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arxiv: 2308.14875 · v4 · submitted 2023-08-28 · 🌀 gr-qc

Thermodynamics and the Joule-Thomson expansion of dilaton black holes in 2+1 dimensions

Leonardo Balart , Sharmanthie Fernando This is my paper

Pith reviewed 2026-05-24 07:31 UTC · model grok-4.3

classification 🌀 gr-qc
keywords dilaton black holes2+1 dimensionsthermodynamicscanonical ensemblegrand canonical ensembleJoule-Thomson expansionHawking-Page transitionstability
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The pith

Dilaton black holes in three dimensions fall into two stability classes depending on the value of the coupling parameter N.

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

The paper examines the thermodynamics of a family of charged dilaton black holes in 2+1 dimensions, parameterized by a dimensionless number N that controls the dilaton coupling. In the canonical ensemble with fixed charge, black holes with N between 2/3 and 1 have locally stable small black holes but unstable large ones, while those with N from 1 to 2 are stable at all sizes. When the cosmological constant is treated as pressure, an extra thermodynamic variable must be added to satisfy the first law. In the grand canonical ensemble with fixed potential, a Hawking-Page phase transition occurs specifically for N equal to 6/5. The thermodynamic volume differs from the geometric one, and the analysis includes Joule-Thomson expansion and checks on the reverse isoperimetric inequality.

Core claim

For the family of static charged dilaton black holes in 2+1 dimensions with horizons existing only for 2/3 ≤ N < 2, the thermodynamic behavior in the canonical ensemble divides into two categories: small black holes are locally stable and large ones are not when 2/3 ≤ N < 1, whereas the black hole remains both locally and globally stable for all horizon radii when 1 ≤ N < 2. In the grand canonical ensemble a Hawking-Page phase transition is present for the case N=6/5. A new thermodynamic parameter is introduced to ensure the first law holds with pressure defined from the cosmological constant.

What carries the argument

The dimensionless parameter N, which is related to the coupling constant for the dilaton with the electromagnetic and gravitational fields, that determines the distinct thermodynamic stability regimes.

If this is right

  • For N in [2/3,1) the specific heat capacity changes sign at some radius, indicating a stability transition.
  • For N in [1,2) both specific heat and Gibbs free energy indicate stability without phase transitions in the canonical ensemble.
  • A Hawking-Page transition appears in the grand canonical ensemble only at N=6/5.
  • The thermodynamic volume is not equal to the geometric volume of the black hole.
  • Joule-Thomson expansion and the reverse isoperimetric inequality can be computed for these solutions.

Where Pith is reading between the lines

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

  • The classification suggests that string-theory motivated cases with N=1 are particularly stable across sizes.
  • Similar stability patterns might appear in other lower-dimensional dilaton models if the same coupling structure is present.
  • Testing the new thermodynamic parameter against other ensembles or rotating versions could reveal if it is universal for these black holes.

Load-bearing premise

An additional thermodynamic variable beyond the usual ones must be introduced so that the first law is satisfied when the cosmological constant acts as pressure.

What would settle it

A direct computation of the specific heat capacity as a function of horizon radius for N=0.8 showing no sign change, or for N=1.2 showing a sign change, would contradict the reported stability categories.

Figures

Figures reproduced from arXiv: 2308.14875 by Leonardo Balart, Sharmanthie Fernando.

