Pith. sign in

REVIEW 3 major objections 5 minor 29 references

Anomalous velocity distributions in slow quantum-tunneling chemical reactions

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that the heavy-tailed velocity distributions in the slow quantum-tunneling reaction $D^-+H_2\to H^-+HD$ are governed by a universal function of buffer-gas density, $q-1=2/[3(n_{\max}/n)^{2/3}+5]$.

desk verdict The central formula is built on a wrong density of states: the paper averages a 4D Maxwellian kernel over χ² fluctuations, so q−1=2/(μ+3) instead of 2/(μ+4), invalidating Eq. (17) and its q→7/5 limit. read the letter →

arxiv 2411.16428 v1 pith:FTKQU5KF submitted 2024-11-25 cond-mat.stat-mech physics.chem-ph

Christian Beck , Constantino Tsallis This is my paper · ORCID
classification cond-mat.stat-mechphysics.chem-ph
keywords nonextensivestatisticalmechanicssuperstatisticsq-Maxwelliandistributionentropicindexiontrapquantumtunnelingreactiontemperaturefluctuationsarealaw
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

This paper claims that the anomalous, heavy-tailed velocity distributions observed in the ion-trap quantum-tunneling reaction $D^-+H_2\to H^-+HD$ are described by a $q$-Maxwellian velocity distribution (a heavy-tailed generalization of the Maxwell-Boltzmann form) whose entropic index $q$ is fixed by the buffer-gas density $n$. The derived formula is $q-1 = 2/[3(n_{\max}/n)^{2/3}+5]$, recovering ordinary Maxwell-Boltzmann statistics ($q=1$) as $n\to 0$ and approaching $q=7/5$ as $n\to\infty$, and it matches the two experimental values reported for this reaction. If the formula is right, a single measured density determines the entire shape of the velocity distribution and the amplitude of temperature fluctuations in the trap. The derivation works by averaging local Maxwellian distributions over a chi-squared distribution of inverse temperatures, with the number of effective degrees of freedom set by a surface-area scaling law.

What carries the argument

The central object is the superstatistical $\chi^2$ model of inverse-temperature fluctuations, combined with a surface-area scaling law for the effective number of degrees of freedom $\mu$. In the model, $\beta = \sum_{i=1}^{\mu} X_i^2$ with Gaussian $X_i$, so the weight function $f(\beta)$ is a chi-squared distribution. Averaging local Maxwellian energy distributions over $f(\beta)$ yields a $q$-exponential (equivalently a $q$-Maxwellian velocity distribution) with $q-1 = 2/(\mu+4)$. The area-law input $\mu(n) = 3(n_{\max}/n)^{2/3}+1$ (or its generalized two-constant form) then turns $q$ into a function of density and fixes the relative variance of temperature fluctuations as $2/\mu(n)$.

What would settle it

Measure the ion velocity distribution at several buffer-gas densities between roughly $0.1\,n_{\max}$ and $n_{\max}$, extract $q$ from the tails, and compare with $q-1 = 2/[3(n_{\max}/n)^{2/3}+5]$; independently measure temperature fluctuations and compare with $\sqrt{2/\mu(n)}$. One measured point that deviates from the predicted curve by more than the error bars, or a scaling exponent clearly different from $2/3$, would refute the claim.

Watch

Extended reading notes

Core claim

The central claim is that the entropic index $q$ is not a free fitting parameter but a deterministic function of the ratio $n/n_{\max}$ given by eq. (17). The paper obtains this by treating the trapped ions superstatistically: in each small spatial region the velocity distribution is locally Maxwellian with inverse temperature $\beta$, but $\beta$ fluctuates as a sum of $\mu$ squared Gaussian variables. With such $\chi^2$ fluctuations, the marginal velocity distribution is a $q$-Maxwellian with $q-1 = 2/(\mu+4)$. The new input is the density dependence: at the smallest accessible volume (density $n_{\max}$), heat loss through the surface fixes $\mu = 4$ (three spatial directions plus time), and as the volume grows the relevant surface scales as $V^{2/3}$, giving $\mu(n) = 3(n_{\max}/n)^{2/3}+1$. Combining these gives the universal curve $q(n)$ and its limits, including the finite-variance boundary $q=7/5$.

