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Generalized thermodynamic relations for perfect spin hydrodynamics

This paper introduces generalized thermodynamic relations into relativistic perfect spin hydrodynamics to ensure a consistent treatment of spin degrees of freedom based on microscopic calculations, thereby establishing a foundation for unifying different formulations and incorporating dissipative corrections.

Original authors: Wojciech Florkowski, Mykhailo Hontarenko

Published 2026-09-09

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CC BY 4.0

Original authors: Wojciech Florkowski, Mykhailo Hontarenko

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

When matter is crushed together with such ferocity that protons and neutrons melt into a soup of their constituent parts, a new state of matter emerges. This is the quark-gluon plasma, a condition that existed for a fleeting moment after the Big Bang and is recreated today in massive particle accelerators by smashing heavy ions together at nearly the speed of light. For decades, physicists have used a set of rules called hydrodynamics to describe how this super-hot, super-dense fluid flows and expands. These rules treat the fluid like a continuous substance, tracking its energy, pressure, and temperature as it evolves through space and time. However, a new layer of complexity has recently come to light. The particles within this fluid, such as hyperons and vector mesons, possess an intrinsic property called spin, which can be thought of as a tiny, internal rotation. Experiments have shown that this spin is not random; it becomes polarized, aligning in specific directions due to the violent collisions. To understand the full story of these cosmic events, scientists must now update their fluid rules to include how this internal spin behaves, a task that has proven surprisingly difficult.

The challenge lies in the fact that the mathematical tools used to describe spinning fluids have been inconsistent. Different research groups have been using different starting points, leading to conflicting pictures of how spin interacts with the fluid's flow. Some approaches rely on simple, intuitive assumptions about how spin behaves, while others are built on detailed calculations from the microscopic world of particle physics. These two paths have led to a gap in understanding, particularly when the spin polarization is strong. In a recent study, researchers Wojciech Florkowski and Mykhailo Hontarenko from the Jagiellonian University in Poland have bridged this gap. They have introduced a new, generalized set of thermodynamic relations that can handle the full complexity of spinning fluids, ensuring that the macroscopic description of the fluid matches the microscopic reality of its particles.

The core of the problem was that previous models treated the relationship between the fluid's motion and its spin in a way that worked only for weak spin effects. When scientists tried to apply these simple rules to situations where the spin was strong, or when they tried to use the detailed spin structures derived from particle physics, the equations broke down. The researchers realized that the standard way of connecting energy, pressure, and temperature to the flow of the fluid was too rigid. It assumed that the fluid's properties were uniform in all directions relative to its motion, but a spinning fluid is inherently different; it has a preferred direction defined by its spin. By revisiting the fundamental laws of thermodynamics, the authors showed that these laws must be expanded into a more complex, multi-dimensional form. Instead of a single equation linking pressure and temperature, the new framework requires a set of interconnected equations that account for the specific geometry of the spin.

This new approach reveals that the fluid carries additional, hidden currents that were previously ignored. In the old models, the flow of particles and energy was described as moving strictly along with the fluid's bulk motion. The new analysis shows that when spin is included, there are subtle, sideways flows that run perpendicular to the main direction of movement. These flows are not signs of friction or energy loss, which are usually associated with such sideways movements, but are instead a natural, perfect feature of a spinning fluid in equilibrium. The researchers demonstrated that these effects are essential for the theory to remain consistent with the laws of physics, specifically the conservation of energy and angular momentum. They argued that without including these new terms, the theory would be inconsistent with the results obtained from the most detailed microscopic calculations available.

The significance of this work extends beyond just fixing a mathematical inconsistency. It provides a solid foundation for the next generation of simulations used to interpret data from heavy-ion collision experiments. By establishing a consistent framework that works for both weak and strong spin polarization, the authors have removed a major obstacle in the field. Their results show that the thermodynamic relations used to describe these fluids must be tensor-based, meaning they must account for direction and orientation in a way that simple numbers cannot. This insight is crucial for developing more accurate models of dissipative spin hydrodynamics, which will eventually allow scientists to extract precise information about the properties of the quark-gluon plasma from experimental observations. The study confirms that the universe's most extreme fluids are governed by rules that are richer and more intricate than previously imagined, requiring a new language to describe their behavior.

Technical Summary: Generalized Thermodynamic Relations for Perfect Spin Hydrodynamics

Problem Statement
Recent experimental observations of non-zero spin polarization in hyperons and vector mesons produced in relativistic heavy-ion collisions have necessitated the inclusion of spin degrees of freedom in relativistic hydrodynamics. However, a significant theoretical inconsistency exists between different formulations of spin hydrodynamics. Specifically, there is a disconnect between approaches based on kinetic theory (which derive spin tensors from microscopic calculations) and phenomenological approaches (which often assume specific forms for the spin tensor and rely on the positivity of entropy production).

The core issue lies in the thermodynamic relations used in "perfect" (non-dissipative) spin hydrodynamics. Standard formulations extend the scalar thermodynamic identities (Gibbs-Duhem relation, first law) by simply multiplying them by the fluid four-velocity $u^\mu$ to include spin terms. This procedure assumes a specific, simplified structure for the spin tensor ( $S^{\mu,\alpha\beta} = u^\mu S^{\alpha\beta}$ ) that is inconsistent with the more complex structures derived from kinetic theory and microscopic calculations. These microscopic results often contain components of the spin tensor orthogonal to the flow vector $u^\mu$ , which the standard scalar-to-tensor conversion fails to capture. Consequently, existing formulations struggle to maintain consistency between the perfect-fluid limit and the kinetic-theory motivated descriptions required for dissipative corrections.

