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arxiv: 2606.08342 · v1 · pith:LLDOWX55new · submitted 2026-06-06 · ❄️ cond-mat.stat-mech · physics.app-ph· physics.class-ph· physics.pop-ph

A spectral model of power-law decay in natural and engineered systems

Bal\'azs S\'andor , M\'ark Honti , Henrique Santos Lima This is my paper

Pith reviewed 2026-06-27 18:55 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.app-phphysics.class-phphysics.pop-ph
keywords power-law decayq-exponential functionspectral decompositiondiffusion transportnonextensive statisticsboundary geometryrelaxation ratesGamma distribution
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The pith

An asymmetric initial tracer profile in a one-dimensional layered diffusion system with an absorbing boundary produces exact q-exponential decay with q=5/3 at vanishing boundary layer thickness.

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

The paper models an incompletely mixed reactor as a layered diffusion matrix with an absorbing boundary and shows that an asymmetric initial concentration projects onto low-wavenumber eigenmodes. This projection creates an emergent Gamma distribution of relaxation rates. In the one-dimensional case, the survival probability becomes exactly the nonextensive q-exponential function with q=5/3 across all times when the boundary layer thickness approaches zero. The result holds invariantly in the one-dimensional limit even when different boundary configurations are considered in higher dimensions. A reader would care because the mechanism derives heavy-tailed relaxation directly from linear diffusion and boundary geometry without additional assumptions from nonextensive statistical mechanics.

Core claim

For a one-dimensional system, an asymmetric, volumetrically distributed initial concentration profile projects onto the low-wavenumber eigenmodes, generating an emergent Gamma distribution of relaxation rates; at an infinitesimal boundary layer thickness (Δz → 0), this profile yields the nonextensive q-exponential decay function exactly across the entire temporal domain with q = 5/3. Extended to d dimensions under a highly localized, boundary-adjacent singular initial condition, the resulting scaling exponents and corresponding q values depend explicitly on the spatial configuration of the absorbing boundaries. However, in the one-dimensional limit (d=1), these distinct initial states and bo

What carries the argument

The spectral projection of an asymmetric initial tracer profile onto the eigenmodes of a layered diffusion matrix with an absorbing boundary, which produces a Gamma distribution of relaxation rates.

If this is right

  • In higher dimensions the q value and scaling exponents depend on the explicit spatial arrangement of the absorbing boundaries.
  • Linear diffusion transport can generate nonextensive statistics solely through the geometric interaction of initial conditions and domain boundaries.
  • The q=5/3 exponent remains fixed and invariant under different boundary formulations once the system is restricted to one dimension.
  • Heavy-tailed dilution and power-law relaxation emerge from the spectrum of the diffusion operator without invoking nonextensive mechanics at the outset.

Where Pith is reading between the lines

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

  • The same spectral mechanism could be tested in two- or three-dimensional reactor geometries to predict how boundary layout changes the observed q.
  • Experimental setups with tunable boundary layer thickness could directly verify the transition to exact q-exponential behavior as the layer approaches zero.
  • Analogous projections might explain power-law tails in other linear transport problems where initial conditions are localized near absorbing surfaces.

Load-bearing premise

The reactor is modeled as a layered diffusion matrix with an absorbing boundary and the initial tracer placement is an asymmetric, volumetrically distributed profile that projects specifically onto low-wavenumber eigenmodes.

What would settle it

Perform a one-dimensional diffusion experiment with a controlled infinitesimal absorbing boundary layer and an asymmetric initial tracer distribution, then measure whether the concentration decay follows the q-exponential form with q=5/3 exactly over the full time range.

Figures

Figures reproduced from arXiv: 2606.08342 by Bal\'azs S\'andor, Henrique Santos Lima, M\'ark Honti.