Load-bearing premise

The prediction collapses if the effective number of degrees of freedom $\mu$ driving temperature fluctuations is not set by surface heat loss through the reactant volume, i.e. if $\mu(n)=3(n_{\max}/n)^{2/3}+1$ is wrong; any other scaling of $\mu$ with density changes the whole curve $q(n)$ including its limiting values.

Editorial extensions

If this is right

  • At any density $n$, the full velocity distribution is a $q$-Maxwellian with $q$ given by eq. (17), so no additional free parameters are needed once the density scale $n_{\max}$ is known.
  • At low density the deviation from Boltzmann-Gibbs statistics scales as $q-1 \propto n^{2/3}$, and the Maxwell distribution is recovered in the limit $n \to 0$.
  • At the highest achievable density $n_{\max}$ the theory predicts $q=5/4$ and an ion temperature twice the buffer-gas temperature; formally, as $n\to\infty$, $q\to 7/5$, the boundary at which the second velocity moment diverges.
  • The relative temperature fluctuation amplitude is predicted to be $\sqrt{2/\mu(n)}$, giving roughly $0.44$ at the density of the left experimental point, consistent in order of magnitude with the reported $\pm 5$ K uncertainty around 15 K.

Reading between the lines

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

  • If the $2/3$ exponent is universal, the same $q$-versus-$n$ curve should describe other ion-trap tunneling reactions when $n$ is rescaled by that experiment's $n_{\max}$; the constants $C_0$ and $C_1$ may shift the offset but not the exponent.
  • The predicted temperature-fluctuation amplitude should be observable as run-to-run scatter in measured ion temperatures that shrinks as the trap volume grows; the present experiment did not report such a systematic scan.
  • Because $q\to 7/5$ marks the finite-variance boundary, experiments pushing densities well beyond the current range should see progressively fatter velocity tails, and near the boundary a single well-defined ion temperature ceases to exist.
  • The formal analogy to area laws in quantum entanglement is suggestive, but the mechanism claimed here is surface heat loss; measuring $q(n)$ in a system without rf heating would test whether the $2/3$ scaling survives when that mechanism is removed.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a superstatistical explanation of the heavy-tailed q-Maxwellian velocity distributions reported by Wild et al. for the D^- + H2 -> H^- + HD tunneling reaction in an ion trap. The authors assume that the inverse temperature beta is chi^2-distributed with mu effective degrees of freedom (Eq. 3), postulate a surface-area scaling mu(n)=3(nmax/n)^(2/3)+1 (Eq. 6), and from a type-B superstatistical average of a local Maxwellian energy distribution derive q-1=2/(mu+4) (Eq. 15). Combining these gives the central formula q-1=2/[3(nmax/n)^(2/3)+5] (Eq. 17), with q->1 as n->0 and q->7/5 as n->infinity. The paper also gives a generalized non-universal form with two constants (Eqs. 24-25), predicts temperature-fluctuation variance delta-beta/beta=sqrt(2/mu), and connects the result to non-additive entropy.

Significance. If the central formula were correct, it would provide a compact, falsifiable prediction for the density dependence of the entropic index and, through Eq. (23), for the amplitude of temperature fluctuations in future ion-trap experiments. The paper is clear about what is assumed and what is derived, and the derivation is short enough to be checked by hand. However, the central derivation rests on the wrong local Maxwellian kernel, and the mu(n) input is a heuristic rather than a derived property of the trap; the generalized formula with fitted constants cannot provide independent validation. The corrected version of the derivation changes the density-dependence formula and destroys the claimed q=7/5 coincidence with the variance threshold.

