⚛️ general relativity

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

This paper establishes a unified thermodynamic framework for self-gravitating non-relativistic fluids that derives the most general non-dissipative fluid model, recovers classical and quantum hydrodynamic equations as special cases, and identifies conditions for "classical holography" that enable a single complex-valued wave-function representation.

Original authors: Mátyás Szücs, Péter Ván

Published 2026-10-27

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CC BY 4.0

Original authors: Mátyás Szücs, Péter Ván

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Fluids are the silent architects of our universe, shaping everything from the swirling clouds in our sky to the massive, rotating stars that light up the night. For nearly two centuries, scientists have relied on a single, powerful set of rules to describe how these fluids move and interact. These rules work brilliantly for everyday situations, from the flow of water in a river to the air moving over an airplane wing. However, when physicists push these rules into extreme or unfamiliar territories—such as the behavior of super-cooled gases, the dynamics of the early universe, or the strange world where quantum mechanics meets gravity—the old rules begin to crack. They require new ingredients, like invisible internal fields or complex mathematical corrections, often added in an ad hoc way to make the equations fit the observations. This leaves a lingering question: is there a single, fundamental principle that can generate all these different fluid behaviors, revealing a hidden unity beneath the surface?

A team of researchers has now answered this question by turning to the most reliable law in physics: the Second Law of Thermodynamics. This law, which dictates that energy naturally spreads out and disorder increases in any isolated system, is usually associated with heat and engines. Here, the researchers used it as a strict filter to test which fluid models are physically possible. They started with a fluid that is not just a simple liquid, but one that carries an extra, invisible internal field and whose behavior depends on how its density changes from one point to the next. By demanding that every possible movement of this fluid must obey the Second Law, they derived the most general set of equations that could possibly exist for such a system. The result is a unified framework that naturally produces the equations for ordinary fluids, the equations for self-gravitating gas clouds like stars, and even the equations that describe quantum particles as waves, all emerging from the same thermodynamic roots.

The most striking discovery in this work is a property the authors call "classical holography." In the context of this study, this does not refer to the exotic, high-dimensional theories of the universe often discussed in theoretical physics. Instead, it describes a specific, elegant mathematical shortcut that occurs when a fluid is perfectly smooth and does not lose energy to friction. Under these conditions, the complex forces pushing and pulling on the fluid from the inside can be replaced by a single, simple number field, much like how a complex landscape can be described by a single map of elevation. This means that the motion of the entire fluid can be described as if it were a single particle moving under the influence of a potential, rather than a chaotic collection of interacting parts. This property holds true for ordinary fluids, for fluids that generate their own gravity, and for fluids that behave like quantum waves, provided they are not rotating and their temperature and entropy are linked in a specific way.

When the researchers applied this framework to fluids that do not rotate, they found that the entire system could be rewritten as a single, complex equation. This equation looks exactly like the famous Schrödinger equation used to describe electrons and other quantum particles. In this view, the density of the fluid and the speed of its flow are not separate things but are two sides of the same coin, encoded in a single wave-like function. This provides a fresh perspective on why quantum mechanics often feels like a fluid dynamics problem; it suggests that the wave-like behavior of particles might be a natural consequence of how a fluid-like system must behave to satisfy the laws of thermodynamics. The study does not claim to prove that the universe is a fluid, but it demonstrates that the mathematical structures of quantum mechanics and gravity can emerge naturally from a thermodynamic description of a fluid with specific internal properties.

Beyond unifying these theories, the research points toward a new possibility that had not been previously recognized. The framework suggests a specific way that the density of a fluid could couple directly to its own gravitational field, creating a modified version of gravity that includes subtle corrections based on how the density changes in space. This "density-gravity coupling" is a thermodynamically allowed interaction that could, in theory, alter how gravity behaves on small scales or in dense environments. The authors are careful to note that while this interaction is mathematically consistent with the laws of thermodynamics, its physical reality and whether it can be observed in the real world remain to be tested. It is a suggestion born from the logic of the equations, a new path to explore in the search for a deeper understanding of how matter and gravity interact.