Figure 1
Figure 1. Figure 1: FIG. 1: Boundary concentration [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Modal weight distribution [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (Top) Boundary concentration [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

We present a first-principles spectral mechanism for the emergence of nonextensive $q$-exponential dilution and power-law relaxation in non-ideal transport systems. By modeling an incompletely mixed reactor as a layered diffusion matrix with an absorbing boundary, we demonstrate that macroscopic power-law tails depend on the geometric interaction between the initial tracer placement and the domain's boundary configuration. For a one-dimensional system, an asymmetric, volumetrically distributed initial concentration profile projects onto the low-wavenumber eigenmodes, generating an emergent Gamma distribution of relaxation rates; at an infinitesimal boundary layer thickness ($\Delta z \to 0$), this profile yields the nonextensive $q$-exponential decay function exactly across the entire temporal domain with $q = 5/3$. Extended to $d$ dimensions under a highly localized, boundary-adjacent singular initial condition, the resulting scaling exponents and corresponding $q$ values depend explicitly on the spatial configuration of the absorbing boundaries. However, in the one-dimensional limit ($d=1$), these distinct initial states and boundary formulations intersect, rendering the $q=5/3$ exponent geometrically invariant. Our approach establishes a clear connection between linear diffusion transport and nonextensive statistical mechanics, showing how heavy-tailed transport can be derived from boundary geometry and spectral dimensionality.

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 claims a first-principles spectral derivation of nonextensive q-exponential decay (q=5/3 exactly) in 1D diffusion with absorbing boundaries. An asymmetric volumetric initial tracer profile in a layered diffusion matrix projects onto low-wavenumber eigenmodes, producing a Gamma distribution of relaxation rates; this yields the q-exponential exactly for all t when the boundary layer thickness Δz→0. The result generalizes to d dimensions with q depending on boundary geometry, but remains invariant at q=5/3 in the 1D limit.

Significance. If the central claim of an exact, parameter-free match to the q-exponential across the full temporal domain holds, the work would be significant: it would derive heavy-tailed relaxation directly from the eigenstructure of the linear diffusion operator and boundary geometry, without fitted parameters or fractional operators, thereby linking classical transport to nonextensive statistics in a geometrically explicit way.

major comments (2)
  1. [Abstract] Abstract: the central claim that the profile 'yields the nonextensive q-exponential decay function exactly across the entire temporal domain with q=5/3' is contradicted by the short-time asymptotics of the diffusion equation. A Gamma-weighted sum over low modes is analytic at t=0 and expands as 1−const·t+…, while any volumetrically distributed initial condition on the 1D diffusion operator with absorbing boundary produces the non-analytic M(t)∼const·√(Dt) mass-loss term from the high-mode tail.
  2. [Abstract] Abstract (1D system paragraph): the assumption that the asymmetric volumetric initial condition projects specifically onto low-wavenumber eigenmodes with high modes negligible is not supported. For the standard sine eigenbasis of the 1D diffusion operator, overlap integrals with a smooth volumetric profile yield c_n∼1/n (or slower), so the spectral tail remains and controls the √t short-time regime; the Δz→0 limit regularizes the boundary but does not eliminate this tail.
minor comments (1)
  1. The abstract invokes 'the nonextensive q-exponential decay function' without writing the explicit functional form or the precise definition of the Gamma distribution of rates used in the projection.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for identifying points that require clarification. We respond to each major comment below and indicate where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the profile 'yields the nonextensive q-exponential decay function exactly across the entire temporal domain with q=5/3' is contradicted by the short-time asymptotics of the diffusion equation. A Gamma-weighted sum over low modes is analytic at t=0 and expands as 1−const·t+…, while any volumetrically distributed initial condition on the 1D diffusion operator with absorbing boundary produces the non-analytic M(t)∼const·√(Dt) mass-loss term from the high-mode tail.

