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A rotating gas of massless fermions admits a local pressure that is thermodynamically consistent, satisfying the Euler relation and the differential thermodynamic identities at once.

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2026-08-04 15:54 UTC pith:VRLCJ2FI

load-bearing objection The GCE construction is solid and the local-pressure extension is a plausible new proposal, but the claimed resolution of the Euler-relation problem rests on an unproven spin-density ansatz, so the abstract overstates what is demonstrated. the 3 major comments →

arxiv 2509.17640 v2 pith:VRLCJ2FI submitted 2025-09-22 hep-th nucl-th

Thermodynamics of rotating fermions

Victor E. Ambrus , Aleksandar Geci\'c This is my paper
classification hep-th nucl-th
keywords RotationDirac fermionsGrand canonical ensembleThermodynamic potentialSpin hydrodynamicsVortical effectsEuler relationMassless fermions
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tries to establish that the long-standing problem of defining a thermodynamic pressure for a rigidly rotating gas of Dirac fermions is solvable, at least for massless fermions. Starting from the grand canonical ensemble, the authors construct a thermodynamic potential current and then pass to a local description in which the state is fixed by local temperature, chemical potential, and spin potential. They propose a closed-form pressure and show that its derivatives give entropy, charge, and spin densities that satisfy both the Euler relation and the standard differential thermodynamic identities. If correct, this removes a previously reported obstruction in spin hydrodynamics and shows that vorticity and spin potential must enter the pressure jointly.

Core claim

The central claim is that, for massless Dirac fermions in rigid rotation, the local thermodynamic pressure can be written as the classical pressure plus explicit quantum corrections involving the spin potential and the kinematic vorticity and acceleration. The paper's main result, Eqs. (52)-(55), gives this pressure together with matching entropy, charge, and spin densities and a dynamic pressure that absorbs the difference between the effective pressure (one third of the energy density) and the thermodynamic pressure. The authors verify that ∂P/∂T = s, ∂P/∂μ = Q_V, and ∂P/∂Ω_{αβ} = S^{αβ}_C, so the Euler relation and the differential relations hold simultaneously. This is presented as a res

What carries the argument

The argument is carried by the thermodynamic potential current φ^μ, built from the grand-canonical potential, whose contraction with the fluid four-velocity defines the thermodynamic pressure. For massless fermions the key identity is the proposed local pressure formula, Eq. (52): P = P_cl − (Ω²+a²)/4 σ^ω_{A;cl} + (Ω⁴+6Ω²a²)/192π² + (44ω²a²+51a⁴)/2880π², where Ω is the magnetic part of the spin potential, ω and a are the kinematic vorticity and acceleration, and P_cl and σ^ω_{A;cl} are classical quantities. Differentiating P with respect to temperature, chemical potential, and spin potential yields entropy, charge, and spin densities that satisfy the Euler relation and the integrability cond

Load-bearing premise

The load-bearing premise is the postulated form of the local pressure in Eq. (52) and the companion energy density in Eq. (54), which for Ω≠ω are not derived from the quantum energy-momentum tensor; if the true local state gives a different pressure or energy density, the claimed consistency fails.

What would settle it

Compute the canonical energy-momentum tensor of massless Dirac fermions in a local equilibrium state with spin potential explicitly different from the vorticity tensor, and compare the resulting energy density with Eq. (54), i.e. check whether ϵ=3(P+Π) holds with Π from Eq. (55). Any mismatch, or any violation of ∂P/∂T=s, would settle that the proposed pressure is not the physical one.

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If this is right

  • If the main result is correct, a local equilibrium state of rotating massless fermions has a well-defined thermodynamic pressure distinct from the effective pressure, with the difference carried by a dynamic pressure.
  • The Euler relation and the differential thermodynamic relations can be satisfied simultaneously, so no subtraction of zero-temperature terms is needed for this system.
  • In spin hydrodynamics, the local pressure is a function of both the spin potential and the kinematic vorticity, not of the vorticity alone.
  • The derived entropy, charge, and spin densities provide concrete benchmark expressions that can be compared against direct quantum field theory mode-sum calculations.
  • The construction gives a quantum-field-theoretical grounding for spin hydrodynamics of rotating fluids, extending classical kinetic-theory results to include quantum corrections.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • When the spin potential and vorticity coincide, the pressure formula reduces to a pure vorticity-dependent expression, so the difference between thermal equilibrium and a generic local state is a concrete prediction that a direct mode-sum calculation could check.
  • The same construction may extend to massive fermions or to states with axial or helical chemical potentials, where analytic expressions already exist; the dynamic-pressure mechanism would then need to absorb additional correction terms.
  • The appearance of dynamic pressure proportional to a² suggests that rotation accompanied by radial acceleration necessarily produces a non-equilibrium pressure contribution, which could be probed in accelerated rotating fluid setups.
  • Treating the spin potential and the vorticity as independent thermodynamic variables may provide a template for other off-equilibrium hydrodynamic theories beyond spin, such as those with a separate heat or shear potential.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper studies the thermodynamics of a rigidly rotating gas of free massless Dirac fermions. In Sec. 3 the authors construct the grand canonical potential and its associated thermodynamic potential current, showing how entropy, charge, angular momentum, and spin densities emerge, and emphasizing the exactness of the global relations. In Sec. 4 they pass to a local thermodynamic description with local temperature T, chemical potential μ, and spin potential Ω_{αβ}. For the equilibrium case where the spin potential coincides with the kinematic vorticity ω_{αβ}, they derive the local thermodynamic pressure and related densities from the known quantum expectation values, Eq. (47). The main new result, Eqs. (52)-(55), extends this by postulating a form for the canonical spin density, Eq. (51), a form for the pressure, Eq. (52), and a form for the energy density, Eq. (54). The authors claim these quantities satisfy the Euler relation and the differential thermodynamic relations (56).