Figure 1
Figure 1. Figure 1: The figure shows the temperature [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The figure shows the temperature CP,Q vs the horizon radius r+ for varying values of N. Left graph is for N < 1 and the right graph is for N > 1. Here −Λ = −0.1, γ = 1 for both graphs. For the left graph, Q = 0.5, β = 17 and for the right graph Q = 0.386, β = 0.5 3.6 Thermodynamics for the black hole with N = 6 7 Here we will analyze details of thermodynamics for the black hole with N = 6 7 . Notice that Λ… view at source ↗
Figure 3
Figure 3. Figure 3: The figure shows f(r) vs r for the dilaton black hole for N = 6 7 . Here Q = 0.376, β = 0.1, γ = 1,Λ = −1. The blue curve corresponds M = 0.144 and is a black hole solution with two horizons. The red curve corresponds to M = 0.104 and does not have horizons. The green curve corresponds to an extreme black hole with M = 0.118. The pressure P vs r+ is plotted in Fig.(4). As is clear from the figure, there is… view at source ↗
Figure 4
Figure 4. Figure 4: The figure shows the pressure P vs the horizon radius r+ for fixed charge. Here N = 6/7, Q = 0.259, T = 0.406 and β = 0.1. 3.6.1 Thermodynamic stability For the black hole to be locally stable, CP,Q > 0. We have plotted CP,Q vs r+ and T vs r+ in Fig.(5) and Fig.(6). Both T and CP,Q are zero when the black hole is extreme at rex = [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The figure shows the specific heat capacity [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The figure shows the specific heat capacity [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The figure shows the Gibbs free energy G vs the temperature T for fixed charge. Here N = 6 7 , Q = 0.5, Λ = −1, γ = 1, and β = 1. 2 4 6 8 10 12 14 r+ -0.1 0.1 0.2 0.3 0.4 T CP,Q < 0 CP,Q > 0 20 40 60 80 100 120 r+ -1.0 -0.5 0.5 1.0 1.5 2.0 G [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The figure shows the temperature T vs r and Gibbs free energy G vs r for fixed charge. Here N = 6 7 , Q = 0.5, Λ = −1, γ = 1, and β = 1. 3.7 Thermodynamics of the black holes for N = 1 case If N = 1 in eq.(9), we obtain the black hole solution given by following metric: ds2 = −f(r)dt2 + f −1 (r)dr2 + r 2R 2 (r)dθ2 , (51) where R(r) = γr−1/2 (52) and f(r) = −16M √ r γ − 8Λβr + 8Q 2 . (53) 11 [PITH_FULL_IMA… view at source ↗
Figure 9
Figure 9. Figure 9: The figure shows P vs r+ for the dilaton black hole for constant temperature. Here, Q = 1, β = 0.1. and T = 0.28. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The figure shows the specific heat capacity [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Gibbs free energy G versus T for N = 1, Q = 0.5, γ = 1 and Λ = −1.0. 4 Thermodynamics for the N = 2 3 case When we set N = 2/3 in the metric function given by eq.(9), we find that it exhibits a singularity. Numerous studies [29, 30, 35–37, 39, 40, 44, 54] have explored models of charged black holes in three￾dimensional Einstein-Dilaton gravity. In these studies, cases leading to singularities are individu… view at source ↗
Figure 12
Figure 12. Figure 12: The metric function is represented for three different values of [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The specific heat capacity CP,Q and Hawking temperature T vs the horizon r+ for N = 2/3 black hole for fixed charge. Here, Q = 0.5,Λ = −1, γ = 1 and β = 0.486, 0.400, 0.360 (red, blue, green). 0.2 0.4 0.6 0.8 1.0 1.2 1.4 r+ -0.4 -0.2 0.0 0.2 0.4 CP,Q 0.5 1.0 1.5 2.0 2.5 3.0 r+ -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 T [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The specific heat capacity CP,Q and Hawking temperature T vs the horizon r+ with amplified scale close to r+ = 0 for the same cases as in Fig.(13) To understand the global stability one has to study the Gibbs free energy as we did for other cases of N. In Fig.(15) we have plotted G vs T and in Fig(16) G vs r+ and T vs r+. Similar to the N = 6/7 case, for T < Tmax there are two branches of black holes: sma… view at source ↗
Figure 15
Figure 15. Figure 15: Gibbs free energy G is represented as a function of horizon r+, for fixed charge. Here, Q = 0.5, β = 0.3, γ = 0.448,Λ = −1. 20 40 60 80 100 120 r+ 0.01 0.02 0.03 0.04 0.05 T CP,Q < 0 CP,Q > 0 20 40 60 80 100 120 r+ 0.02 0.04 0.06 0.08 0.10 G [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The figure shows the temperature T vs r and Gibbs free energy G vs r for fixed charge. Here Q = 0.5, β = 0.3, γ = 0.448,Λ = −1. 5 Joule-Thomson expansion First, let us describe the Joule-Thomson expansion for a thermodynamical system as follows: In a Joule-Thomson experiment, gas will flow constantly along a thermally insulated tube which is divided into two compartments. There would be a porus plug in be… view at source ↗
Figure 17
Figure 17. Figure 17: The figure shows the Joule-Thomson coefficient [PITH_FULL_IMAGE:figures/full_fig_p021_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The figure shows the Joule-Thomson coefficient [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The figure shows the inversion curves, Ti vs Pi for the dilaton black hole for various N values. Here β = 1.88, Q = 1.636. For N = 1 case, the above relations simplifies to, Ti = 2Q2 πr+ (103) and Ti = 16β 3 Pi (104) It can be observed that for N = 1 the relation is independent of the charge Q or N. It only depends on β. The inversion temperature increases monotonically with the inversion pressure Pi as s… view at source ↗
Figure 20
Figure 20. Figure 20: The figure shows the inversion curves, Ti vs Pi for the dilaton black hole for N = 1 case. Here β = 1.00, 1.50, 2.00. For a given N and Q, there is only one inversion curve for the dilaton black hole. In contrast, for the Van der Waals fluid, there are two inversion curves as shown in [52]. Furthermore, the dilaton black hole does not have critical behavior similar to Van der Waals fluids as shown in sect… view at source ↗
Figure 21
Figure 21. Figure 21: The figure shows isenthalpic and inversion curves for the dilaton black hole for [PITH_FULL_IMAGE:figures/full_fig_p024_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The figure shows the Joule-Thomson coefficient [PITH_FULL_IMAGE:figures/full_fig_p024_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: The figure shows the inversion curves, Ti vs Pi for the dilaton black hole for N = 2/3 case. Here Q = 1 and β = 1.17, 1.25, 1.36. Isenthalpic curves for N = 2/3 is plotted in Fig.(24) and they ar every similar to the 2/3 < N < 2 case. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: The figure shows isenthalpic and inversion curves for the dilaton black hole for [PITH_FULL_IMAGE:figures/full_fig_p026_24.png] view at source ↗
read the original abstract

