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arxiv: 2508.01609 · v2 · pith:KA6EDYCHnew · submitted 2025-08-03 · ❄️ cond-mat.stat-mech

Momentum distribution and correlation function of free particles in the Tsallis statistics using conventional expectation value and equilibrium temperature

Masamichi Ishihara This is my paper

Pith reviewed 2026-05-19 01:47 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords Tsallis statisticsmomentum distributioncorrelation functionfree particlesnonextensive entropyequilibrium temperatureentropic parameter qfinite particle number
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The pith

In Tsallis statistics using conventional expectation value and equilibrium temperature, free particles have momentum distributions and correlations that depend on q and N and differ from Boltzmann-Gibbs statistics.

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

This paper applies Tsallis statistics to a system of free particles while using the conventional linear expectation value and defining temperature as the equilibrium temperature. The standard relation between energy and temperature from Boltzmann-Gibbs statistics continues to hold under these choices. The momentum distribution is derived and shown to depend on the entropic parameter q and the particle number N. As a result, a correlation function exists between the momenta of free particles, which is absent in standard statistics. The value of q is constrained to lie in the interval from 1 minus 1 over (3N/2 plus 1) to 1.

Core claim

Adopting the conventional expectation value and the equilibrium temperature in the Tsallis statistics for free particles preserves the energy-temperature relation of the Boltzmann-Gibbs case, yet produces a momentum distribution that depends on both q and N, resulting in nonzero correlations among the particles and restricting q to 1-1/(3N/2+1) < q < 1.

What carries the argument

The Tsallis momentum distribution obtained from the nonadditive entropy with the linear expectation value and equilibrium temperature for finite N.

If this is right

  • The energy and temperature are related in the same way as in Boltzmann-Gibbs statistics.
  • The momentum distribution differs from the Maxwell-Boltzmann form and depends on q and N.
  • Nonzero correlations appear in the momentum of free particles.
  • The entropic parameter q must satisfy 1-1/(3N/2 +1) < q < 1.
  • These results hold specifically when the equilibrium temperature is used.

Where Pith is reading between the lines

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

  • Similar q and N dependence might appear in other thermodynamic quantities if the same approach is applied.
  • Finite particle number effects in nonextensive statistics could be probed through correlation measurements in dilute gases.
  • Extensions to systems with interactions may show how the correlations modify standard behaviors.

Load-bearing premise

The conventional linear expectation value together with the equilibrium temperature definition keeps the usual energy-temperature relation intact for finite particle numbers in Tsallis statistics.

What would settle it

Calculate or measure the momentum correlation function for free particles with a specific small N and a q value inside the allowed range, then check if it matches the derived expression or reduces to zero as in standard statistics.

read the original abstract

We applied the Tsallis statistics with the conventional expectation value to a system of free particles, adopting the equilibrium temperature which is often called the physical temperature. The entropic parameter $q$ in the Tsallis statistics is less than one for power-law-like distribution. The well-known relation between the energy and the temperature in the Boltzmann--Gibbs statistics holds in the Tsallis statistics, when the equilibrium temperature is adopted. We derived the momentum distribution and the correlation in the Tsallis statistics. The momentum distribution and the correlation in the Tsallis statistics are different from those in the Boltzmann--Gibbs statistics, even when the equilibrium temperature is adopted. These quantities depend on $q$ and $N$, where $N$ is the number of particles. The correlation exists even for free particles. The parameter $q$ satisfies the inequality $1-1/(3N/2+1) < q < 1$.

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 / 1 minor

Summary. The manuscript applies Tsallis statistics with the conventional (linear) expectation value to a system of N free particles, adopting the equilibrium (physical) temperature. It asserts that the standard energy-temperature relation E = (3/2) N kT from Boltzmann-Gibbs statistics continues to hold. The authors derive the momentum distribution and the two-particle correlation function, which are reported to differ from the Maxwell-Boltzmann forms, to depend explicitly on both the entropic index q and N, and to exhibit non-zero correlations even for free particles. The admissible range is given as 1 - 1/(3N/2 + 1) < q < 1.

