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arxiv: 2606.23039 · v1 · pith:PJ33HQ7Znew · submitted 2026-06-22 · 🧮 math-ph · hep-th· math.MP

Statistical Physics of Planar Carroll Systems

Fei Huang , Lo\"ic Marsot This is my paper

Pith reviewed 2026-06-26 06:46 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.MP
keywords Carroll algebracentral extensionsstatistical physicsSouriau geometric thermodynamicsrotating discangular momentumentropy scalingideal gas law
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The pith

Central extensions of the Carroll algebra in the plane enable consistent statistical physics for Carrollian systems with angular momentum.

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

The paper shows that the Carroll limit of Poincaré statistical physics, which fails to converge in general dimensions, does converge in two dimensions when central extensions of the Carroll algebra are included and systems carry angular momentum. Using Souriau's geometric thermodynamics, the partition function is computed for particles on a uniformly rotating disc, showing that rotation is required for equilibrium with one central charge fixing its direction. Thermodynamic quantities follow from symmetry alone, with entropy scaling logarithmically with disc area and pressure obeying the two-dimensional ideal gas law. An effective Hamiltonian is also derived. A reader would care because this supplies a well-defined thermodynamics precisely where the Carroll limit is physically relevant.

Core claim

Thanks to the central extensions of the Carroll algebra in the plane, and by considering systems with angular momentum, there exists a well defined notion of planar Carrollian statistical physics. For particles on a uniformly rotating disc the partition function is computed via Souriau's geometric thermodynamics, with one central charge determining the rotation direction. Entropy scales logarithmically with the disc area, pressure follows the two-dimensional ideal gas law, and an effective Hamiltonian is derived from symmetry considerations.

What carries the argument

Central extensions of the Carroll algebra in two dimensions together with Souriau's geometric thermodynamics applied to angular-momentum-carrying particles on a uniformly rotating disc.

If this is right

  • Rotation is inevitable for thermal equilibrium of planar Carroll systems.
  • One of the central charges fixes the direction of rotation.
  • Entropy scales logarithmically with the disc area.
  • Pressure obeys the two-dimensional ideal gas law.
  • An effective Hamiltonian follows directly from the symmetry algebra.

Where Pith is reading between the lines

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

  • Dimensionality appears decisive for convergence of the Carroll limit, suggesting similar behavior may hold in other low-dimensional Carrollian models.
  • The logarithmic entropy-area relation invites comparison with area-law phenomena in information or gravitational contexts where Carroll symmetry emerges.
  • Symmetry-based methods used here could be applied to derive thermodynamics for Carrollian systems on other manifolds without explicit Hamiltonians.

Load-bearing premise

The central extensions of the Carroll algebra in two dimensions, together with the inclusion of angular momentum, suffice to make the Carroll limit of Poincaré statistical physics converge.

What would settle it

Explicit computation of the partition function for a planar Carroll system on a rotating disc that diverges even after including the central extensions and angular momentum.

read the original abstract

In this article, we define and study the statistical physics of planar Carrollian systems. While it has been shown recently that, for general dimensions, the Carroll limit of Poincar\'e statistical physics typically does not converge, we show that thanks to the central extensions of the Carroll algebra in the plane, and by considering systems with angular momentum, there exists a well defined notion of planar Carrollian statistical physics. Using Souriau's geometric thermodynamics, we compute the partition function for particles on a uniformly rotating disc, and show that rotation is inevitable for thermal equilibrium of planar Carroll systems, with one of the central charges determining the direction of rotation. We derive thermodynamic quantities in particular entropy, which scales logarithmically with the disc area, and pressure, which follows the two-dimensional ideal gas law. Though all results are obtained from symmetry considerations, we also derive the corresponding effective Hamiltonian.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript defines and studies the statistical physics of planar Carrollian systems. It argues that while the Carroll limit of Poincaré statistical physics fails to converge in general dimensions, the central extensions of the Carroll algebra in two dimensions, combined with the inclusion of angular momentum, allow a well-defined notion of planar Carrollian statistical physics. Using Souriau's geometric thermodynamics, the partition function is computed for particles on a uniformly rotating disc; the authors show that rotation is required for thermal equilibrium, with one central charge fixing the sense of rotation. Thermodynamic quantities are derived, including an entropy that scales logarithmically with disc area and a pressure obeying the two-dimensional ideal-gas law. An effective Hamiltonian is also obtained from symmetry considerations.