major comments (3)
  1. [Eqs. (7)–(15) and End Matter] Eq. (7) states p(E|beta)=beta^2 E e^{-beta E}, which is the Gamma(2) energy distribution. That is not the Maxwell-Boltzmann energy density associated with the 3D velocity distribution in Eq. (1): for E=mv^2/2, the normalized density from p(v|beta) proportional to v^2 e^{-beta E} is p(E|beta)=beta^{3/2} E^{1/2} e^{-beta E}/Gamma(3/2), so in the End-Matter notation Z(beta) is proportional to beta^{-3/2}, not beta^{-2}. Repeating the type-B superstatistical integral with the correct kernel yields q-1=2/(mu+3), not 2/(mu+4). Hence Eq. (17), the limits in the bullet list, Eq. (21), and Eq. (22) do not follow from the stated premises. Concretely, at n=nmax (mu=4) the universal curve gives q=9/7 instead of 5/4, and in the formal n->infinity limit mu=1 gives q=3/2 instead of 7/5, so the claimed coincidence with the <v^2> threshold in Eqs. (18)-(20) is lost. The red fit in Fig. 2 cannot arbitrate, because C0 and C1 are free and absorb the changed denominator.
  2. [Eq. (6) and Fig. 2] All density dependence of the central result enters through the postulate mu(n)=3(nmax/n)^(2/3)+1. The surface-area argument is plausible but it is not derived from the trap dynamics, the collision kinetics, or the quantum process, and it is the only source of the exponent 2/3 and of the anchor mu(nmax)=4; without it, Eq. (17) is a conjecture rather than a consequence of superstatistics. In addition, the experimental validation is weak: only two data points from Wild et al. are used, and the red curve in Fig. 2 is a two-parameter fit (C1=2.14, C0=2.56) to exactly those points. Such a fit cannot serve as independent evidence for Eqs. (24)-(25), and no goodness-of-fit or error-bar analysis is given.
  3. [End Matter, Eqs. (34)-(36) and Fig. 2] The manuscript oscillates between a 'universal' formula and a non-universal one. Eq. (17) is introduced with C1=3, C0=1 and called universal, while Eqs. (24)-(25) allow C1 and C0 to depend on experimental details such as Mathieu parameters. If the constants are non-universal, the statement that the n-dependence in Eq. (17) is universal needs a precise definition of the idealized limit in which C1=3 and C0=1 apply; otherwise the central claim is not well posed. In particular, the fitted values C1=2.14 and C0=2.56 sum to 4.70, which differs by about 18% from the stated expectation C0+C1 approximately equal to 4, so the generalized model is not tightly constrained by the 4-degree-of-freedom anchor.
minor comments (5)
  1. [End Matter, Eq. (23)] The density 'n = 4.8 * 10^{14} cm^{-1}' should read cm^{-3}; the comparison of the predicted delta-beta/beta=0.44 with the experimental uncertainty of 5 K out of 15 K is only an order-of-magnitude consistency check and should be labeled as such.
  2. [Main text, temperature discussion] The sentence containing 'roughly given given by T = (15 +/- 5) Kelvin' has a duplicated word, and the comparison with delta-beta/beta should not be described as a coincidence without a more careful accounting of how the experimental temperature uncertainty relates to the variance of the superstatistical temperature distribution.
  3. [Non-additivity section] The text refers to 'Supplementary Material' for details of the non-additivity relation, but no supplementary material is included in this version; please add it or remove the pointer.
  4. [References and fitting constants] Reference [24] is listed as a private communication; the source of the fitted values C1=2.14 and C0=2.56 should be documented in an accessible way so that the fit is reproducible.
  5. [Non-additivity section] The sentence comparing the bilinear entropy term to a 'dot-product of entropic vectors' is not defined and should be either made precise or removed.

Circularity Check

1 steps flagged · score 6.0 of 10

Red-curve agreement with the two data points is a two-parameter fit by construction; the blue q(n) curve is assumption-based but not circular.

  1. fitted input called prediction [End Matter, Eqs. (35)-(36) and Fig. 2 caption]
    "We obtain an optimum fit of the data of the experiment of Wild et al. [5, 24] if we choose C1 = 2.14 and C0 = 2.56."

    Eq. (36) sets q−1 = 2/(μ(n)+4), with μ(n)=C1(nmax/n)^(2/3)+C0 from Eq. (35). The two constants C1 and C0 are selected by an 'optimum fit' to the two experimental q-values of Wild et al., and the resulting red curve is described in the Fig. 2 caption as 'the theoretical prediction of eq. (24) and (25), adjusting the two non-universal fitting constants.' With two free parameters and two data points, the red curve passes through the data by construction. Its agreement is therefore not an independent test of the universal formula (17); it is a two-parameter interpolation of the very points it is said to confirm.

