Thermodynamic compatibility, holographic property and wave-function representation of generalized weakly nonlocal self-gravitating non-relativistic fluids

A thermodynamic analysis of weakly nonlocal non-relativistic fluids is presented under the assumption that an additional scalar field also contributes to the dynamics. The most general evolution of this field and the constitutive relations for the pressure tensor and energy current density are determined by the Liu procedure. Classical holography of perfect (\ie non-dissipative) fluids is generally proven, according to which the divergence of the pressure tensor can be given by the gradient of a corresponding scalar potential. A consequence of holographic property is vorticity conservation, which opens the way toward wave-function representation of hydrodynamic equations to obtain the Schr\"odinger equation. Another special case of the derived fluid model is Newtonian gravity, when the internal variable is the gravitational potential itself. Coupled phenomena, such as the Schr\"odinger--Newton system, are also discussed. The presented thermodynamic framework can shed light on some connections between formulations of classical and quantum physics.