Bose-Einstein condensation at constant pressure

Yu.M. Poluektov

REVIEW 2 major objections 5 minor 2 references

Bose-Einstein condensation at constant pressure

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Constant-pressure Bose-Einstein condensation has a finite heat capacity.

desk verdict A compact, algebraically consistent extension of a self-imported mean-field model to constant pressure, with new results that inherit the model's free-particle below-T_c ansatz. read the letter →

arxiv 2411.16226 v1 pith:VQZRXYDD submitted 2024-11-25 cond-mat.quant-gas quant-ph

Yu.M. Poluektov This is my paper · ORCID

classification cond-mat.quant-gasquant-ph PACS 03.75.Hh03.75.Nt05.30.Jp05.70.-a67.85.Jk67.10.Fj

keywords Bose-Einsteincondensationconstantpressureweaklynon-idealBosegastemperatureheatcapacityentropyparticlenumberdensitymean-fieldinteraction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Bose-Einstein condensation is normally studied at fixed volume, but some experimental conditions hold pressure instead, and the gas must expand as it cools. This paper treats that constant-pressure case in a weakly non-ideal Bose gas with repulsive interactions. It finds the transition temperature $T_P$ as the solution of a single algebraic equation in the ratio $T_P/T_{P0}$, where $T_{P0}$ is the ideal-gas value; because the interaction constant is positive, $T_P$ always lies below $T_{P0}$. Below the transition it computes the total and condensate densities, energy, entropy, and heat capacities, and shows that the isobaric heat capacity stays finite even though the temperature derivatives of both heat capacities jump.

What carries the argument

The engine of the calculation is the weakly non-ideal gas replacement: the chemical potential is shifted to $\mu^* = \mu - 2\upsilon n$ and the pressure gains a mean-field term, $p = \upsilon n^2 + gT B_{5/2}(t)/\Lambda^3$, where $B_{5/2}$ is the Bose function and $\upsilon$ is the repulsive interaction constant. The transition-temperature equation $\eta y^3 + y^{5/2} - 1 = 0$, with $y = T_P/T_{P0}$, is the central two-term identity that carries the argument; it fixes $T_P$ and, through the dimensionless parameter $\eta$, all the density ratios and heat-capacity coefficients both below and above the transition.

What would settle it

Measure the isobaric heat capacity of a dilute, weakly interacting Bose gas at fixed pressure through the transition; the paper predicts a finite $C_p/N$ that grows linearly in $T$ near $T_P$, whereas the ideal-gas model predicts a divergence. A measured divergence, or a nonlinear temperature dependence whose slope does not match Eq. (24), would falsify the model.

Watch

Extended reading notes

Core claim

The central claim is that constant-pressure condensation in this model is governed by the equation $\eta (T_P/T_{P0})^3 + (T_P/T_{P0})^{5/2} - 1 = 0$, with $\eta = (25/4)\,\upsilon p /(T_{P0}^2\sigma_0^2)$, so the condensation temperature is depressed below the ideal-gas value whenever $\upsilon>0$. Below $T_P$, the pressure law $p = \upsilon n^2 + gT\zeta(5/2)/\Lambda^3$ fixes the total density, the over-condensate density keeps the free-particle form $n' = g\zeta(3/2)/\Lambda^3$, and the condensate fills the difference. From those relations the paper obtains explicit temperature laws for $S/N$, $C_V/N$, and $C_p/N$, with $C_p/N = (3/2)(S/N)[1 + (5/6)g\zeta(5/2)T/(\upsilon n^2\Lambda^3)]$ below the transition. The paper concludes that energy, entropy, and their first derivatives are continuous at $T_P$, while the temperature derivatives of the heat capacities jump.

Load-bearing premise

The calculation rests on the assumption that below $T_P$ the non-condensed particles keep the non-interacting ideal-gas density $n' = g\zeta(3/2)/\Lambda^3$ and that the condensate contributes exactly $\upsilon n^2$ to the pressure.

