On the local thermodynamic relations in relativistic spin hydrodynamics

This paper demonstrates through rigorous quantum statistical analysis of free fermions that the commonly assumed local differential thermodynamic relations in relativistic spin hydrodynamics fail even at global equilibrium, revealing unavoidable corrections to the spin density-pressure relationship that cannot be resolved by entropy-gauge transformations.

The Gist

Imagine a bustling city where traffic flows smoothly. Physicists use a set of rules called "hydrodynamics" to predict how this traffic (fluids) moves. For decades, they've had a perfect rulebook for normal traffic. But recently, scientists discovered that in the extreme conditions of heavy-ion collisions (like smashing atoms together at near-light speed), the particles don't just move; they also spin, like tiny tops.

To describe this, physicists created a new rulebook called "Relativistic Spin Hydrodynamics."

The Old Assumption: The "Perfect Recipe"

In the past, when building this new rulebook, scientists made a very educated guess. They assumed that the relationship between the fluid's pressure, temperature, and spin was just a simple extension of the old rules.

Think of it like baking a cake. You know that if you add more sugar, the cake gets sweeter. The scientists assumed that if you add more "spin" to the fluid, the pressure would change in a perfectly predictable, linear way. They wrote down a "recipe" (a mathematical equation) that said:

"The change in pressure caused by spin is exactly equal to the amount of spin present."

They used this recipe to build the rest of their theory, assuming it was the solid foundation for everything else.

The Discovery: The Recipe is Wrong

In this paper, Francesco Becattini and Rajeev Singh act like rigorous food critics who decided to taste-test that recipe under a microscope. They didn't just guess; they used a powerful quantum statistical method (a very precise way of counting how particles behave) to check if the recipe held up in two specific scenarios:

1. Massless particles (like photons or ultra-fast electrons).
2. Massive particles (heavier particles).

They looked at these particles in a state of perfect global balance (Global Thermodynamic Equilibrium), where the fluid is rotating and accelerating.

The Result: The recipe failed.

When they calculated the actual pressure change caused by spin, it did not match the amount of spin present.

• The Analogy: Imagine you add a cup of sugar to a cake, and you expect the sweetness to go up by exactly one "sweetness unit." But when you taste it, the sweetness has gone up by one unit plus a mysterious extra sprinkle of something else.
• The Reality: The change in pressure had an "extra term." It wasn't just the spin; it was the spin plus a correction factor related to how the fluid was accelerating and rotating. This correction was just as big as the spin itself.

The "Magic Wand" Attempt

The authors then asked: "Is there a way to fix the recipe? Maybe we just defined 'entropy' (a measure of disorder) slightly wrong?"

In physics, there's a concept called a "gauge transformation," which is like a magic wand that lets you redefine how you measure things without changing the physical reality. They tried using this "entropy-gauge transformation" to see if they could tweak the definitions of pressure and entropy to make the old recipe work again.

The Result: The magic wand didn't work. No matter how they redefined the entropy current (the flow of disorder), the extra "mysterious sprinkle" in the pressure equation remained. The fundamental relationship they had been relying on simply does not exist in the real quantum world.

The Conclusion

The paper concludes that the traditional method of assuming these simple differential thermodynamic relations is incorrect for relativistic spin hydrodynamics.

• What this means: If scientists continue to use the old, simple recipe, their models of how spinning fluids behave will be missing important pieces. They might be overlooking specific "dissipative terms" (ways energy is lost or friction occurs) that are crucial for an accurate description.
• The Takeaway: To get the physics right, we cannot just guess the rules based on old intuition. We must use the rigorous quantum statistical method to derive the rules from scratch, because the universe is more complex than our simple "recipes" suggest.

In short: The old math for spinning fluids was a good guess, but it turns out to be wrong. We need to rewrite the rulebook using the actual quantum laws of nature.

Technical Summary

Technical Summary: On the local thermodynamic relations in relativistic spin hydrodynamics

Problem Statement
The development of relativistic hydrodynamic frameworks incorporating spin as an additional degree of freedom has accelerated following experimental observations of spin polarization in heavy-ion collisions. A central challenge in this field is the derivation of constitutive relations for the energy-momentum and spin tensors. Traditionally, these derivations rely on the assumption that local differential thermodynamic relations hold, specifically the relation $dp = s dT + n d\mu + \frac{1}{2}S_{\mu\nu}d\omega^{\mu\nu}$, where $p$ is pressure, $s$ is entropy density, $n$ is charge density, $S_{\mu\nu}$ is spin density, and $\omega_{\mu\nu}$ is the spin potential.

