Generalized Hydrodynamics Revisited

A number of attempts to formulate a continuum description of complex states have been proposed to circumvent more cumbersome many-body and simulation methods. Typically these have been quantum systems and the resulting phenomenologies frequently called "quantum hydrodynamics". This objective is placed in a formally controlled context using the exact macroscopic conservation laws for number density, energy density, and momentum density, together with standard tools of non-equilibrium statistical mechanics. These continuum equations entail perfect fluid fluxes that are functionals of the chosen fields, plus unknown irreversible energy and momentum fluxes. Typically, the latter are obtained from a solution to the Liouville-von Neumann equation for small space and time variations. Instead, here we avoid that restriction by requiring that the unknown irreversible fluxes deliver the exact linear response functions for the fields. In this way, a non-linear generalized hydrodynamic description is obtained, valid across a broad range of length and time scales. The example of electrons in a given ion configuration is described to make contact with current phenomenological "quantum hydrodynamics".