The power of this approach lies in its rigor. Instead of guessing which new terms to add to the equations, the researchers let the Second Law of Thermodynamics do the work. By treating the fluid's internal state and its spatial variations as fundamental ingredients, they showed that the laws governing everything from the flow of water to the collapse of a star are not separate, unrelated discoveries. They are different faces of the same thermodynamic truth. The study confirms that when a fluid is free of friction and rotation, its behavior simplifies in a profound way, allowing it to be described by a single potential field. This "holographic" reduction is not a trick of the math but a fundamental feature of how non-dissipative systems must behave. It bridges the gap between the macroscopic world of stars and the microscopic world of quantum particles, showing that both can be understood through the same lens of thermodynamic consistency.

While the paper provides a robust theoretical foundation, it also highlights the limits of what can be known without further observation. The new density-gravity coupling, for instance, is a valid possibility within the theory, but its stability and observational consequences are still open questions. The researchers emphasize that their work is a construction of possibilities, a map of what is thermodynamically allowed, rather than a final verdict on what the universe is doing. They have shown that the equations of quantum mechanics and gravity are not arbitrary inventions but are deeply rooted in the same principles that govern the flow of heat and the spread of energy. By revealing this common ground, the study offers a clearer path forward for physicists trying to understand how the different layers of reality fit together, suggesting that the key to unification may lie not in new forces, but in a deeper understanding of the old ones.

Technical Summary: Thermodynamic Compatibility, Holographic Property, and Wave-Function Representation of Generalized Weakly Nonlocal Self-Gravitating Non-Relativistic Fluids

Problem Statement
Physics traditionally formulates gravity and quantum mechanics in distinct languages. While Newtonian gravity is often modeled as a self-gravitating fluid and the Schrödinger equation as a quantum-pressure-carrying fluid, these connections have historically been treated separately or through specific, ad-hoc generalizations. The central problem addressed is whether there exists a single, principled thermodynamic framework capable of unifying these diverse continuum models. Specifically, the authors seek to derive the most general fluid model involving density, velocity, energy, and an internal scalar field (and their gradients) that is strictly constrained by the Second Law of Thermodynamics, without relying on variational principles or specific microscopic backgrounds.

Methodology
The authors employ the methodology of nonequilibrium thermodynamics, specifically the Liu procedure, to evaluate the entropy inequality for a weakly nonlocal continuum.

Constitutive State Space: The study defines a state space spanned by density ( $\varrho$ ), velocity ( $\mathbf{v}$ ), specific total energy ( $e$ ), and an internal scalar field ( $\phi$ ). Crucially, the theory is "weakly nonlocal," meaning the constitutive functions (entropy, pressure tensor, energy current, and evolution of the internal field) depend not only on these state variables but also on their spatial gradients (up to second order for density and the scalar field, and first order for velocity and energy).
Thermodynamic Constraints: The Second Law is applied as a conditional inequality. The Liu procedure utilizes Lagrange–Farkas multipliers to separate the entropy production rate density from the balance equations. This rigorous approach determines the specific form of the entropy function, the entropy current, and the constitutive relations for the pressure tensor and energy current.
Energy Representation: The analysis transitions to an energy representation, separating the fluid internal energy from an "extra energy density" ( $\varepsilon$ ) arising from the internal variable and gradients. This allows for the identification of thermodynamic forces and fluxes.
Perfect Fluid Limit: The study focuses on the dissipation-free (perfect fluid) limit, where entropy production is zero. This restricts the constitutive functions to their reversible components, leading to specific conditions for the pressure tensor and the evolution of the internal field.

