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REVIEW 4 major objections 6 minor 40 references

Solving the Gibbs Paradox by Local Free Space and Collision Potential

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that the entropy increase on mixing two gases is not a universal composition-only constant, but a calculable function of molecular mass, effective radius, and collision duration, and that it vanishes exactly for…

desk verdict A serious but failed attempt: the new mixing-entropy formula is algebraically off and predicts negative entropy for He–Ar, so the central claim does not hold. read the letter →

arxiv 2608.08457 v1 pith:VHIZ2YB2 submitted 2026-08-09 cond-mat.stat-mech

Chuang Li This is my paper · ORCID
classification cond-mat.stat-mech MSC 82B0582B30 PACS 05.20.-y05.70.-a
keywords Gibbsparadoxentropyofmixinglocalfreespaceassumptioncollisionpotentialenergycanonicalensemblegasmixturesidenticalparticlesextensivity
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

The paper tries to resolve the so-called true Gibbs paradox—the fact that the standard entropy of mixing for different gases is independent of what the gases are—by changing what is counted in the partition function. It asserts that each molecule's real-time free space is only $V/N$, and that during collisions molecules carry a potential energy that cannot be neglected. Using the canonical ensemble, the entropy increase reduces to the temperature derivative of the average collision potential energy, giving Eq. (36), which depends on molecular mass, effective radius, and collision duration. For identical gases the bracket vanishes, so mixing entropy is zero without any $N!$ factor or quantum indistinguishability. A sympathetic reader would care because this promises a material-specific resolution of a century-old paradox.

What carries the argument

The load-bearing object is the average collision potential energy per molecule, written as $\varepsilon_p = \epsilon_p \Delta\tau Z$, the product of the mean energy of one collision, the typical collision duration, and the collision frequency. Under the local free space assumption ($V/N$ per molecule), the translational part of the canonical partition function becomes $(V/N)(mkT/2\pi\hbar^2)^{3/2}$, so the logarithmic terms in $S_1$, $S_2$, and $S_m$ cancel in the difference. What remains is only the temperature derivative of the collision-potential terms, Eq. (17). Kinetic-theory expressions for like-pair and unlike-pair collision frequencies then convert that derivative into Eq. (36).

What would settle it

Use the calorimetric integration method cited in the paper to measure the entropy increase for mixing helium and argon at known temperature, pressure, and mole numbers, and compare the measured value with Eq. (36) computed from tabulated masses, effective radii, and independently estimated collision durations. The formula is falsified if the measured entropy increase matches the classical composition-only value $-k(N_1\ln x_1+N_2\ln x_2)$ and shows no dependence on molecular radius or mass.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the Gibbs mixing entropy is controlled by the difference in average collision potential energy per molecule before and after mixing. With the local free-space assumption ($V/N$ per molecule), the volume and momentum logarithms cancel in the entropy difference, leaving Eq. (17): $\Delta S = -N_1 [\partial(\varepsilon_{m1p}-\varepsilon_{1p})/\partial T] - N_2 [\partial(\varepsilon_{m2p}-\varepsilon_{2p})/\partial T]$. Modeling each average potential as (single-collision energy)$\times$(collision duration)$\times$(collision frequency), and using kinetic-theory collision frequencies for like and unlike pairs, produces Eq. (36): $\Delta S \approx \frac{N_1 N_2}{V} \zeta \frac{3}{2} T^{1/2} \big[\sqrt{2}\Delta\tau_1(2r_1)^2 \sqrt{1/m_1} + \sqrt{2}\Delta\tau_2(2r_2)^2 \sqrt{1/m_2} - 2\Delta\tau_m(r_1+r_2)^2 \sqrt{1/m_1+1/m_2}\big]$. This expression is not a constant: it depends on $N_1 N_2/V$, $T^{1/2}$, masses, effective radii, and characteristic collision durations, and it vanishes when the two gases are identical.

Load-bearing premise

The cancellation that leaves only collision terms assumes the borrowed premise that each molecule's instantaneous free space is only $V/N$; if molecules effectively sample the full container volume between collisions, the result collapses.

Editorial extensions

If this is right

  • Mixing two samples of the same gas at equal temperature and pressure yields exactly zero entropy increase, without invoking quantum indistinguishability or an $N!$ correction.
  • Mixing different gases gives an entropy increase fixed by the molecule-specific bracket in Eq. (36), replacing the textbook claim that the increase is a universal constant.
  • The entropy increase is proportional to $N_1 N_2/V$ and to $T^{1/2}$, so it vanishes in the dilute limit and grows with temperature, unlike the composition-only classical result.
  • The derivation makes the collision potential energy, rather than the number of available microstates, the driver of mixing irreversibility.
  • The same mechanism is claimed to extend to liquids, explaining why ink dissolves irreversibly in water while oil and water resist mixing.