In this paper, we study thermodynamics and its applications of a family of static charged dilaton black holes in 2+1 dimensions found by Chan and Mann \cite{Chan:1994qa} and Xu \cite{Xu:2019pap}. There is a dimensionless parameter $N$ in the black hole solutions presented: it is related to the coupling constant for the dilaton with the electromagnetic field and the gravitational field. Black hole horizons exist only for $ \frac{2}{3} \leq N < 2$. $N =1$ black hole is a solution to low energy string theory. Thermodynamics is studied in the canonical ensemble where charge is constant as well as in grand canonical ensemble where the potential is constant. The cosmological constant is considered as a thermodynamical variable where the pressure $P = -\frac{\Lambda}{ 8 \pi}$. We computed the first law for the black hole and introduced new thermodynamical parameter in order to satisfy the first law. We computed temperature, thermodynamic volume, specific heat capacities, Gibbs free energy and studied local and global stability of the black hole. Thermodynamic volume differs from the geometric volume. In the canonical ensemble, we noticed that thermodynamic behavior falls into two broad categories: For $\frac{2}{3} \leq N < 1$, small black holes are locally stable and large black holes are not. For $ 1 \leq N < 2$ the black hole is locally and globally stable for all values of the horizon radius. In order to demonstrate the two broad categories, we have presented $N =1, \frac{2}{3}$ and $N = \frac{6}{7}$ black holes in detail. There were no phase transitions for the above values of $N$. In the grand canonical ensemble, we noticed that there is a Hawking-Page phase transition for the black hole with $N=6/5$. We have also studied the Joule-Thomson expansion and the Reverse Isoperimetric Inequality of these black holes...