    Authors: The referee correctly identifies that generic volumetric initial conditions produce a non-analytic short-time mass loss due to the high-mode tail. In the present model, however, the asymmetric initial profile is not generic: the Δz → 0 limit of the layered diffusion matrix produces a specific projection whose coefficients decay rapidly enough that the high-mode contribution to the √t term is eliminated. The resulting discrete weights exactly discretize the Gamma distribution of rates, so the q-exponential holds for all t including the short-time regime. We will add an explicit calculation of the overlap integrals in the revised manuscript to demonstrate this suppression. revision: yes

  2. Referee: [Abstract] Abstract (1D system paragraph): the assumption that the asymmetric volumetric initial condition projects specifically onto low-wavenumber eigenmodes with high modes negligible is not supported. For the standard sine eigenbasis of the 1D diffusion operator, overlap integrals with a smooth volumetric profile yield c_n∼1/n (or slower), so the spectral tail remains and controls the √t short-time regime; the Δz→0 limit regularizes the boundary but does not eliminate this tail.

    Authors: We agree that a generic smooth volumetric profile yields c_n ∼ 1/n. The manuscript, however, employs a geometrically constrained asymmetric profile whose functional form in the Δz → 0 limit is dictated by the boundary-layer construction; this form is orthogonal to the high-n sine modes to leading order. Consequently the spectral tail is suppressed and the Gamma distribution emerges exactly. We will revise the text to include the explicit overlap integrals that confirm the rapid decay of c_n for n ≫ 1. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard eigenmode projection of linear diffusion operator

full rationale

The paper models the system via the standard spectral decomposition of the 1D diffusion operator with absorbing boundary (eigenvalues λ_n ∝ n²). The initial asymmetric volumetric profile yields coefficients c_n via overlap integrals; the abstract states that this projection generates an emergent Gamma distribution of rates whose Laplace transform is the q=5/3 exponential in the Δz→0 limit. No parameter is fitted to the target q-exponential, no self-citation supplies a uniqueness theorem or ansatz, and the Gamma weights are not defined in terms of the final functional form. The central claim therefore reduces to an explicit (if possibly approximate) calculation within linear algebra rather than to a self-referential definition or renamed input. This is the normal non-circular outcome for a spectral derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the validity of eigenfunction expansion for the diffusion operator in a bounded domain with absorbing boundaries and on the specific choice of initial condition that produces the Gamma distribution of rates; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption Transport obeys the linear diffusion equation in a bounded domain with absorbing boundaries.
    Core modeling premise stated in the abstract.
  • domain assumption The initial concentration is asymmetric and volumetrically distributed so that it projects onto low-wavenumber eigenmodes.
    Required for the Gamma distribution of relaxation rates to emerge.

pith-pipeline@v0.9.1-grok · 5777 in / 1296 out tokens · 27835 ms · 2026-06-27T18:55:29.305857+00:00 · methodology

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Forward citations

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Reference graph

Works this paper leans on

37 extracted references · cited by 1 Pith paper

  1. [1]

    Scher and E

    H. Scher and E. W. Montroll.Anomalous transit-time disper- sion in amorphous solids.Physical Review B12(6), 2455– 2477 (1975)

    1975

  2. [2]

    Havlin and D

    S. Havlin and D. Ben-Avraham.Diffusion in disordered me- dia.Advances in Physics36(6), 695–798 (1987)

    1987

  3. [3]

    Bouchaud and A

    J.-P. Bouchaud and A. Georges.Anomalous diffusion in dis- ordered media: Statistical mechanisms, models and physical applications.Physics Reports195(4–5), 127–293 (1990)

    1990

  4. [4]

    Y. W. Tsang.Study of alternative tracer tests in charac- terizing transport in fractured rocks.Geophysical Research Letters22(11), 1421–1424 (1995)

    1995

  5. [5]

    Metzler and J

    R. Metzler and J. Klafter.The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports339(1), 1–77 (2000)

    2000

  6. [6]

    Metzler and J

    R. Metzler and J. Klafter.The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics.Journal of Physics A: Mathematical and General37(31), R161–R208 (2004)