Editorial extensions

If this is right

A gas held at fixed pressure should enter the condensate at a temperature lower than the ideal-gas estimate whenever repulsion is present.
The isobaric heat capacity remains finite at the transition, so the unphysical divergence of the ideal-gas constant-pressure treatment is removed by the mean-field interaction term.
Below $T_P$, the total density decreases with temperature, and the condensate density at zero temperature is $\sqrt{p/\upsilon}$.
The energy and entropy are continuous at $T_P$, so the transition is of the same continuous type as in the constant-density case, but with a jump in the slopes of $C_V$ and $C_p$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the free-particle depletion law $n' = g\zeta(3/2)/\Lambda^3$ is replaced by the Bogoliubov depletion of a dilute gas, the constant-pressure transition temperature and all heat-capacity formulas would shift quantitatively; the algebraic structure of the transition equation would likely survive but with modified exponents.
Because $\eta \sim p^{1/5}$, the relative suppression of $T_P$ below $T_{P0}$ depends only weakly on pressure, which is a simple signature that could be tested in a tuned-pressure cold-atom or helium experiment.
The same two-term power-law structure should reappear in related geometries or with anisotropic traps, where the pressure law changes but the dimensional balance still produces an equation of the form $a y^3 + b y^{5/2} - 1 = 0$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I'll give you the short version first. This is a compact theory note that extends Poluektov's earlier weakly non-ideal Bose gas model from constant volume to constant pressure. The constant-pressure interacting results are new relative to the two self-cited references, and I verified a large fraction of the algebra by hand. Within the model, the derivation holds: Eq. (12) for the interacting condensation temperature follows, the density law (16) integrates correctly and matches the boundary value, and the entropy and heat capacities in Eqs. (20)-(24) satisfy the thermodynamic identities. The figures match the formulas. This is real, reproducible formal work, not a hand-wave.

The main soft spot is exactly the one the stress-test note identifies. The central results all inherit the assumption that below T_P the thermal cloud is a non-interacting ideal gas pinned at t=0, so n' = g zeta(3/2)/Lambda^3 (Eq. 15). That ansatz is imported from Ref. [1] without derivation or validation. In a real dilute Bose gas the low-energy excitations are Bogoliubov phonons, so the T^{3/2} depletion law -- and everything built on it, including T_P, the density laws, and the heat-capacity jumps -- would change quantitatively. The paper never acknowledges this. That doesn't make the paper wrong on its own terms, but it makes the results conditional on a model premise rather than a direct statement about constant-pressure BEC in a real gas. The paper should have stated this limitation plainly. Also, the slope coefficients and derivative-jump formulas (Eqs. 29, 37, 39-41) are stated without derivation. A referee should ask for them. The bibliography is extremely thin -- two self-cited papers -- and the interaction constant upsilon is never anchored to a scattering length. For a mean-field model paper that's a weakness, not a fatal one. The figure captions in the arXiv version are garbled, which doesn't affect the math but makes the figures unusable as is.

On the positive side, the paper is transparent and honest about the ideal-gas pathology at fixed pressure and confronts it directly. No data are fit, no circular parameter extraction is involved, and the eta -> 0 limit correctly recovers the ideal-gas T_P0 and standard above-T_P heat capacities. The author clearly knows the model's structure.

Who should read it? Specialists in mean-field BEC models and anyone teaching the thermodynamics of Bose gases. I'd send it to a referee for a specialist journal, with a request to verify the slope algebra and to force a clear statement of the model's domain of validity. My own verdict: conditional acceptance, not unconditional, but it's a legitimate paper, and the referee's time is justified.

Referee Report

2 major / 5 minor

Summary. The paper considers Bose-Einstein condensation at constant pressure using the weakly non-ideal Bose gas model of Ref. [1]. In this model the chemical potential is shifted by the mean-field interaction, mu* = mu - 2 upsilon n, and the pressure and energy acquire interaction terms proportional to upsilon n^2. The paper derives a closed equation for the condensation temperature T_P (Eq. 12) that reduces to the ideal-gas value T_P0 when the interaction constant vanishes, and then computes the total and condensate densities, entropy, energy, and isochoric and isobaric heat capacities in the condensed phase (T < T_P) and above T_P. It finds that the entropy and energy are continuous at T_P, the heat capacities are finite, and the derivatives of the heat capacities jump at the transition. All derivations are algebraic and no data are fitted.

Significance. The paper is a self-contained thermodynamic calculation within an explicitly stated model. The algebra is internally consistent: Eq. (12) follows from the model equation of state, Eq. (16) integrates the density law and obeys the boundary condition at T_P, and the eta -> 0 limit recovers the ideal-gas results. The model has no fitted parameters (the free parameters are the interaction constant upsilon and the external pressure p, which define eta). If the model is accepted as a phenomenological description, the paper provides a complete and useful constant-pressure thermodynamics that fills a gap in the literature. However, the physical significance is heavily conditional: the key premise, Eq. (15), treats the non-condensed fraction below T_P as a non-interacting ideal gas with zero effective chemical potential. This is not derived from a microscopic theory and is in tension with Bogoliubov phonon physics for a real dilute Bose gas. The paper does not define the regime of validity of the ansatz or compare with existing many-body results, which limits the strength of the claims.

major comments (2)