full rationale

The central universal formula (17) is not obtained by fitting the experimental q-values: it follows from the assumed area-law form μ(n)=3(nmax/n)^(2/3)+1 in Eq. (6), together with the superstatistical relation q−1=2/(μ+4). Eq. (6) is an input hypothesis about heat loss through a surface, not a quantity defined in terms of q(n), so the blue curve has independent, though assumption-laden, content. The red curve, by contrast, uses C1 and C0 fitted to the two measured points, so its agreement is by construction; calling it a 'theoretical prediction' makes this a partial circularity in the comparison offered as support. The dotted Tsallis curve is explicitly 'by construction' and is a self-citation, but it is not load-bearing for the main formula. The possible local-kernel inconsistency (Gamma(2) versus Gamma(3/2), i.e. a 4D rather than 3D Maxwellian kernel) is a correctness issue rather than a circularity, since it makes the derivation inconsistent with Eq. (1) rather than equivalent to its input. No load-bearing self-citation chain, uniqueness argument, or ansatz-smuggled-via-citation was found. Hence score 6: one prediction reduces by construction, while the central universal formula retains independent content.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central q(n) formula rests on the geometry of heat loss and on the superstatistical framework imported from prior work. The two fitted constants C1 and C0 make the red curve pass through the same two data points used for validation, and the so-called universal blue curve uses hand-chosen values mu_min=4 and an exponent 2/3. No new physical entity is introduced.

free parameters (4)
  • C1 (multiplicative constant in mu(n)) = 2.14
    Optimized to fit the two experimental q points from Wild et al. Fig. 3(c); controls the red curve in Fig. 2.
  • C0 (additive constant in mu(n)) = 2.56
    Same fit; controls the q(n) offset and qmax = 1 + 2/(C0+4).
  • mu_min = 4 at n = nmax = 4
    Chosen as three spatial plus one temporal degree of freedom for heat loss; sets q(nmax)=1.25. No microscopic derivation.
  • Surface-area exponent 2/3 in mu(n) = 2/3
    Assumed from heat loss through surface scaling as V^(2/3); not derived from trap dynamics.
assumptions (6)
  • domain assumption Superstatistical time-scale separation (local equilibrium faster than temperature fluctuation variation)
    Required for eq. (10) to be a valid marginalization; referenced to [18].
  • domain assumption Inverse temperature fluctuations follow a chi-squared distribution with mu degrees of freedom
    Adopted from Rouse and Willitsch [3]; the paper notes log-normal superstatistics at T=0, so this choice is a finite-T modeling assumption.
  • domain assumption Locally, the ion gas is Maxwellian with density of states giving Z(beta) ~ beta^(-2)
    Used in type-B superstatistics integration in End Matter; assumes ideal-gas-like single-particle states in each local region.
  • ad hoc to paper The effective number of fluctuation degrees of freedom scales as surface area, mu ~ V^(2/3) ~ n^(-2/3)
    Introduced in eqs. (5)-(6) as the core density dependence, without derivation from the trap dynamics.
  • ad hoc to paper At the maximum achievable density, mu_min = 4 (three spatial plus one temporal degree of freedom)
    Fixes q(nmax)=1.25; no microscopic justification.
  • ad hoc to paper In the formal infinite-density limit, only one temporal degree of freedom remains (mu=1)
    Yields q_infinity=7/5, which the paper highlights as the variance-existence boundary; the limit is not experimentally achievable.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Anomalous velocity distributions in slow quantum-tunneling chemical reactions." pith.science (2026). https://pith.science/paper/FTKQU5KF

@misc{pith2026241116428,
  author       = {Pith},
  title        = {Pith review of: Anomalous velocity distributions in slow quantum-tunneling chemical reactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FTKQU5KF}},
  note         = {Machine review of arXiv:2411.16428}
}
abstract

Recent work [Wild et al., Nature 615, 425 (2023)] has provided an experimental break-through in the realization of a quantum-tunneling reaction involving a proton transfer. The reaction $D^-+H_2 \to H^-+HD$ has an extremely slow reaction rate as it can happen only via quantum tunneling, thus requiring an extremely large density of the reactants in the ion trap. At these high densities strong deviations from Maxwell-Boltzmann statistics are observed. Here we develop a consistent generalized statistical mechanics theory for the above nonequilibrium situation involving quantum effects at high densities. The trapped ions are treated in a superstatistical way and a $q$-Maxwellian velocity distribution with a universal dependence of the entropic index $q$ on the density $n$ of the buffer gas is derived. We show that the velocity distribution of the ions is non-Maxwellian, more precisely $q$-Gaussian, i.e., $p(v) \propto v^2 [1+(q-1)\tilde{\beta} v^2]^{1/(1-q)}$, with entropic index $q>1$ depending on the density $n$ of $H_2$ molecules, in excellent agreement with the experimental observations of Wild et al. Our theory also makes predictions on the statistics of temperature fluctuations in the ion trap which can be tested in future experiments. Through the superstatistical approach, we obtain an analytical expression for $q(n)$ which is consistent with the available experimental data, and which yields $\lim_{n\to 0}q(n)=1$, i.e. recovering the Maxwell-Boltzmann distribution in the ideal gas limit, as well as $\lim_{n\to\infty}q(n)=7/5$.