    2004

  7. [7]

    A. V. Chechkin, V. Yu. Gonchar, J. Klafter, and R. Metzler. Fundamentals of L´ evy flight processes.Advances in Chemical Physics133, 439–496 (2004)

    2004

  8. [8]

    I. M. Sokolov and J. Klafter.From diffusion to anomalous diffusion: A century after Einstein.In Anomalous Diffusion: From Basics to Applications, Springer (2005)

    2005

  9. [9]

    Berkowitz, A

    B. Berkowitz, A. Cortis, M. Dentz, and H. Scher.Modeling non-Fickian transport in geological formations as a continu- ous time random walk.Reviews of Geophysics44(2), RG2003 (2006)

    2006

  10. [10]

    Koren, M

    T. Koren, M. A. Lomholt, A. V. Chechkin, J. Klafter, and R. Metzler.First passage times of L´ evy flights coexisting with subdiffusion.Physical Review E76(3), 031129 (2007)

    2007

  11. [11]

    Dentz, A

    M. Dentz, A. Cortis, H. Scher, and B. Berkowitz.Time be- havior of solute transport in heterogeneous media: Transition from non-Fickian to Fickian dispersion.Physical Review E 70(1), 010101(R) (2004)

    2004

  12. [12]

    Haggerty and S

    R. Haggerty and S. M. Gorelick.Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity.Water Resources Research31(10), 2383–2400 (1995)

    1995

  13. [13]

    J. D. Hyman, H. Rajaram, S. Srinivasan, N. Makedonska, S. Karra, H. Viswanathan, and G. Srinivasan.Matrix diffusion in fractured media: New insights into power law scaling of breakthrough curves.Geophysical Research Letters46(23), 13785–13795 (2019)

    2019

  14. [14]

    De Simone, O

    S. De Simone, O. Bour, and P. Davy.Impact of matrix diffusion on heat transport through heterogeneous fractured aquifers.Water Resources Research59(2) (2023)

    2023

  15. [15]

    L. Wang, S. Yoon, L. Zheng, T. Wang, X. Chen, and P. K. Kang.Flux exchange between fracture and matrix dictates late-time tracer tailing.Journal of Hydrology627, 130480 (2023)

    2023

  16. [16]

    Tsallis.Possible generalization of Boltzmann-Gibbs statis- tics.Journal of Statistical Physics52(1–2), 479–487 (1988)

    C. Tsallis.Possible generalization of Boltzmann-Gibbs statis- tics.Journal of Statistical Physics52(1–2), 479–487 (1988)

    1988

  17. [17]

    Tsallis, S

    C. Tsallis, S. V. F. Levy, A. M. C. Souza, and R. Maynard. Statistical-mechanical foundation of the ubiquity of L´ evy dis- tributions in nature.Physical Review Letters75(20), 3589– 3593 (1995)

    1995

  18. [18]

    A. R. Plastino and A. Plastino.Non-extensive statistical me- chanics and the Fokker-Planck equation.Physica A: Statisti- cal Mechanics and its Applications222(1–4), 347–354 (1995)

    1995

  19. [19]

    Tsallis.Nonextensive statistics: Theoretical, experimen- tal and computational evidences and connections.Brazilian Journal of Physics29(1) (1999)

    C. Tsallis.Nonextensive statistics: Theoretical, experimen- tal and computational evidences and connections.Brazilian Journal of Physics29(1) (1999)

    1999

  20. [20]

    T. D. Frank.Nonlinear Fokker-Planck equations: Fundamen- tals and applications.Springer, Berlin (2004)

    2004

  21. [21]

    Borland.Microscopic dynamics of the nonlinear Fokker- Planck equation: A phenomenological model.Physical Re- view E57(6), 6634–6642 (1998)

    L. Borland.Microscopic dynamics of the nonlinear Fokker- Planck equation: A phenomenological model.Physical Re- view E57(6), 6634–6642 (1998)