[III-IV (Eqs. 9-16)] The entire condensed-phase calculation is built on Eq. (15), which asserts that below T_P the non-condensed density is that of a non-interacting ideal gas pinned at t = 0, n'(T) = g zeta(3/2)/Lambda^3. This relation is imported from Ref. [1] without derivation, and it is not a controlled approximation for a weakly interacting Bose gas: in such a gas the low-energy excitations are phonons, and the thermal depletion of the condensate (beyond the zero-temperature quantum depletion) scales as T^2, not T^(3/2). Because Eqs. (14), (16), (20), (23)-(24), and (40)-(41) all descend directly from Eq. (15), the quantitative predictions of the paper are conditional on the validity of the free-particle ansatz. The authors should provide a microscopic justification for Eq. (15), specify the parameter regime in which it can be expected to hold, and compare the resulting T_P and heat capacities with the predictions of a Bogoliubov/Popov treatment or, where available, with Monte Carlo results. Without such a justification, the paper's title and abstract overstate the generality of the results.
[IV (Eqs. 20, 23)] A direct consequence of the same ansatz is the low-temperature behavior S proportional to T^(3/2) and C_V proportional to T^(3/2) (Eqs. (20) and (23)). For a real weakly interacting Bose condensate, the phonon branch gives S proportional to T^3 and C_V proportional to T^3 at T much less than T_c. The paper should acknowledge this limitation and indicate whether the model is intended to describe only a temperature interval near T_P, where the free-particle approximation may be less inaccurate, or whether it is a pure toy model. This distinction affects how the finite heat capacities and derivative jumps reported in Section IV should be interpreted.

minor comments (5)

[Figure captions] The figure captions (Figures 1, 2, 3, 4) contain garbled text (e.g., '/s48/s46/s48/s48') that obscures the axis labels; these should be repaired before publication.
[IV (Eqs. 21-23)] The derivation of Eq. (23) from Eqs. (21)-(22) is not shown; a few intermediate steps would make the paper more readable.
[III (Eq. 13)] The symbol eta is introduced only through Eq. (13); a brief physical interpretation (e.g., the ratio of the interaction-pressure scale to the ideal-gas pressure at T_P0) would help.
[IV-V] The paper does not comment on the order of the transition according to the Ehrenfest classification; since the heat capacities are continuous but their temperature derivatives jump, the transition is third-order in this model. This could be stated explicitly.
[Introduction] The paper would benefit from a sentence relating the interaction constant upsilon to the s-wave scattering length for a dilute gas, to facilitate contact with experimental literature.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all results are analytic consequences of the explicitly stated weakly non-ideal gas model, with self-citations that are transparent and not circular.

full rationale

The paper's derivation chain is self-contained relative to its declared starting point. The constant-pressure condensation temperature is obtained by setting t=0 in the model pressure law (Eq. 9), giving Eq. (11), which is algebraically rewritten as Eq. (12); the η→0 limit recovers the ideal-gas value T_P0. The below-T_P densities, entropy, and heat capacities are closed-form solutions of Eqs. (14), (15), and the entropy formula imported from Ref. [1]. No parameter is fitted to data, and no predicted quantity is identical to an input by construction. The self-citations to Refs. [1] and [2] are load-bearing as sources of the model and of the ideal-gas isobaric heat capacity, but they are not circular: Ref. [1] is a parameter-free model with stated assumptions that do not include the constant-pressure conclusion, and the ideal-gas entropy/heat-capacity results are standard and independently derivable from Eqs. (1)-(3). The physical limitation noted by a skeptical reader—that the thermal cloud below T_c is treated as a free ideal gas rather than with a Bogoliubov spectrum—is a modeling assumption, not a circularity; the paper explicitly frames all results within the weakly non-ideal gas model of Ref. [1]. Therefore the circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the author's own weakly non-ideal gas model (Ref. [1]) taken wholesale: one free interaction constant υ, no fitted values, no invented entities. The genuinely free parameter is υ; η and ξ_p are reparametrizations of υ and p that organize the formulas. The below-T_P entropy law and the t = 0 pinning of the effective chemical potential are domain assumptions imported from [1]. Compared with a full dilute-gas theory, the ledger is lean: one coupling constant plus standard ideal-gas statistics, but the physical content beyond the mean-field term is thin (no Bogoliubov spectrum, no quantum depletion).

free parameters (2)

υ (interaction constant) = unspecified
The single input of the weakly non-ideal gas model, introduced via the replacement μ* = μ - 2υn in Section III (Eqs. 9-10). All results are re-expressed through η = (25/4)υp/(T_P0²σ₀²) (Eq. 13) and ξ_p = (2/5)σ₀η(T_P/T_P0)^(1/2) (Eq. 25). υ is never fitted or anchored to a scattering length or to data in this paper.
η (dimensionless interaction-pressure parameter) = 0.5 in Figures 3 and 4
Defined by Eq. (13) as a reparametrization of υ, p, T_P0, and σ₀. It is not an independent input, but it is the parameter in which all results are expressed and the one chosen by hand for the figures (η = 0.5).

