Comment on Statistical mechanics from quantum envariance and exchange symmetry

Technical Summary: Comment on Statistical Mechanics from Quantum Envariance and Exchange Symmetry

Problem Statement
This paper critiques a recent proposal by Ojha, Sardana, and Ghosh (Ref. [1]) which attempts to derive the Gibbs factor ($1/N!$) and modify the Saha equilibrium relation using "environment-assisted invariance" (envariance). The authors of Ref. [1] argue that tracing environmental records of particle permutations generates a von Neumann entropy of $k_B \ln N!$, which they claim explains the Gibbs factor and necessitates a multiplicative correction of $1/(N_e! N_p!)$ to the Saha equation. The present work demonstrates that these conclusions are mathematically inconsistent and thermodynamically invalid. Specifically, the critique identifies errors in the definition of the starting entropy, the calculation of the partial trace, the interpretation of environmental decoherence, and the scaling behavior of the proposed corrections.

Methodology and Analysis
The paper employs a rigorous analysis of quantum statistical mechanics, focusing on density matrices, partial traces, and thermodynamic limits. The methodology involves:

1. Re-evaluating Entropy Definitions: Comparing the entropy expressions used in Ref. [1] against the standard Sackur–Tetrode formula to identify the omission of the $-N \ln N$ term required for indistinguishable particles.
2. Correcting the Partial Trace: Re-deriving the reduced density matrix from the global state proposed in Ref. [1] to show that the printed equation is not a valid partial trace and that the resulting entropy depends critically on the orthogonality of system branch states.
3. Counterexample Construction: Using explicit two-particle calculations (bosons in the same state and bosons/fermions in orthogonal states) to disprove the claim that environmental records universally yield an entropy of $k_B \ln N!$ or enforce physical exchange symmetry sectors.
4. Thermodynamic Limit Analysis: Examining the scaling of the proposed factorial corrections with volume and particle number to test for extensivity and the correct dilute-gas limit.
5. Chemical Potential Consistency: Analyzing the derivation of the Saha equation to demonstrate that the proposed correction double-counts indistinguishability already accounted for in standard chemical potentials.

Key Contributions and Results

• Invalidity of the Entropy Derivation: The paper shows that the entropy $S_{ent} = k_B \ln N!$ claimed in Ref. [1] is not generally obtained. The reduced density matrix yields this entropy only if the system branch states $|\Phi_\pi\rangle$ are mutually orthogonal. For bosons, where multiple permutations often map to the same physical state, the entropy is zero, not $k_B \ln N!$. Furthermore, orthogonal environmental records destroy coherence but do not project the system onto the required symmetric (bosonic) or antisymmetric (fermionic) Hilbert subspace; they merely create an incoherent mixture of labeled states.
• Incorrect Subtraction of Entropy: The proposal to subtract the positive von Neumann entropy of the reduced state from a classical entropy to derive the Gibbs factor is thermodynamically unjustified. The Gibbs factor arises from a reduction in the number of accessible microstates (counting), which corresponds to a negative entropy correction ($-\Delta S = k_B \ln N!$), whereas tracing an environment produces a positive mixing entropy.
• Mathematical Inconsistency in Scaling: The paper refutes the claim that $\ln N!$ is sublinear or vanishes in the thermodynamic limit. Using Stirling's formula, it is shown that $\ln N! / N \sim \ln N$, which diverges. Consequently, the proposed correction factor $\Gamma_{ind} = 1/(N_e! N_p!)$ does not approach unity in the dilute limit; instead, it drives the system toward zero or introduces a volume-dependent divergence that violates the requirement for an intensive thermodynamic limit.
• Double-Counting in the Saha Equation: The critique demonstrates that the standard Saha equation already incorporates indistinguishability via the $1/N!$ factors in the partition function, which lead to the logarithmic density dependence in the chemical potential ($\mu \propto \ln n$). Multiplying the Saha relation by an additional $1/(N_e! N_p!)$ term constitutes a double-counting of indistinguishability, leading to non-intensive results that depend arbitrarily on the chosen observation volume.
• Correct Formulations: The paper provides the correct generalizations for degenerate quantum gases. Instead of factorial corrections to total particle numbers, the proper treatment uses fugacities ($z_i$) and polylogarithm functions ($Li_{3/2}$) to describe Bose-Einstein or Fermi-Dirac statistics, which smoothly recover the classical Saha equation in the non-degenerate limit.

Significance and Claims
The paper concludes that the central claims of Ref. [1]—that environmental permutation records dynamically enforce indistinguishability and that a new factorial correction is required for the Saha equation—are unsupported.

• The "Gibbs factor" is not derived from envariance but is a consequence of restricting the physical state space to symmetric or antisymmetric sectors (or correcting the counting in the classical limit).
• Environmental decoherence generates a positive entropy that cannot be simply subtracted to recover classical thermodynamic results.
• The proposed modification to the Saha equation is thermodynamically inconsistent, failing to yield a finite intensive limit and double-counting indistinguishability.

The author asserts that while standard combinatorial limits (binomial to Poisson/Gaussian) and the standard mode sums for Bose-Einstein and Fermi-Dirac statistics presented in Ref. [1] remain valid, the specific arguments linking these to envariance to derive the Gibbs factor and modify the Saha equation are fundamentally flawed. The paper advocates for the retention of standard canonical and fugacity-based formulations for statistical mechanics.