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

3 major / 2 minor

Summary. The manuscript studies the thermodynamics of static charged dilaton black holes in 2+1 dimensions (solutions of Chan-Mann and Xu) with dimensionless parameter N (horizons for 2/3 ≤ N < 2). Treating the cosmological constant as pressure P = -Λ/8π, the authors state that a new thermodynamic parameter is introduced to satisfy the first law. They compute temperature, thermodynamic volume (distinct from geometric volume), specific heats, and Gibbs free energy. Local/global stability is classified in the canonical ensemble into two categories depending on whether 2/3 ≤ N < 1 or 1 ≤ N < 2; a Hawking-Page transition is reported for N=6/5 in the grand canonical ensemble. Joule-Thomson expansion and the reverse isoperimetric inequality are also examined.

Significance. If the thermodynamic framework is internally consistent, the results would delineate N-dependent stability regimes and a specific phase transition, adding to the catalog of lower-dimensional black-hole thermodynamics with variable Λ. The explicit distinction between thermodynamic and geometric volumes and the Joule-Thomson analysis would be useful if the underlying first-law construction is justified from the action or Wald entropy.

major comments (3)
  1. [Abstract and first-law computation] Abstract and first-law computation section: the statement that 'a new thermodynamical parameter' is introduced 'in order to satisfy the first law' when P = -Λ/8π indicates that the standard thermodynamic identity does not close on the usual variables. No derivation from the Einstein-dilaton action, Wald entropy, or a consistent Smarr relation is indicated; without an explicit conjugate variable and verification that dM = T dS + Φ dQ + V dP + (new term) holds identically, the signs of C_Q and G used for all subsequent stability and phase-transition claims are unreliable.
  2. [Canonical ensemble stability analysis] Canonical-ensemble stability classification (paragraphs reporting the two broad categories for 2/3 ≤ N < 1 and 1 ≤ N < 2): these conclusions rest on the sign of the specific heat at constant charge. Because the heat capacity is obtained only after the ad-hoc parameter is added, the reported division into 'small black holes locally stable, large not' versus 'stable for all radii' cannot be taken as established until the first-law construction is shown to be non-circular.
  3. [Grand canonical ensemble] Grand-canonical ensemble, N=6/5 Hawking-Page transition: the existence of the transition is inferred from the sign change in Gibbs free energy. The same first-law adjustment affects G, so the reported transition requires independent confirmation that the thermodynamic potential is correctly normalized once the new parameter is included.
minor comments (2)
  1. [Thermodynamic volume] The abstract states that 'thermodynamic volume differs from the geometric volume' but does not give the explicit expression for the thermodynamic volume in terms of the new parameter; this relation should be stated clearly.
  2. [Numerical examples] The range of N for which horizons exist is given as 2/3 ≤ N < 2, yet the detailed examples are only N=1, 2/3, 6/7, and 6/5; a brief remark on why other values in 1 ≤ N < 2 were not plotted would improve completeness.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below, focusing on the first-law construction and its consequences for the stability and phase-transition results.

read point-by-point responses

  1. Referee: Abstract and first-law computation section: the statement that 'a new thermodynamical parameter' is introduced 'in order to satisfy the first law' when P = -Λ/8π indicates that the standard thermodynamic identity does not close on the usual variables. No derivation from the Einstein-dilaton action, Wald entropy, or a consistent Smarr relation is indicated; without an explicit conjugate variable and verification that dM = T dS + Φ dQ + V dP + (new term) holds identically, the signs of C_Q and G used for all subsequent stability and phase-transition claims are unreliable.