    1998

  22. [22]

    Bologna, C

    M. Bologna, C. Tsallis, and P. Grigolini.Anomalous diffu- sion associated with nonlinear fractional derivative Fokker- Planck-like equation: Exact time-dependent solutions.Phys- ical Review E62(2), 2213–2218 (2000)

    2000

  23. [23]

    Beck.Dynamical foundations of nonextensive statistical mechanics.Physical Review Letters87(18), 180601 (2001)

    C. Beck.Dynamical foundations of nonextensive statistical mechanics.Physical Review Letters87(18), 180601 (2001)

    2001

  24. [24]

    Rangarajan and M

    G. Rangarajan and M. Ding.First passage time distribution for anomalous diffusion.Physics Letters A273(5–6), 322– 330 (2000)

    2000

  25. [25]

    R. K. Niven.Exact q-exponential solutions of the logistic equation and related reaction-diffusion models.Physical Re- 8 view E74(3), 031111 (2006)

    2006

  26. [26]

    Beck and E

    C. Beck and E. G. D. Cohen.Superstatistics.Physica A: Sta- tistical Mechanics and its Applications322, 267–275 (2003)

    2003

  27. [27]

    D. P. Rao and L. L. Edwards.Mixing effects in stirred tank reactors: A comparison of models.Chemical Engineering Science28(5), 1179–1192 (1973)

    1973

  28. [28]

    E. B. Nauman and B. A. Buffham.Mixing in continuous flow systems.John Wiley & Sons, New York (1983)

    1983

  29. [29]

    Villermaux.The role of hydrodynamics in the design of chemical reactors.In Encyclopedia of Fluid Mechanics, Vol

    J. Villermaux.The role of hydrodynamics in the design of chemical reactors.In Encyclopedia of Fluid Mechanics, Vol. 4, Gulf Publishing (1986)

    1986

  30. [30]

    Martelli, A

    F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti.Solution of the time-dependent diffusion equa- tion for layered diffusive media by the eigenfunction method. Physical Review E67(5), 056606 (2003)

    2003

  31. [31]

    E. J. Carr and I. W. Turner.A semi-analytical solution for multilayered diffusion with a moving boundary.International Journal of Heat and Mass Transfer102, 1169–1179 (2016)

    2016

  32. [32]

    Egolf and K

    P. Egolf and K. Hutter.Tsallis extended thermodynamics ap- plied to 2-d turbulence: L´ evy statistics and q-fractional gen- eralized Kraichnanian energy and enstrophy spectra.Entropy 20(2), 109 (2018)

    2018

  33. [33]

    Redner.A Guide to First-Passage Processes.Cambridge University Press (2001)

    S. Redner.A Guide to First-Passage Processes.Cambridge University Press (2001)

    2001

  34. [34]

    J. C. Crittenden, R. R. Trussell, D. W. Hand, K. J. Howe, G. Tchobanoglous, B. Ward, and J. H. Borchardt.Stantec’s Water Treatment: Principles and Design.Wiley (2022)

    2022

  35. [35]

    F. J. Hurtado, A. S. Kaiser, and B. Zamora.Fluid dynamic analysis of a continuous stirred tank reactor for technical op- timization of wastewater digestion.Water Research71, 282– 293 (2015)

    2015

  36. [36]

    Sz¨ oll¨ osi-Nagy.The discretization of the continuous linear cascade by means of state space analysis.Journal of Hydrol- ogy58, 223–236 (1982)

    A. Sz¨ oll¨ osi-Nagy.The discretization of the continuous linear cascade by means of state space analysis.Journal of Hydrol- ogy58, 223–236 (1982)

    1982

  37. [37]

    Szilagyi and A

    J. Szilagyi and A. Sz¨ oll¨ osi-Nagy.Recursive Streamflow Fore- casting: A State-Space Approach.Taylor & Francis (2010)

    2010