However, previous studies (e.g., Ref. [27]) have argued that while the Gibbs relation $Ts = \rho + p - \mu n - \frac{1}{2}\omega_{\mu\nu}S^{\mu\nu}$ can be satisfied by definition, the differential relation linking the derivative of pressure with respect to the spin potential to the spin density ($\partial p / \partial \omega_{\mu\nu} = S_{\mu\nu}$) is not generally guaranteed. This paper investigates whether this differential relation holds at global thermodynamic equilibrium for free fermion systems and whether it can be restored via entropy-gauge transformations (redefinitions of the entropy current).

Methodology
The authors employ a rigorous quantum statistical mechanics framework to derive the local equilibrium density operator and the entropy current. The analysis proceeds as follows:

1. Formalism: The local equilibrium density operator $\hat{\rho}_{LE}$ is constructed by maximizing entropy subject to constraints on energy-momentum, charge, and spin densities. This introduces Lagrange multiplier fields: the four-temperature vector $\beta_\mu$, reduced chemical potential $\zeta$, and reduced spin potential $\Omega_{\mu\nu}$.
2. Entropy Current Derivation: The entropy current $s^\mu$ is derived from the partition function, ensuring it vanishes in the zero-temperature limit (pure state). The pressure $p$ is defined via the thermodynamic potential current $\phi^\mu$ as $p = T \phi^\mu u_\mu$.
3. Specific Models: The authors calculate thermodynamic functions (energy density $\rho$, charge density $n$, spin density $S_{\mu\nu}$, and pressure $p$) up to second order in thermal vorticity for two specific systems at global equilibrium with rotation and acceleration:
   • Massless free fermions: Using exact solutions available in the canonical pseudo-gauge.
   • Massive free fermions: Using second-order expansions in thermal vorticity within the canonical pseudo-gauge.
4. Verification: The authors explicitly compute the derivative of the pressure function with respect to the spin potential, $\partial p / \partial \omega_{\mu\nu}$, and compare it against the calculated spin density $S_{\mu\nu}$.
5. Gauge Transformation Analysis: The paper investigates whether an "entropy-gauge transformation" (adding the divergence of an antisymmetric field to the entropy current) can redefine the pressure and entropy to restore the validity of the differential thermodynamic relations.

Key Results
The study demonstrates that the standard differential thermodynamic relation $\partial p / \partial \omega_{\mu\nu} = S_{\mu\nu}$ fails for both massless and massive free fermions at global thermodynamic equilibrium in the canonical pseudo-gauge.

• Massless Fermions: For massless Dirac fermions, the derivative of the pressure with respect to the spin potential yields:
  $$\frac{\partial p}{\partial \omega_{\lambda\nu}} \bigg|_{T,\mu} = S_{\lambda\nu} + \frac{T^2}{12}(a_\nu u_\lambda - a_\lambda u_\nu)$$
  where $a_\mu$ is the acceleration field and $u_\mu$ is the fluid four-velocity. The term proportional to acceleration is of the same order as the spin density itself.
• Massive Fermions: A similar calculation for massive fermions in the non-relativistic limit confirms the presence of an additional term proportional to $(a_\lambda u_\nu - a_\nu u_\lambda)$ in the derivative of the pressure, which does not match the spin density.
• Entropy-Gauge Transformation: The authors prove that no redefinition of the entropy current (entropy-gauge transformation) can eliminate these correction terms. The necessary condition for such a transformation to exist requires a specific mixed-derivative equality ($\partial S_{\lambda\nu}/\partial T = \partial s / \partial \omega_{\lambda\nu}$) which is violated by the calculated thermodynamic functions.

Significance and Claims
The paper concludes that the traditional method of assuming differential thermodynamic relations (specifically Eq. 6 in the text) to derive constitutive relations in relativistic spin hydrodynamics is inappropriate. The failure of these relations is not an artifact of a specific pseudo-gauge choice that can be fixed by redefining the entropy current; rather, it is a fundamental feature of the thermodynamics of spinning systems at equilibrium.

The authors assert that relying on the incorrect differential relations may lead to the omission of dissipative terms for the spin tensor. Consequently, accurate determination of constitutive relations in relativistic spin hydrodynamics requires the use of rigorous quantum-statistical methods (as proposed in Refs. [27, 33]) rather than educated guesses based on standard thermodynamic differentials. The work underscores the necessity of moving beyond traditional thermodynamic assumptions when incorporating spin into relativistic hydrodynamic frameworks.