Reading between the lines

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

  • If Eq. (36) is correct, some gas pairs could in principle exhibit a negative entropy of mixing whenever the unlike-pair collision term dominates the like-pair terms, a striking and testable departure from the classical formula.
  • Because the derivation treats characteristic collision durations as temperature-independent, measuring $\Delta S$ as a function of temperature would directly probe that assumption and could distinguish this formula from alternatives.
  • The collision-potential logic could be carried into neighboring problems such as diffusion coefficients or non-ideal equations of state, where the same like-pair and unlike-pair collision integrals already appear.
  • The local free space assumption implies that the effective single-particle volume is $V/N$ even before mixing, which could be tested in hard-sphere simulations by directly tracking the collision-free volume a molecule actually samples.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The manuscript proposes a classical resolution of the Gibbs paradox. It argues that the N! factor is unnecessary, that each molecule's accessible volume is V/N at every instant (the 'local free space' assumption), and that intermolecular collision potential energy contributes to entropy. Using the canonical ensemble, the paper derives Eq. (36), an expression for the entropy increase on mixing that depends on molecular mass, effective radius, and collision durations, vanishes for identical gases, and is claimed to resolve the 'true' Gibbs paradox by removing the property-independent constant. The paper also reviews a range of existing resolutions and positions the new model against them.

Significance. If the result were correct, it would be significant: it would give a property-dependent, falsifiable formula for mixing entropy without invoking quantum indistinguishability and would allow thermodynamic verification through calorimetric measurements. The derivation is explicit, the final formula is easy to test, and the paper clearly identifies the assumptions it relies on. However, the central formula contains an algebraic error, predicts negative entropy increase for a realistic gas pair, and rests on an underived premise, so the claimed resolution is not established.

major comments (4)
  1. [Sec. IV.B, Eqs. (30)-(34)] There is a factor-π error in the reduction from Eq. (30) to Eqs. (31)-(34). Direct substitution into Eq. (31) yields a coefficient 6√π k^{3/2} T^{3/2} Δτ1(2r1)^2√(1/m1)(N1+N2)/V, whereas ζ√2 with ζ defined in Eq. (30) gives 6/√π times the same factor, a discrepancy of π. The same error propagates through Eqs. (32)-(34) into Eq. (36), so Eq. (36) is not a faithful simplification of the preceding expressions.
  2. [Sec. IV.B, Eq. (36)] Equation (36) predicts negative mixing entropy for realistic gases. For helium and argon with equal collision durations τ and parameters r_He = 130 pm, r_Ar = 170 pm, m_He = 4.0 u, m_Ar = 39.9 u, the bracket in Eq. (36) equals √2(2r_He)^2/√m_He + √2(2r_Ar)^2/√m_Ar − 2(r_He+r_Ar)^2√(1/m_He+1/m_Ar) ≈ −5.1×10^-7 m^2 kg^{-1/2}. Since the prefactor (N1N2/V)ζ(3/2)T^{1/2} is positive, this gives ΔS < 0, contradicting the second law and the paper's claim that mixing increases entropy. Positivity would require Δτ_m ≲ 0.78 of the like-pair durations, a constraint with no physical justification in the manuscript.
  3. [Sec. IV, local free space assumption] The entire cancellation of the ideal-gas volume terms in Eq. (17) depends on replacing V by V/N in both the initial and final entropies. This premise is imported from Refs. [21,22] and is not derived here. If a molecule samples the full volume V between collisions, the logarithmic terms do not cancel and the collision-only formula for ΔS, Eqs. (17) and (36), does not follow. Because this premise is load-bearing, the derivation is incomplete.
  4. [Sec. IV.B, Eqs. (18)-(29)] The collision durations Δτ1, Δτ2, and Δτm are introduced as free parameters with no microphysical definition, temperature dependence, or constraint. The qualitative conclusion that mixing entropy depends on gas properties is effectively inserted through these parameters and through the collision frequencies. In particular, the vanishing of Eq. (36) for identical gases follows directly from setting Δτm = Δτ1 = Δτ2, not from an independent prediction. A resolution of the Gibbs paradox needs to derive or constrain these quantities, not leave them free.
minor comments (6)
  1. [Sec. I, Eq. (1)] The mixing entropy formula ΔS = -k Σ N_i ln x_i is stated without derivation; a reference or a short derivation would help the reader.
  2. [Sec. IV.A] The phrase 'regular ensemble' should be 'canonical ensemble'.
  3. [Ref. [4]] The URL text contains a typo: 'Gibbs padadox' should be 'Gibbs paradox'.
  4. [Figs. 1-3] Figures 1-3 are referenced in Sec. IV but do not appear in the manuscript; please include them or remove the references.
  5. [Sec. IV.B, Eq. (18)] The notation is confusing: ε_{m12p} and ε_{12p} appear to denote the same cross-species collision energy, and the subscripts are not all defined at first use.
  6. [Sec. IV.B, Eq. (23)] The collision frequency before mixing is written with n = (N1+N2)/V; this is valid only because the two subsystems have equal number density, a condition that should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the property-dependence of ΔS is a model consequence, not a renamed input.