Significance. If the main result were actually derived from the underlying quantum state, it would provide an explicit thermodynamically consistent local pressure for rotating massless fermions with an independent spin potential, thereby resolving the tension reported in Refs. [17,18]. The paper has clear strengths: the global grand-canonical treatment in Sec. 3 is exact and well organized, and the equilibrium QFT calculation in Sec. 4.2 is a useful baseline. The proposed ansatz is simple and correctly reduces to the equilibrium result when Ω=ω. However, as it stands the central off-equilibrium claim is an assumption rather than a derivation; the consistency conditions in Eq. (56) are built into the ansatz by construction. The significance of the paper is therefore conditional on deriving or independently testing Eq. (51).

major comments (3)
  1. [§4.3, after Eq. (50) and before Eq. (52)] The central result rests on an unproven ansatz. Eq. (51) is introduced with 'We now assume', and Eq. (52) with 'We therefore postulate'. The spin density S_C^{αβ} and pressure P are not computed from the expectation values of the spin tensor and energy-momentum tensor for a state with independent spin potential Ω≠ω. Consequently the relations (56) are automatic: s and Q are defined through derivatives of P in (53), and S_C^{αβ} is chosen in (51) to equal ∂P/∂Ω_{αβ}. This does not amount to an independent check. If the true spin density for a rotating fermion state with Ω≠ω differs from (51), the claimed resolution of the Refs. [17,18] tension collapses. Please derive (51) from a QFT state with spin potential, or explicitly reframe the result as a conjecture and provide a nontrivial test (e.g., a calculation of the canonical spin density from the density operator).
  2. [§4.3, Eq. (54)] The energy density is stated as 'given by' but is not derived from u_μΘ^{μν}u_ν for the state with Ω≠ω. It is instead fixed by requiring consistency with the Euler relation and with the equilibrium limit Ω=ω. The sentence after Eq. (56) saying that ϵ is 'derived from the local energy-momentum tensor' is therefore not supported. Please show how Eq. (54) follows from the quantum expectation values, or clearly state that it is part of the postulated thermodynamic scheme rather than an independent QFT result.
  3. [§4.3, final paragraph] The comparison with Refs. [17,18] is only a list of two differences. To substantiate the claim that the tension is resolved, the authors should identify which assumption of Refs. [17,18] is abandoned, and show explicitly with equations why the no-go argument no longer applies. The present text leaves the reader to infer that allowing both Ω and ω dependence and a dynamic pressure is sufficient; this should be demonstrated rather than asserted.
minor comments (3)
  1. [§4.2, Eq. (45)] The sentence 'while ˜ϕ^ρ = ˜ϕ^φ = 0' appears to contain a typo: the φ component was just given as nonzero; it should probably read '˜ϕ^ρ = ˜ϕ^z = 0'.
  2. [§4.3, around Eqs. (50)-(52)] The signs and definitions of the squares Ω², κ², ω², and a² are not spelled out before they are used. Since many coefficients in (52)-(55) depend on these signs, please define the conventions explicitly (e.g., Ω² = Ω_μ Ω^μ with the metric signature used in the paper).
  3. [Abstract and Sec. 4.3] The abstract says 'We find the thermodynamic pressure', but the pressure in Eq. (52) is a postulate. Please moderate the wording so that the ansatz nature is visible, for instance 'We propose and verify the thermodynamic consistency of a local pressure...'.

Circularity Check

1 steps flagged

Main result reduces to unproven spin-density ansatz; thermodynamic consistency is built in by construction.

specific steps
  1. fitted input called prediction [Sec. 4.3, Eqs. (51)–(56)]
    "We now assume that the spin density S_C^{αβ} is, to leading-order, dependent only on the spin potential, namely S_C^{αβ} = 1/2(Ω^{αβ}+u^α κ^β − u^β κ^α)σ_ω^A. ... We therefore postulate that the local pressure has the following form: P=... (52). ... It can be checked that the above quantities are compatible with the Euler relation, (32), and thermodynamically consistent, in the sense that ∂P/∂T = s, ∂P/∂μ = Q_V, ∂P/∂Ω_{αβ} = S_C^{αβ}."

