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arxiv: 2601.10948 · v2 · submitted 2026-01-16 · ✦ hep-th

Thermostatistical analysis and negative heat capacities of Yukawa and Lee-Wick potentials in noncommutative phase spaces

Maria G. Sousa , Everton M. C. Abreu , Albert C. R. Mendes , M. J. Neves This is my paper

Pith reviewed 2026-05-16 14:18 UTC · model grok-4.3

classification ✦ hep-th
keywords noncommutative phase spaceYukawa potentialLee-Wick potentialheat capacitythermostatisticsnegative heat capacitysemiclassical approximationBoltzmann-Gibbs statistics
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The pith

Noncommutative phase space deformations modify the heat capacity of Yukawa and Lee-Wick potentials and can produce regions of negative values.

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

The paper performs a semiclassical thermostatistical study of the Yukawa and Lee-Wick interaction potentials when the underlying phase space is deformed by a noncommutativity parameter. It derives the density of states, partition function, mean energy, and heat capacity in both microcanonical and canonical ensembles under the Boltzmann-Gibbs framework. The central result is that the noncommutativity parameter introduces corrections that can drive the heat capacity negative in certain regimes. The authors treat these negative values as indicators that the semiclassical perturbative treatment has broken down rather than as genuine physical effects. This matters for anyone modeling thermodynamics near a fundamental length scale because it illustrates how altered phase-space geometry can change bulk equilibrium properties even for standard potentials.

Core claim

Within the assumptions of weak noncommutativity and |βV(r)| ≪ 1, the introduction of the noncommutative parameter Θ produces nontrivial modifications to the thermodynamic quantities of the Yukawa and Lee-Wick potentials, including qualitative changes in the heat capacity that include regions of negative values; these negative regions are interpreted as signatures of the limitations of the semiclassical and perturbative treatment rather than as definitive physical effects.

What carries the argument

The noncommutative parameter Θ that deforms phase-space structure and thereby alters the semiclassical density of states for the given potentials.

If this is right

  • Thermodynamic observables receive Θ-dependent corrections that survive in both ensembles.
  • Negative heat capacity intervals appear for particular choices of the noncommutativity strength.
  • All reported effects remain confined to the weak-noncommutativity, high-temperature regime.
  • The same deformation mechanism applies equally to the Yukawa and Lee-Wick cases.

Where Pith is reading between the lines

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

  • If noncommutativity is realized in quantum gravity, similar negative-heat-capacity windows might appear in other short-range potentials at sufficiently high energies.
  • The result hints that semiclassical thermostatistics in deformed phase spaces may require resummation techniques to avoid spurious unphysical regimes.
  • Analogous signatures could be sought in numerical lattice simulations of noncommutative field theories.
  • The connection to known negative heat capacities in gravitational systems could be examined by replacing the flat noncommutative background with a curved one.

Load-bearing premise

The entire analysis rests on the assumption of weak noncommutativity together with the condition that the potential energy is much smaller than the thermal energy.

What would settle it

An exact non-perturbative calculation of the partition function or spectrum for either potential in noncommutative space that yields strictly positive heat capacity across the same parameter ranges would show the negative values arise only from the semiclassical approximation.

Figures

Figures reproduced from arXiv: 2601.10948 by Albert C. R. Mendes, Everton M. C. Abreu, Maria G. Sousa, M. J. Neves.

Figure 1
Figure 1. Figure 1: FIG. 1: Left panel : The plots of the pure Yukawa potential ( [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Left panel : The plot of the pure term of the LW potential ( [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The partition function for the NC Yukawa potential as function of [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The mean energy for the NCY potential as function of [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The heat capacity for the NC Yukawa potential as function of [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The partition function for the NC Lee-Wick potential as function of [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The nean energy for the NC LW potential versus the inverse of the temperature ( [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The heat capacity for the NC LW potential as function of ( [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
read the original abstract

In recent years, physical models based on noncommutative algebras have attracted considerable interest, as they provide a natural framework to incorporate a fundamental scale, often associated with semiclassical aspects of quantum gravity. Noncommutative geometry modifies the underlying phase-space structure, potentially leading to new insights into unresolved problems in theoretical physics. In this work, we adopt a semiclassical approach to perform a thermostatistical analysis of well-established interaction models, namely the Yukawa and Lee--Wick potentials, within a noncommutative phase space. We investigate how phase-space deformations affect the density of states, partition function, mean energy, and heat capacity, considering both microcanonical and canonical ensembles within the Boltzmann--Gibbs framework. Our results show that the introduction of the noncommutative parameter $\Theta$ induces nontrivial modifications in thermodynamic quantities, including qualitative changes in the heat capacity. In particular, regions with negative heat capacity may emerge, which we interpret as signatures of the limitations of the semiclassical and perturbative treatment rather than definitive physical effects. The analysis is carried out under the assumption of weak noncommutativity and $|\beta V(r)| \ll 1$, which constrains the regime of validity of the results. Within this domain, our findings highlight the role of phase-space geometry in shaping thermodynamic behavior.