Figures

Figures reproduced from arXiv: 2411.16428 by the authors.

Figure 1
Figure 1. FIG. 1. The basic idea of superstatistics. The spatial region [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. ) the value µ ≈ 10.5 and thus δβ/β = δT /T ≈ 0.44. The authors of [5] do not provide precise temperature or temperature fluctuation measurements but just mention in their paper that the temperature of the ions in their ex￾periments is roughly given given by T = (15 ± 5) Kelvin, i.e. they indicate a rather large uncertainty in tempera￾ture given by ±5 Kelvin. Apparently the order of magni￾tude of their stated δT /T ≈… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 28 canonical work pages

  1. [1]

    This makes physical sense

    For n → 0 we have q → 1. This makes physical sense. For small densities of reactants, ordinary statistical mechanics described by q = 1 is recov- ered

    work page

  2. [2]

    This is the case where the densityn takes on the largest achiev- able value nmax that is reachable in the experiment

    For n = nmax, we obtain q = 5 4 = 1.25. This is the case where the densityn takes on the largest achiev- able value nmax that is reachable in the experiment. In this case there are strong temperature fluctua- tions in the system, and heat losses occur through the 3 directions of space of the smallest possible elementary reaction volume Vmin. We also allow...

    work page

  3. [3]

    Although not experimentally achievable, we may formally consider the limit n → ∞of infinite den- sity. In this case formally the system is reduced to a 1-dimensional point system that is infinitely con- centrated and has only one degree of freedom left, the explicit time dependence of the temperature. This limit leads to µ = 1 and the resulting q-value is...

    work page

  4. [4]

    This can be seen as follows: We have for the q-Maxwellian distributions pq(v) ∼ v2e − ˜β 1 2 mv2 q ∼ v2− 2 q−1 (v → ∞)

    above, is precisely the limit value up to which the variance of the velocity v of the particles still exists. This can be seen as follows: We have for the q-Maxwellian distributions pq(v) ∼ v2e − ˜β 1 2 mv2 q ∼ v2− 2 q−1 (v → ∞). (18) Hence for the integral defining the second moment of ve- locity v M2 = ⟨v2⟩ = Z ∞ 0 v2pq(v)dv (19) the integrand decays as...

    work page

  5. [5]

    Voe, Power-law distributions for a trapped ion in- teracting with a classical buffer gas, Phys

    R.G. Voe, Power-law distributions for a trapped ion in- teracting with a classical buffer gas, Phys. Rev. Lett 102, 063001 (2009)

    work page 2009

  6. [6]

    Asvani and S

    O. Asvani and S. Schlemmer, Numerical simulations of kinetic ion temperature in a cryogenic linear multipole trap, Int. J. Mass spectrom. 279, 147 (2009)

    work page 2009

  7. [7]

    Rouse and S

    I. Rouse and S. Willitsch, Superstatistical energy distri- butions of an ion in an ultracold buffer gas , Phys. Rev. Lett. 118, 143401 (2017)

    work page 2017

  8. [8]

    Noetzold, S.Z

    M. Noetzold, S.Z. Hassan, J. Tauch, E. Endres, R. Wester and M. Weidem¨ uller, Thermometry in a multiple ion trap, Appl. Sci. 10, 5264 (2020)

    work page 2020

Show all 29 references
  1. [9]

    R. Wild, M. Notzold, M. Simpson, T.D. Tran and R. Wester, Tunnelling measured in a very slow ion-molecule reaction, Nature 615, 425 (2023)

    2023

  2. [10]

    Deiss, S

    M. Deiss, S. Willitsch, J. Hecker Denschlag, Cold trapped molecular ions and hybrid platforms for ions and neutral particles, Nat. Physics 20, 713 (2024)