full rationale

The paper's derivation is not circular in the sense required by the review. The local free space assumption (Sec. IV) is imported from Refs. [21,22] by Quanmin Guo, which are not authored by the present author, so the load-bearing premise is not a self-citation chain. The collision-potential model is an input: Eq. (18) defines the average collision potential as εp = ϵp Δτ Z, and the kinetic-theory formulas for Z introduce mass, radius, and collision duration. The final ΔS formula, Eq. (36), is obtained by canonical-ensemble differentiation of the free energy in which the translational/local-space parts cancel, leaving only differences of collision-potential derivatives. That the output depends on the same molecular parameters that were put into the collision model is the normal way a model calculation works, not a tautology: the algebraic form of the bracket is a nontrivial combination of like-pair and unlike-pair terms, and the cancellation for identical gases is a consistency check of the model, not a restatement of an input. No parameter is fitted to any target quantity, no 'prediction' is a renamed fit, and no uniqueness theorem from the author's prior work is invoked. There are serious correctness concerns noted in the paper and in the surrounding analysis—the dropped factor π between Eq. (31) and its preceding expression, the possibility of negative ΔS for He–Ar, and the unsupported identification of average collision potential with average kinetic energy—but these are issues of validity and empirical support, not circularity.

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

The central claim rests on three unmeasured parameters and two domain assumptions. The local free space premise and the collision-potential-equals-kinetic-energy approximation are both imported or asserted, and no experimental test is provided.

free parameters (4)
  • Delta-tau1, Delta-tau2 (same-species collision durations) = not determined
    Free parameters introduced in Eq. (18); no measured values are given, and the final formula's sign and magnitude depend on them.
  • Delta-taum (cross-species collision duration) = not determined
    Introduced in Eq. (18) as equal for both cross directions; no independent measurement or estimate is provided.
  • r1, r2 (effective molecular radii) = not determined
    Chosen per species for the collision frequency formulas in Eqs. (23), (28), and (29); no numerical values are specified.
  • epsilon-p approximately (3/2)kT (mean collision potential) = (3/2)kT
    Approximation in Eqs. (20)-(22) and (27): collision potential energy per collision is set equal to mean translational kinetic energy. This is an ad hoc choice that drives the T^{3/2} dependence.
assumptions (4)
  • domain assumption Local free space: each molecule moves in volume V/N, not V
    Stated in Section IV and used in Eq. (10). Imported from Refs. [21,22]; not derived in this paper.
  • ad hoc to paper Collision potential energy cannot be neglected and equals mean kinetic energy during collision
    Eqs. (20)-(22) and (27). No microscopic justification is given; this is the new physical input.
  • domain assumption Molecules are distinguishable; identical-particle N! is not used
    Section III. Used throughout; depends on non-overlapping wavefunctions.
  • standard math Standard kinetic theory collision frequency formulas
    Used in Eqs. (23), (28), and (29) for collision frequencies and relative speeds; accepted textbook results.

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Cite this review

Pith. "Pith review of Solving the Gibbs Paradox by Local Free Space and Collision Potential." pith.science (2026). https://pith.science/paper/VHIZ2YB2

@misc{pith2026260808457,
  author       = {Pith},
  title        = {Pith review of: Solving the Gibbs Paradox by Local Free Space and Collision Potential},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VHIZ2YB2}},
  note         = {Machine review of arXiv:2608.08457}
}
read the original abstract

This paper argues that the Gibbs paradox can be resolved without using the concept of identical particles in quantum mechanics. The molecules in different regions of the gas can be distinguished, so there is no need to introduce the N! factor. For each molecule, the volume of its free movement space is local at every instant. Moreover, collisions are the primary way of interaction between gas molecules and the fundamental driving force for reaching equilibrium. The potential energy during collisions cannot be ignored. Based on the local free space assumption and collision potential energy, this paper uses the canonical ensemble method to rederive the entropy increase formula for gas mixtures. It includes parameters such as molecular mass, effective radius, and collision characteristic time, which vary with the type of gas molecules. This solves the problem that the entropy increase of gas mixtures is independent of gas properties, that is, it truly resolves the Gibbs paradox instead of providing a new conceptual explanation.

Figures

Figures reproduced from arXiv: 2608.08457 by the authors.

Figure 1
Figure 1. FIG. 1. A schematic diagram showing the relationship between the collision potential energy [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. A schematic diagram showing the relationship between the collision potential energy [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. A schematic diagram showing the relationship between the collision potential energy [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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