Significance. If the convergence of the Carroll limit is rigorously established, the work supplies the first consistent framework for Carrollian statistical mechanics in the plane and yields concrete, symmetry-derived predictions (logarithmic entropy, ideal-gas pressure) that are falsifiable. The reliance on geometric thermodynamics and the derivation of an effective Hamiltonian from first principles are methodological strengths that could extend to Carrollian analogs in condensed-matter or gravitational settings.

major comments (2)
  1. [Abstract and opening paragraph] Abstract and §1 (opening paragraph): the central claim that the two-dimensional central extensions plus angular momentum suffice to make the Carroll limit converge is load-bearing, yet the visible text supplies neither an explicit partition-function expression nor an error estimate or convergence proof for the limit; without these the thermodynamic results cannot be verified.
  2. [Section on Souriau geometric thermodynamics] Section on Souriau geometric thermodynamics (presumably §3–4): the statement that one central charge determines the direction of rotation requires an explicit relation between the sign of the central charge, the equilibrium condition, and the measure on the coadjoint orbit; the current symmetry argument alone does not demonstrate uniqueness or stability of the rotating equilibrium.
minor comments (2)
  1. The effective Hamiltonian is stated to be derived from symmetry; placing its explicit form in an appendix or main-text equation would allow direct comparison with the geometric-thermodynamics partition function.
  2. Notation for the two central charges should be introduced once and used consistently; the current text occasionally refers to “one of the central charges” without a numbered label.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the two major comments point by point below, clarifying the role of the central extensions and geometric thermodynamics while indicating revisions that will strengthen the explicitness of the arguments.

read point-by-point responses

  1. Referee: [Abstract and opening paragraph] Abstract and §1 (opening paragraph): the central claim that the two-dimensional central extensions plus angular momentum suffice to make the Carroll limit converge is load-bearing, yet the visible text supplies neither an explicit partition-function expression nor an error estimate or convergence proof for the limit; without these the thermodynamic results cannot be verified.

    Authors: We agree that the convergence claim is central and benefits from greater explicitness. The partition function is computed explicitly via Souriau's geometric thermodynamics on the coadjoint orbits of the centrally extended Carroll algebra (including angular momentum) in Sections 3–4; the central extensions regularize the phase-space integrals that diverge in the Carroll limit without them. The resulting thermodynamic quantities (logarithmic entropy, ideal-gas pressure) are direct consequences. To make this fully verifiable from the opening sections, we will insert the explicit partition-function expression together with a short convergence argument and error estimate in the revised manuscript. revision: yes

  2. Referee: [Section on Souriau geometric thermodynamics] Section on Souriau geometric thermodynamics (presumably §3–4): the statement that one central charge determines the direction of rotation requires an explicit relation between the sign of the central charge, the equilibrium condition, and the measure on the coadjoint orbit; the current symmetry argument alone does not demonstrate uniqueness or stability of the rotating equilibrium.

    Authors: We thank the referee for this precise observation. The geometric construction already ties the sign of one central charge to the orientation of the coadjoint orbit and hence to the sense of rotation that yields a normalizable equilibrium measure; the second central charge is fixed by the requirement of a well-defined thermal state. Nevertheless, the referee is correct that an explicit formula linking the central-charge sign, the equilibrium condition, and the orbit measure would strengthen the uniqueness and stability statements. We will add this relation, together with a brief stability remark, in the revised version of Sections 3–4. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from the external fact of central extensions in the 2D Carroll algebra plus angular momentum, which the abstract states suffice for convergence of the Carroll limit (contrasted with the non-convergent general-dimensional case shown elsewhere). It then invokes the independent Souriau geometric-thermodynamics framework to obtain the partition function on a rotating disc, from which entropy (logarithmic in area) and pressure (2D ideal-gas law) follow by direct symmetry algebra without any fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations. All steps remain self-contained against external mathematical and thermodynamic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that central extensions of the Carroll algebra permit a convergent statistical limit in two dimensions and that Souriau's geometric thermodynamics applies directly to these systems. No free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Central extensions of the Carroll algebra in the plane allow the Carroll limit of Poincaré statistical physics to converge when angular momentum is included.
    Invoked in the abstract to justify existence of a well-defined planar Carrollian statistical physics.
  • domain assumption Souriau's geometric thermodynamics can be applied to Carrollian systems on a rotating disc.
    Used to compute the partition function and thermodynamic quantities.

pith-pipeline@v0.9.1-grok · 5673 in / 1277 out tokens · 35048 ms · 2026-06-26T06:46:57.364016+00:00 · methodology

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