assumptions (4)

ad hoc to paper Weakly non-ideal gas model of Ref. [1]: effective chemical potential t = (μ - 2υn)/T, pressure p = υn² + gT B_{5/2}(t)/Λ³, and energy E = (N/n)[υn² + (3/2)gT B_{5/2}(t)/Λ³] (Eqs. 9-10).
The entire calculation is built on the author's own prior model, taken over without derivation or independent validation in this paper; the condensate is assumed to contribute pressure υn² and the thermal component the ideal-gas term. The transition temperature and all thermodynamic functions below T_P follow from this constitutive law.
domain assumption For T < T_P the effective chemical potential is pinned at t = 0, so the over-condensate density and pressure are n' = gζ(3/2)/Λ³ and p' = gTζ(5/2)/Λ³ (Eqs. 14-15).
Standard BEC ansatz: the non-condensate saturates at the ideal-gas t = 0 value at every T below T_P, even though the total density n(T) varies with temperature; this is what makes the density law (16) and every downstream formula take their simple closed forms.
domain assumption Entropy below T_P is carried only by the over-condensate particles: S/N = (5/2)gζ(5/2)/(nΛ³) (Eq. 20), imported from Ref. [1].
No derivation is given in this paper; the condensate is assigned zero entropy, which is the standard ideal-condensate assumption but contributes directly to the C_V = (3/2)S/N result (Eq. 23).
standard math Ideal Bose gas relations above T_P: n = gB_{3/2}(t)/Λ³, p = gT B_{5/2}(t)/Λ³, S/N = (5/2)[B_{5/2}/B_{3/2} - t], C_V/N = (15/4)[B_{5/2}/B_{3/2} - (3/5)B_{3/2}/B_{1/2}] (Eqs. 2-3, 32-33).
Standard statistical mechanics of the ideal Bose gas, including the analytic continuation of the Bose functions used for the near-T_P expansions (Eq. 37).

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Cite this review

Pith. "Pith review of Bose-Einstein condensation at constant pressure." pith.science (2026). https://pith.science/paper/VQZRXYDD

@misc{pith2026241116226,
  author       = {Pith},
  title        = {Pith review of: Bose-Einstein condensation at constant pressure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VQZRXYDD}},
  note         = {Machine review of arXiv:2411.16226}
}

read the original abstract

In the weakly non-ideal gas model [1], the Bose-Einstein condensation at constant pressure is considered. The temperature of transition to the state with condensate is found. Temperature dependences of the total density and condensate density, the energy, entropy and heat capacities are calculated.

Figures

Figures reproduced from arXiv: 2411.16226 by the authors.

**Figure 2.** Dependence of the ratio of the condensation temper [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Temperature dependences of 1 – the total density n(T )/np, 2 – the over-condensate density n ′ (T )/np, 3 – the condensate density nB(T )/np at η = 0.5. Let us consider the temperature dependences of entropy and heat capacities in the state with condensate. The entropy per one particle is given by the formula S/N = (5/2) gζ(5/2)/nΛ 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Temperature dependences of the entropy, heat capa [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Poluektov, A simple model of Bose-Einstein condensation of interactin g particles , J

Yu.M. Poluektov, A simple model of Bose-Einstein condensation of interactin g particles , J. Low Temp. Phys. 186, 347 – 362 (2017). doi:10.1007/s10909-016-1715-5; arXiv:1 602.02746v1 [cond-mat.stat-mech]

work page doi:10.1007/s10909-016-1715-5 2017
[2]

Poluektov, Isobaric heat capacity of an ideal Bose gas , Russ

Yu.M. Poluektov, Isobaric heat capacity of an ideal Bose gas , Russ. Phys. J. 44(6), 627 – 630 (2001). doi:10.1023/A:1012599929812

work page doi:10.1023/a:1012599929812 2001

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Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Poluektov, A simple model of Bose-Einstein condensation of interactin g particles , J

Yu.M. Poluektov, A simple model of Bose-Einstein condensation of interactin g particles , J. Low Temp. Phys. 186, 347 – 362 (2017). doi:10.1007/s10909-016-1715-5; arXiv:1 602.02746v1 [cond-mat.stat-mech]

work page doi:10.1007/s10909-016-1715-5 2017

[2] [2]

Poluektov, Isobaric heat capacity of an ideal Bose gas , Russ

Yu.M. Poluektov, Isobaric heat capacity of an ideal Bose gas , Russ. Phys. J. 44(6), 627 – 630 (2001). doi:10.1023/A:1012599929812

work page doi:10.1023/a:1012599929812 2001