  We study the equation of motion for the Noether current in an electron gas
within the framework of the Schwinger-Keldysh Closed-Time-Path formalism. The
equation is shown to be highly non-linear and irreversible even for a
non-interacting, ideal gas of electrons at non-zero density. We truncate the
linearised equation of motion, written as the Laurent series in Fourier space,
so that the resulting expressions are local in time, both at zero and at small
finite temperatures. Furthermore, we show that the one-loop Coulomb
interactions only alter the physical picture quantitatively, while preserving
the characteristics of the dynamics that the electric current exhibits in the
absence of interactions. As a result of the composite nature of the Noether
current, {\it composite sound} waves are found to be the dominant IR collective
excitations at length scales between the inverse Fermi momentum and the mean
free path that would exist in an interacting electron gas. We also discuss the
difference and the transition between the hydrodynamical regime of an ideal
gas, defined in this work, and the hydrodynamical regime in phenomenological
hydrodynamics, which is normally used for the description of interacting gases.
