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

TL;DR

The authors develop a thermodynamically consistent, weakly nonlocal fluid framework with an internal scalar variable using the Liu procedure, deriving constitutive relations and energy representations that respect the Second Law. They show that perfect fluids exhibit a classical holographic property that recasts momentum balance in terms of a scalar potential, which in turn permits vorticity conservation and a Madelung-type wave-function formulation. This approach yields Schrödinger–Madelung and Schrödinger–Newton dynamics as natural limits or specializations, including nonlocal gravity-inspired couplings, thereby linking gravity, thermodynamics, and quantum-like dynamics within a single formalism. The work provides a unified, thermodynamically grounded route to classical-quantum connections and outlines multiple physically relevant reductions (Euler, Korteweg, Newtonian gravity, and generalized SN equations) with explicit field equations.

Abstract

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 Schrödinger 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 Schrödinger--Newton system, are also discussed. The presented thermodynamic framework can shed light on some connections between formulations of classical and quantum physics.