Methodology
The authors address this inconsistency by critically re-examining the thermodynamic relations for particles with spin 1/2. Their methodology proceeds as follows:

Kinetic Theory Analysis: The authors utilize a classical kinetic model treating spin via an internal angular momentum tensor $s^{\alpha\beta}$ . They define the equilibrium distribution function for particles and antiparticles, incorporating the spin polarization tensor $\omega_{\alpha\beta}$ and the spin chemical potential.
Derivation of Macroscopic Currents: By integrating the equilibrium distribution functions over momentum and spin degrees of freedom, they derive the explicit forms of the baryon current ( $N^\mu$ ), energy-momentum tensor ( $T^{\mu\nu}$ ), spin tensor ( $S^{\lambda,\mu\nu}$ ), and the entropy current ( $S^\mu$ ).
Identification of Orthogonal Components: The integration reveals that the baryon current and entropy current contain "transverse" components proportional to a vector $t^\mu$ (orthogonal to $u^\mu$ , $k^\mu$ , and $\omega^\mu$ ). These components vanish in the spinless limit or at linear order in polarization but are non-zero for general polarization values.
Generalization of Thermodynamic Relations: Instead of relying on the standard scalar relations multiplied by $u^\mu$ , the authors derive generalized tensor forms of the thermodynamic relations directly from the kinetic definitions. They decompose the currents and tensors into components parallel and orthogonal to the flow vector to establish consistent scalar relations for these distinct sectors.

Key Contributions and Results

Generalized Tensor Thermodynamic Relations: The primary result is the derivation of a new set of thermodynamic relations (Eqs. 23, 25, and 26) that replace the standard perfect-fluid forms (Eqs. 11–13).
- The entropy current is given by $S^\mu = T^{\mu\alpha}\beta_\alpha - \frac{1}{2}\omega_{\alpha\beta}S^{\mu,\alpha\beta} - \xi N^\mu + N^\mu$ , where $N^\mu$ is a particle four-current distinct from the standard baryon current $N^\mu$ in the presence of spin.
- These relations hold for any value of the spin polarization tensor $\omega_{\mu\nu}$ , not just in the linear approximation.
- Crucially, these relations naturally incorporate the complex structure of the spin tensor derived from kinetic theory (Eq. 15), including terms orthogonal to the flow vector.
Scalar Decomposition and New Thermodynamic Variables: The authors demonstrate that the standard single scalar thermodynamic identity (Eq. 7) is insufficient. Instead, the generalized relations lead to two coupled scalar equations (Eqs. 32 and 33) describing the "longitudinal" (parallel to $u^\mu$ ) and "transverse" (orthogonal to $u^\mu$ ) sectors of the fluid.
- This reveals that in spin-polarized media, thermodynamic quantities like energy density and pressure acquire transverse corrections that depend on the spin polarization tensor.
- The structure of the energy-momentum tensor is shown to share features with anisotropic magnetohydrodynamics (MHD), possessing terms proportional to $k^\mu k^\nu$ , $\omega^\mu \omega^\nu$ , and mixed terms involving $t^\mu$ .
Consistency with Microscopic Calculations: The new formalism resolves the gap between phenomenological models and kinetic theory. The generalized relations are consistent with the spin tensor structures obtained from microscopic calculations (where $S^{\lambda,\mu\nu}$ contains parts orthogonal to $u^\mu$ ), whereas the traditional approach (multiplying scalar relations by $u^\mu$ ) is shown to be inconsistent with these microscopic results.
Foundation for Dissipative Spin Hydrodynamics: The paper establishes that terms typically associated with dissipative corrections (orthogonal components to the flow) appear naturally at the perfect-fluid level in spin hydrodynamics due to the tensor nature of spin polarization. This finding is critical for constructing consistent dissipative theories, as these terms must be accounted for in the equilibrium reference state before adding non-equilibrium corrections.

Significance and Claims
The authors claim that their work provides the necessary theoretical foundation to unify different formulations of spin hydrodynamics. By introducing generalized thermodynamic relations that respect the tensor structure of spin degrees of freedom, they remove the inconsistency between kinetic-theory-based approaches and phenomenological constructions.

The paper asserts that:

The standard scalar thermodynamic relations must be replaced by tensor forms to correctly describe spin-polarized fluids.
The presence of orthogonal corrections in the perfect-fluid limit is a fundamental feature of spin hydrodynamics, analogous to anisotropic MHD, and not merely a dissipative effect.
These results are universal, as they hold for both Boltzmann and Fermi-Dirac statistics (as demonstrated in the Supplemental Material).
The derived relations form a suitable starting point for introducing dissipative corrections, ensuring that future developments in dissipative spin hydrodynamics are consistent with microscopic physics.

The authors position their work as a "bottom-up" approach that complements "top-down" quantum statistical methods, with the expectation that both frameworks will converge in practical applications to experimental data.

Technical Summary: Generalized Thermodynamic Relations for Perfect Spin Hydrodynamics

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