Key Contributions and Results

Unified Derivation of Fluid Models:
The framework successfully recovers several distinct physical theories as special cases of the same thermodynamically consistent continuum model by selecting specific forms for the extra energy density $\varepsilon$ :
- Euler Fluids: Recovered when $\varepsilon = 0$ .
- Korteweg Fluids: Recovered when $\varepsilon$ depends on density gradients. This includes the hydrodynamic formulation of quantum mechanics (Schrödinger–Madelung fluids) and the Gross–Pitaevskii equation.
- Newtonian Self-Gravitating Fluids: Recovered when the internal field $\phi$ is identified with the gravitational potential $\varphi$ . The derivation naturally yields the Ohanian energy density (sum of interaction and field energy terms).
- Schrödinger–Newton System: Recovered by combining the quantum gradient terms with the gravitational potential.
Classical Holographic Property:
A primary theoretical result is the identification of conditions under which "perfect" fluids exhibit classical holography. This property is defined as the ability to represent the divergence of the pressure tensor ( $\nabla \cdot \mathbf{P}$ ) as the gradient of a scalar potential ( $\varrho \nabla \Phi$ ).
- The authors prove that weak nonlocality (gradient terms) does not obstruct this property; the obstruction arises solely from classical thermal effects.
- The necessary and sufficient condition for classical holography is the integrability condition $\nabla s \times \nabla T = 0$ (coinciding level surfaces of entropy and temperature), which corresponds to the absence of baroclinic vorticity generation.
- When this condition holds, the field-theoretic momentum balance reduces to a particle-like Newton equation: $\dot{\mathbf{v}} = -\nabla \Phi$ .
Wave-Function Representation:
Under the conditions of classical holography and irrotational flow, the authors demonstrate that the mass continuity equation and the Bernoulli equation can be combined into a single complex-valued evolution equation.
- By introducing a wave function $\Psi = \sqrt{\varrho} e^{iS/S_0}$ , the system transforms into a Schrödinger-like equation.
- The "Bohm potential" (quantum potential) emerges naturally as a specific term in the holographic potential derived from the thermodynamic extra energy density.
- The paper clarifies that this is a thermodynamic derivation of the hydrodynamic form of quantum mechanics, not a derivation of quantum mechanics itself; the constant $S_0$ (identified as $\hbar/m$ ) remains a constitutive choice rather than a thermodynamic necessity.
Novel Density–Gravity Coupling:
The framework predicts a previously unrecognized, thermodynamically admissible coupling between density gradients and the gravitational potential.
- This arises from a cross-term in the extra energy density involving $\nabla \varrho \cdot \nabla \varphi$ .
- This coupling leads to a modified Poisson equation where the source term includes a nonlinear gradient correction ( $\nabla \cdot (\nabla \varrho / \varrho)$ ).
- The authors note that while thermodynamically admissible, the stability of this coupling (specifically regarding the convexity of the gradient block) requires further investigation, as it may introduce instabilities at certain density scales.

Significance and Claims
The paper claims to provide a constructive, unified framework based solely on the Second Law of Thermodynamics that explains the structural similarities between classical fluid dynamics, Newtonian gravity, and quantum hydrodynamics.

Unification: It demonstrates that these seemingly disparate theories share a common thermodynamic origin, specifically in how the Second Law constrains non-dissipative continua with internal variables and gradients.
Classical Holography: It establishes that the "holographic" reduction of field equations to particle-like equations is a general property of perfect continua under specific thermal conditions, independent of the specific nature of the internal forces (whether quantum or gravitational).
Methodological Rigor: Unlike approaches relying on variational principles or statistical mechanics, this work derives constitutive relations directly from the entropy inequality using the Liu procedure, ensuring thermodynamic compatibility without assuming a microscopic background.
New Physics: The identification of a thermodynamically consistent density–gravity coupling offers a potential extension to standard Newtonian gravity and the Schrödinger–Newton system, though the paper explicitly states that the physical viability and observational consequences of this coupling remain to be explored.

The authors emphasize that while the framework unifies the structure of these equations, it does not derive the fundamental constants of quantum mechanics (like $\hbar$ ) from thermodynamics; rather, it shows that the Schrödinger equation is a valid constitutive specialization within a broader class of thermodynamically admissible fluid models.

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