches or quasi-adiabatic changes of $\mu$ to the critical
value $\mu_c$, unveiling out-of-equilibrium scaling behaviors of the particle
density, which arise from the critical modes, within a dynamic finite-size
scaling framework.