    2024

  3. [11]

    McMahon, Chemical reactions involving quantum tunneling, Science 299, 833–834 (2003)

    R.J. McMahon, Chemical reactions involving quantum tunneling, Science 299, 833–834 (2003)

    2003

  4. [12]

    Shannon, M.A

    R.J. Shannon, M.A. Blitz, A. Goddard and D.E. Heard, Accelerated chemistry in the reaction between the hy- droxyl radical and methanol at interstellar temperatures facilitated by tunnelling Nat. Chem. 5, 745-749 (2013)

    2013

  5. [13]

    Tizniti et al, The rate of the F + H2 reaction at very low temperatures, Nat

    M. Tizniti et al, The rate of the F + H2 reaction at very low temperatures, Nat. Chem. 6, 141 (2014)

    2014

  6. [14]

    Yang et al

    T. Yang et al. Enhanced reactivity of fluorine with para- hydrogen in cold interstellar clouds by resonance-induced quantum tunnelling , Nat. Chem. 11, 744-749 (2019)

    2019

  7. [15]

    Yuen et al, Quuantum-tunneling isotope-exchange reaction H2 +D− → HD +H −, Phys

    C.H. Yuen et al, Quuantum-tunneling isotope-exchange reaction H2 +D− → HD +H −, Phys. Rev. A 97, 022705 (2018)

    2018

  8. [16]

    Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J

    C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52, 479-487 (1988)

    1988

  9. [17]

    Tsallis, Introduction to Nonextensive Statistical Me- chanics - Approaching a Complex World , Second Edition (Springer-Nature, 2023)

    C. Tsallis, Introduction to Nonextensive Statistical Me- chanics - Approaching a Complex World , Second Edition (Springer-Nature, 2023)

    2023

  10. [18]

    Beck, Generalised information and entropy measures in physics , Contemp

    C. Beck, Generalised information and entropy measures in physics , Contemp. Phys. 50, 495 (2009)

    2009

  11. [19]

    Beck, Dynamical foundations of nonextensive statisti- cal mechanics, Phys

    C. Beck, Dynamical foundations of nonextensive statisti- cal mechanics, Phys. Rev. Lett. 87, 180601 (2001)

    2001

  12. [20]

    Beck and E.G.D

    C. Beck and E.G.D. Cohen, Superstatistics, Physica A 322, 267 (2003)

    2003

  13. [21]

    Beck, E.G.D

    C. Beck, E.G.D. Cohen, and H.L. Swinney, From time series to superstatistics , Phys. Rev. E 72, 056133 (2005)

    2005

  14. [22]

    Van der Straeten and C

    E. Van der Straeten and C. Beck, Superstatistical fluctua- tions in time series: Application to share-price dynamics and turbulence, Phys. Rev. E 80, 036108 (2009)

    2009

  15. [23]

    Tunnelling measured in a very slow ion-molecule reaction

    C. Tsallis, Online Comment on “Tunnelling measured in a very slow ion-molecule reaction” , Nature (17 May 2023), doi: 10.1038/s41586-023-05727-z

    2023 doi

  16. [24]

    Eisert, M

    J. Eisert, M. Cramer, M.B. Plenio, Colloqium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010)

    2010

  17. [25]

    Caruso and C

    F. Caruso and C. Tsallis, Nonadditive entropy reconciles the area law in quantum systems with classical thermo- dynamics, Phys. Rev. E 78, 021102 (2008)

    2008

  18. [26]

    Jizba and J

    P. Jizba and J. Korbel, Maximum entropy principle in statistical inference, Phys. Rev. Lett. 122, 120601 (2019)

    2019

  19. [27]

    Regula and L

    B. Regula and L. Lami, Reversibility of quantum re- sources through probabilistic protocols, Nature Commun. 15, 3096 (2024) 6

    2024

  20. [28]

    Private communication of R. Wester

  21. [29]

    Penrose, Foundations of Statistical Mechanics: A De- ductive Treatment (Pergamon, Oxford, 1970), page 167

    O. Penrose, Foundations of Statistical Mechanics: A De- ductive Treatment (Pergamon, Oxford, 1970), page 167. END MA TTER Type-A versus Type-B superstatistics The averaging of the Boltzmann factors can be done in two different ways, either excluding the normalization constant ...

    1970

Pith tools

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