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We study the equation of motion for the Noether current in an electron gaswithin the framework of the Schwinger-Keldysh Closed-Time-Path formalism. Theequation is shown to be highly non-linear and irreversible even for anon-interacting, ideal gas of electrons at non-zero density. We truncate thelinearised 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 smallfinite temperatures. Furthermore, we show that the one-loop Coulombinteractions only alter the physical picture quantitatively, while preservingthe characteristics of the dynamics that the electric current exhibits in theabsence of interactions. As a result of the composite nature of the Noethercurrent, {\it composite sound} waves are found to be the dominant IR collectiveexcitations at length scales between the inverse Fermi momentum and the meanfree path that would exist in an interacting electron gas. We also discuss thedifference and the transition between the hydrodynamical regime of an idealgas, defined in this work, and the hydrodynamical regime in phenomenologicalhydrodynamics, which is normally used for the description of interacting gases.

Dynamics of the electric current in an ideal electron gas: a sound mode inside the quasi-particles