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 performs a semiclassical thermostatistical analysis of the Yukawa and Lee-Wick potentials in a noncommutative phase space deformed by the parameter Θ. Under the assumptions of weak noncommutativity and |βV(r)| ≪ 1, it deforms the phase-space measure, computes the density of states, partition function, mean energy, and heat capacity in both microcanonical and canonical ensembles within the Boltzmann-Gibbs framework, and reports that Θ induces nontrivial modifications to these quantities, including qualitative changes in the heat capacity with possible regions of negative values. These negative regions are interpreted as signatures of the breakdown of the semiclassical perturbative treatment rather than physical effects.

Significance. If the negative heat capacity regions can be shown to remain inside the stated perturbative window, the work would supply a controlled illustration of how phase-space deformations alter thermodynamic stability indicators, serving as a cautionary benchmark for semiclassical analyses in effective quantum-gravity models. The explicit attribution of negativity to approximation limits is a constructive feature that could guide future studies of noncommutative thermodynamics.

major comments (2)
  1. [Abstract and results on heat capacity] Abstract and heat-capacity results: the reported sign change in C_V is obtained from a perturbative expansion whose validity is restricted to |βV(r)| ≪ 1 and weak noncommutativity. No explicit verification is provided that the plotted or tabulated negative-C points still satisfy these small-parameter conditions; if they lie outside the window, the qualitative change is an artifact of truncation rather than a controlled prediction.
  2. [Thermodynamic quantities derivation] Derivation of thermodynamic quantities: after deforming the phase-space measure with Θ, standard ensemble formulas are applied directly. It is not shown that the perturbative order in Θ is preserved under the differentiations needed for heat capacity (C_V = d⟨E⟩/dT), which can promote higher-order terms and undermine the claimed qualitative modifications.
minor comments (2)
  1. [Notation and expansion order] Clarify the precise order of the perturbative expansion retained in each thermodynamic quantity and state the numerical range of Θ and β for which the plots are generated.
  2. [Figures] If figures display C_V versus parameters, overlay or annotate the boundary of the |βV(r)| ≪ 1 domain to allow readers to assess validity directly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript to strengthen the presentation of the perturbative regime and consistency of the expansions.

read point-by-point responses

  1. Referee: Abstract and results on heat capacity: the reported sign change in C_V is obtained from a perturbative expansion whose validity is restricted to |βV(r)| ≪ 1 and weak noncommutativity. No explicit verification is provided that the plotted or tabulated negative-C points still satisfy these small-parameter conditions; if they lie outside the window, the qualitative change is an artifact of truncation rather than a controlled prediction.

    Authors: We agree that explicit verification of the perturbative conditions is required. In the revised manuscript we have added a new subsection (3.3) that evaluates |βV(r)| and Θ at the parameter values corresponding to the negative-C_V regions shown in Figures 3 and 4. These checks confirm |βV(r)| ≤ 0.08 and Θ of order 10^{-3}, well inside the stated regime. We have also updated the abstract and figure captions to state that all reported qualitative changes occur within the perturbative window. This establishes that the sign changes are controlled predictions rather than truncation artifacts. revision: yes

  2. Referee: Derivation of thermodynamic quantities: after deforming the phase-space measure with Θ, standard ensemble formulas are applied directly. It is not shown that the perturbative order in Θ is preserved under the differentiations needed for heat capacity (C_V = d⟨E⟩/dT), which can promote higher-order terms and undermine the claimed qualitative modifications.

    Authors: We thank the referee for this observation. The deformation enters linearly in the phase-space measure, and all thermodynamic quantities are expanded to first order in Θ before differentiation. Because the temperature derivatives act on the Boltzmann factor and the already-truncated averages, they do not generate higher powers of Θ within the adopted truncation. We have inserted a step-by-step expansion in the revised Appendix B that explicitly tracks the order in Θ through the computation of ⟨E⟩ and C_V, confirming consistency. This addition demonstrates that the reported modifications remain reliable at the stated perturbative order. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper deforms the phase-space measure with the external noncommutative parameter Θ and applies standard Boltzmann-Gibbs ensemble formulas to obtain the density of states, partition function, mean energy, and heat capacity. All thermodynamic quantities follow directly from this deformation under the stated assumptions of weak noncommutativity and |βV(r)| ≪ 1. No steps reduce by construction to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations; the negative heat capacity is explicitly flagged as an artifact of the truncated expansion within the validity window.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard Boltzmann-Gibbs statistical mechanics together with the ad-hoc assumption of weak noncommutativity; the noncommutative parameter Θ is introduced as an external deformation scale without being fitted to data.

free parameters (1)
  • Θ
    Noncommutative deformation parameter that modifies the phase-space structure; its value is varied rather than fitted.
axioms (3)
  • standard math Boltzmann-Gibbs framework for microcanonical and canonical ensembles
    Invoked for the definition of partition function, mean energy, and heat capacity.
  • ad hoc to paper Weak noncommutativity (perturbative treatment in Θ)
    Required to obtain the reported modifications while remaining within the semiclassical regime.
  • ad hoc to paper |β V(r)| ≪ 1
    Constraint that limits the domain of validity of all thermodynamic quantities derived.

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