    The pressure (52) is not derived from the quantum state for Ω≠ω. Its coefficients are chosen so that, using (50), ∂P/∂Ω_{αβ} reproduces exactly the assumed spin density (51) with σ_ω^A as given in (53). Meanwhile, s and Q are defined in (53) as the T- and μ-derivatives of the same P. Thus the relations (56) are identities satisfied by construction, not independent checks against the energy-momentum tensor, spin tensor, or charge currents of a state with independent spin potential. The energy density (54) is likewise fixed so that the Euler relation holds. Hence the claimed 'thermodynamic consistency' is built into the ansatz, and the central result carries no physical content beyond the unproven assumption (51) plus the equilibrium limit (47).

full rationale

The paper's GCE construction (Sec. 3) and the equilibrium local pressure (47) are derived from QFT and are externally corroborated (Refs. [37–40] by other groups), so they are not circular. The circularity lies in Sec. 4.3: the off-equilibrium extension replaces derivation by a postulate. The spin density (51) is assumed, and the pressure (52) is postulated with coefficients tuned so that its spin-potential derivative equals that assumed spin density, while s and Q are defined as its T and μ derivatives. The 'consistency' relations (56) therefore hold by definition, not because they have been verified against the quantum fields. The paper honestly uses the words 'assume' and 'postulate,' but the abstract and the 'main result' framing present the outcome as a finding ('We find the thermodynamic pressure') and as a resolution of the tension in Refs. [17,18]. That tension is only resolved if (51) is true for independent spin potential, which is not derived. This is a partial, construction-level circularity: the equilibrium limit anchors the coefficients, but the off-equilibrium content reduces to the ansatz. Score 6 reflects that the central claim is a fit renamed as a prediction, though the construction is not vacuous because it is consistent and reduces correctly when Ω=ω.

Axiom & Free-Parameter Ledger

0 free parameters · 8 axioms · 0 invented entities

No data are fitted and no new entities are introduced. The load-bearing assumptions are the postulates in Sec. 4.3: the spin-density ansatz (51), the pressure form (52), and the energy density (54). These are ad hoc to this paper and are not derived from the quantum field theory, which elevates the circularity burden.

axioms (8)
  • standard math Free Dirac field modes on a rotating background with Fermi-Dirac occupation numbers f_j = [exp(β0(E_j - Ω0 m_j - σ_j μ)) + 1]^{-1}
    Mode decomposition and occupancy in the rotating frame, Sec. 2, Eq. (5).
  • domain assumption The grand canonical density operator ρ ∝ exp(-β0(H - μ0 Q - Ω0·J)) defines the thermal state
    Standard GCE assumption for a rigidly rotating body, Sec. 2, Eq. (1).
  • domain assumption Tolman-Ehrenfest relations T = ΓT0 and μ = Γμ0 hold locally
    Local temperature and chemical potential are red-shifted by Γ, Sec. 3.1, Eq. (23).
  • domain assumption The equilibrium expressions for P_eff, σ^ω_A, Q_V and the stress tensor from Ref. [20] (self-cited) are correct
    Eqs. (40)-(43) are taken from the authors' previous paper and treated as the starting point for the local construction.
  • ad hoc to paper Spin density depends only on the spin potential: S^{αβ}_C = (1/2)(Ω^{αβ}+u^ακ^β-u^βκ^α) σ^ω_A
    Eq. (51), introduced with 'We now assume'; no derivation from QFT is given.
  • ad hoc to paper The local pressure takes the postulated form (52) with specific coefficients
    Eq. (52), introduced with 'We therefore postulate'; this is the central assumption.
  • ad hoc to paper The energy density is given by Eq. (54)
    Stated as 'given by' without derivation from the QFT energy-momentum tensor for Ω≠ω; needed for the Euler relation.
  • domain assumption The dynamic pressure satisfies Π = -1/6 ω_{αβ} ∂P/∂ω_{αβ}
    Eq. (55) compatibility relation, a standard relation in relativistic hydrodynamics but used here to fix the ω²a² coefficient.

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Cite this review

Pith. "Pith review of Thermodynamics of rotating fermions." pith.science (2026). https://pith.science/paper/VRLCJ2FI

@misc{pith2026250917640,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics of rotating fermions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VRLCJ2FI}},
  note         = {Machine review of arXiv:2509.17640}
}
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read the original abstract

We consider the thermodynamic properties of a rotating gas of fermions. We begin by constructing the thermodynamic potential $\Phi$ and its associated current $\phi^\mu$ within the grand canonical ensemble of a macroscopic rigidly rotating body, where the ensemble parameters are the temperature $T_0$ and chemical potential $\mu_0$ on the rotation axis, as well as the rotation angular velocity $\Omega_0$. We then consider the problem of local thermodynamics, where the thermodynamic state is defined by the local temperature $T$ and chemical potential $\mu$, as well as the local spin potential tensor, $\Omega_{\mu\nu}$. We find the thermodynamic pressure $P$, given as the sum of the usual classical (non-quantum) pressure and other corrections due to the spin potential and the kinematic state of the fluid. We compute the associated entropy, charge and spin densities, and show they are consistent with the Euler relation.

discussion (0)

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.