Significance. If the central derivations are verified, the result would establish that Tsallis statistics with the physical temperature can preserve the exact thermodynamic relation for the ideal gas while generating q- and N-dependent corrections to the momentum distribution and inducing correlations among free particles. This finite-N dependence and the explicit cutoff handling could be relevant for non-extensive modeling of small systems or power-law tails in statistical mechanics.

major comments (2)
  1. [Abstract] Abstract and the derivation of the energy-temperature relation: the claim that <E> = (3/2) N kT holds exactly for q < 1 is load-bearing, yet the Tsallis probability for q < 1 imposes a hard cutoff E_max = U + 1/[(1-q)β]. For the unbounded free-particle spectrum H = ∑ p_i²/2m this truncation must be shown explicitly not to alter the normalization or the linear expectation value; the manuscript should supply the truncated integrals and the resulting marginal momentum distribution to confirm that β remains independent of q and N while still satisfying the equilibrium-temperature definition.
  2. [Derivation of momentum distribution] Derivation of the momentum distribution: the abstract states that the distribution depends on q and N and differs from the Maxwellian, but without the explicit normalized single-particle marginal or the limit q → 1 shown in a dedicated section or equation, it is impossible to verify that the deviation is physical rather than an artifact of the cutoff or the N-dependent normalization.
minor comments (1)
  1. The abstract refers to 'the correlation' without specifying whether it is the two-particle position or momentum correlation function; a brief clarification of the observable would aid readability.

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 will revise the manuscript to incorporate the requested clarifications while preserving the core derivations.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the derivation of the energy-temperature relation: the claim that <E> = (3/2) N kT holds exactly for q < 1 is load-bearing, yet the Tsallis probability for q < 1 imposes a hard cutoff E_max = U + 1/[(1-q)β]. For the unbounded free-particle spectrum H = ∑ p_i²/2m this truncation must be shown explicitly not to alter the normalization or the linear expectation value; the manuscript should supply the truncated integrals and the resulting marginal momentum distribution to confirm that β remains independent of q and N while still satisfying the equilibrium-temperature definition.

    Authors: We agree that explicit verification strengthens the presentation. The admissible q-range 1-1/(3N/2+1) < q < 1 is fixed by the requirement that the normalization integral over the truncated phase space remains finite and positive. Within this range the conventional (linear) expectation value is evaluated directly and yields exactly <E> = (3/2) N kT with the physical temperature T entering only through β = 1/(kT). The cutoff is enforced uniformly on the total Hamiltonian, so the marginal distributions inherit the same β. We will add a dedicated subsection containing the explicit truncated integrals for both normalization and <E>, together with the resulting single-particle marginal, to demonstrate that β is independent of q and N. revision: yes

  2. Referee: [Derivation of momentum distribution] Derivation of the momentum distribution: the abstract states that the distribution depends on q and N and differs from the Maxwellian, but without the explicit normalized single-particle marginal or the limit q → 1 shown in a dedicated section or equation, it is impossible to verify that the deviation is physical rather than an artifact of the cutoff or the N-dependent normalization.

    Authors: The joint distribution is the normalized q-exponential of the total kinetic energy. The single-particle marginal is obtained by integrating over the remaining N-1 momenta subject to the global cutoff; the resulting N-dependence is therefore physical and originates from the finite-N normalization. Direct expansion of the q-exponential for q → 1 recovers the Maxwell-Boltzmann distribution with the same β. We will insert an explicit formula for the normalized marginal momentum distribution and a short paragraph (or appendix) that takes the q → 1 limit analytically, thereby confirming that the reported q- and N-dependence is not an artifact. revision: yes

Circularity Check

0 steps flagged

No circularity in claimed derivation chain

full rationale

The paper applies the standard Tsallis formalism with the conventional (linear) expectation value and adopts the equilibrium temperature definition that preserves the E = (3/2)NkT relation for the free-particle Hamiltonian. It then computes the momentum distribution and two-particle correlation explicitly from the resulting normalized Tsallis probabilities at finite N. The q-range 1-1/(3N/2+1) < q < 1 follows directly from normalization and positivity constraints on the distribution, and the claimed differences from the Maxwellian case emerge from the explicit functional form rather than from any fitted parameter or self-referential definition. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior work are invoked; the derivation remains self-contained against the external benchmark of the Tsallis probability expression and the stated temperature convention.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central results rest on the standard Tsallis entropy form, the definition of the conventional expectation value, and the identification of equilibrium temperature with the physical temperature that recovers the usual energy-temperature relation.

free parameters (1)
  • q
    Entropic index; constrained to the interval 1-1/(3N/2+1) < q < 1 but chosen externally for each system.
axioms (2)
  • domain assumption Tsallis entropy functional with conventional (linear) expectation value
    Invoked throughout the derivation of the distribution and correlation functions.
  • domain assumption Equilibrium temperature equals the physical temperature that satisfies the standard energy-temperature relation
    Stated as the temperature choice that makes the energy-temperature relation hold.

pith-pipeline@v0.9.0 · 5690 in / 1286 out tokens · 21308 ms · 2026-05-19T01:47:31.247596+00:00 · methodology

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 1 internal anchor

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