    Authors: We agree that the first-law construction requires explicit verification. In the manuscript the new conjugate pair is introduced so that the differential identity holds upon direct substitution of the metric-derived expressions for M, T, S, Φ, Q, V and P; we will revise the relevant section to display the explicit form of the first law, name the new variable, and show the algebraic verification that the identity is satisfied identically. A derivation from the Einstein-dilaton action via Wald entropy is not supplied in the present work. revision: partial

  2. Referee: Canonical-ensemble stability classification (paragraphs reporting the two broad categories for 2/3 ≤ N < 1 and 1 ≤ N < 2): these conclusions rest on the sign of the specific heat at constant charge. Because the heat capacity is obtained only after the ad-hoc parameter is added, the reported division into 'small black holes locally stable, large not' versus 'stable for all radii' cannot be taken as established until the first-law construction is shown to be non-circular.

    Authors: Once the first law is satisfied with the additional term, the specific heat C_Q is obtained from the standard thermodynamic relation C_Q = T (∂S/∂T)_Q and its sign is fixed by the explicit metric functions. The two stability regimes therefore follow directly from that consistent framework. We will add a short paragraph after the first-law verification that recalls this relation and confirms that the sign changes are unaltered by the extra term. revision: yes

  3. Referee: Grand-canonical ensemble, N=6/5 Hawking-Page transition: the existence of the transition is inferred from the sign change in Gibbs free energy. The same first-law adjustment affects G, so the reported transition requires independent confirmation that the thermodynamic potential is correctly normalized once the new parameter is included.

    Authors: The Gibbs free energy is constructed as the Legendre transform consistent with the extended first law (including the new term). The sign change at N = 6/5 is read off from the resulting explicit expression. We will include a brief check that the potential satisfies the expected thermodynamic differential relation after the revision of the first-law section. revision: yes

standing simulated objections not resolved
  • Derivation of the extended first law (including the new conjugate pair) from the Einstein-dilaton action or via Wald's Noether-charge entropy formula.

Circularity Check

1 steps flagged

New thermodynamic parameter introduced solely to enforce first law makes stability and phase-transition claims reduce to construction

specific steps
  1. self definitional [Abstract]
    "We computed the first law for the black hole and introduced new thermodynamical parameter in order to satisfy the first law."

    The new parameter is added explicitly to make the first law hold when cosmological constant is treated as pressure. Thermodynamic quantities (T, C, G) are therefore defined such that dM = T dS + ... + V dP holds by construction; stability conclusions derived from those quantities inherit the same definitional closure rather than emerging from an independent calculation.

full rationale

The paper's central results on local/global stability in canonical ensemble (two N-ranges) and Hawking-Page transition in grand canonical ensemble rest on temperature, heat capacities, and Gibbs free energy. These quantities are obtained only after explicitly adding an unspecified new thermodynamic parameter whose sole stated purpose is to close the first law when P = -Λ/8π. Because the first law is satisfied by this addition rather than derived from the action or Wald entropy, the signs of C_Q and G used for stability are forced by the construction itself. No independent derivation or external benchmark is indicated in the provided text. This matches the self-definitional pattern and justifies a score of 6; the remainder of the analysis (Joule-Thomson, RII) inherits the same thermodynamic variables.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

Abstract-only review limits visibility into explicit assumptions; the new parameter and the pressure interpretation of Lambda are the main additions whose justification is not visible.

free parameters (2)
  • N
    Dimensionless coupling parameter already present in the Chan-Mann/Xu solutions; horizon existence restricted to 2/3 ≤ N < 2.
  • new thermodynamical parameter
    Introduced explicitly to satisfy the first law when Lambda is treated as pressure.
axioms (2)
  • domain assumption Cosmological constant is a thermodynamic pressure P = -Λ/8π
    Standard extended-phase-space assumption invoked when first law is written.
  • domain assumption Black-hole thermodynamics applies in canonical and grand-canonical ensembles
    Background assumption used to define stability and phase transitions.
invented entities (1)
  • new thermodynamical parameter no independent evidence
    purpose: To close the first law for these dilaton solutions
    No independent evidence or derivation supplied in abstract; introduced to make the first law hold.

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