Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

Energy Balance in Avalanche Models for Solar Flares

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The power-law distribution of solar flare energies in three avalanche models comes from the transition rates between energy states, not from the distribution of stored energy.

desk verdict A useful application of the master-equation idea to CA flare models, but the transition-rate attribution is based on an assumed functional form rather than a direct measurement. read the letter →

arxiv 1909.00195 v1 pith:4JLGPRMM submitted 2019-08-31 astro-ph.SR

Nastaran Farhang , Michael S. Wheatland , Hossein Safari This is my paper · ORCID
classification astro-ph.SR
keywords solarflarescellularautomataself-organizedcriticalityavalanchemodelsflareenergydistributionwaiting-timeprobabilitybalanceequationq-exponential
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 asks why the energies of simulated solar flares follow a power law in cellular automaton avalanche models: is the power law inherited from the distribution of stored magnetic energy, or from the rates at which the system jumps between energy states? Simulating flares with a reduced Lu-Hamilton model and with two variants of an energy-optimized model, the authors estimate these transition rates through a probability balance equation. In every simulation the distribution of lattice energies is a narrow, quasi-normal peak, so it cannot produce a scale-free distribution of flare sizes on its own. The authors conclude that the power law lives in the transition rates, and that the waiting-time distribution is essentially Poisson in two of the three models, with one model showing a modest departure. If true, this matters because observed power laws would then tell us about the physics of energy release in flares rather than about how active regions store magnetic energy.

What carries the argument

The central object is the probability balance equation (a master equation for the energy of the active region), which describes a continuous range of energy states with driving and with transitions between energies at rates $\alpha(E,E')$. The paper couples this with the assumed power-law transition-rate form $\alpha(E,E') = \alpha_0 E^{\delta}(E-E')^{-\gamma}\theta(E-E'-E_0)$, which lets the released energy and the current system energy enter separately. Fitting the resulting expression for $N(E)$ to simulation data separates the roles of $P(E)$ and $\alpha(E,E')$; because $P(E)$ is peaked, any power law in $N(E)$ is attributed to $\alpha$. The waiting-time analysis uses the $q$-exponential distribution as the fitting family.

What would settle it

Track the lattice energy immediately before and after every avalanche in a long simulation and directly tabulate $\alpha(E,E')$; if the directly measured transition rates are not a power law of the released energy while $N(E)$ still is, the claimed origin in transition rates fails.

Watch

Extended reading notes

Core claim

Using a probability balance equation for a single scalar energy $E$, the paper shows that in the reduced Lu-Hamilton cellular automaton and in two variants of an energy-optimized automaton, the stationary distribution $P(E)$ of lattice energy is a narrow, approximately Gaussian peak located near the mean energy. Since such a peaked $P(E)$ cannot generate a power law over many decades, the observed $N(E) \propto E^{-\gamma}$ must be carried by the transition rates $\alpha(E,E')$. Fitting $N(E)=\alpha_0 E^{-\gamma} \int_E^\infty (E')^{\delta} P(E') \, dE'$ gives $\gamma = 1.40 \pm 0.02$ for the Lu-Hamilton model and $\gamma = 1.65 \pm 0.02$ for both optimized variants, with $\delta$ near zero to $0.07$. The paper therefore concludes that the power-law flare size distribution originates in the distribution of transition rates between energy states, not in the energy distribution of the active region. It also finds waiting times described by a $q$-exponential: $q$ is $1.00001 \pm 0.00001$ for the reduced Lu-Hamilton model, $1.28 \pm 0.02$ for the anisotropic optimized model, and $1.00010 \pm 0.00001$ for the isotropic optimized model, meaning two of the three are consistent with a simple Poisson process.

Load-bearing premise

The whole argument assumes that the state of the model can be described by one number — its total energy — and that each flare is an instant change from one energy to another; if how long a flare lasts or how the energy is arranged matters, the calculated jump rates may not be meaningful.

Editorial extensions

If this is right

  • In these models, the scale-free flare size distribution is a property of the release mechanism (how much energy is emitted in a jump), not of how much magnetic energy the lattice holds.
  • The same master-equation decomposition gives a route to modeling flare energy and waiting-time statistics together, with waiting times mostly Poisson except when the redistribution rule is anisotropic and globally driven.
  • Observed power-law flare size distributions should be interpreted as constraints on transition-rate functions, so different active regions can show different $\gamma$ because their rate functions differ, not because their stored-energy distributions differ.
  • The fitted parameter sets provide baselines that future, more physical avalanche models can be tested against: compare their fitted transition rates against the observed frequency-size distributions rather than comparing $P(E)$ directly.

Reading between the lines

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

  • A direct test of the paper's conclusion would be to measure the energy just before and after each avalanche in the simulations and tabulate $\alpha(E,E')$ empirically, without assuming the power-law ansatz; this would show whether the fitted $\gamma$ is an artifact of the assumed functional form.
  • The same balance-equation decomposition could be applied to observed active regions if their free magnetic energy can be estimated over time, asking whether real flares also obey rate-dominated power laws.
  • The different $\gamma$ values for the Lu-Hamilton model (1.40) versus the optimized models (1.65) suggest that the power-law index is not universal even within this model family, which may matter for interpreting observed variations in flare power-law slopes.
  • The $q=1.28$ waiting-time result for the anisotropic optimized model predicts measurable flare clustering; checking long active-region observed series for $q>1$ would test whether that redistribution rule is physically relevant.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper applies a probability balance (master) equation, previously used by Wheatland and collaborators for solar flare energetics, to cellular automaton (CA) avalanche models of solar flares. Flare events are simulated with a reduced Lu-Hamilton model and two versions of an 'optimized' model, and the authors measure the frequency-size distribution N(E) of released energies and the lattice energy distribution P(E). They then substitute an assumed power-law transition rate, α(E,E') = α0 E^δ (E−E')^−γ, into the balance equation and fit the resulting expression for N(E) to the simulated distributions using a genetic algorithm. From the quasi-normal shape of P(E) and the quality of the fits, the paper concludes that the power-law form of N(E) originates in the distribution of transition rates between energy states rather than in the distribution of stored lattice energy. The paper also fits the waiting-time distributions with a q-exponential and reports that the reduced LH and isotropic optimized models are consistent with a simple Poisson process, while the anisotropic optimized model deviates.

Significance. If established, the paper's conclusion would provide a useful conceptual clarification for CA flare models: the scale-free avalanche size distribution is inherited from the energy-jump probability distribution, not from the stored-energy distribution. The mathematical decomposition in Eq. (11) is correct, and the authors' observation that P(E) is quasi-normal while N(E) is a power law is a clean consistency argument that the power law cannot come from P(E) alone. The paper also usefully extends the probability balance framework to two recent CA models and provides a quantitative test of the waiting-time distribution. However, the significance is limited by the fact that the quantitative evidence for the central claim is a fit of an assumed transition-rate form, not an independent measurement of the transition rates from the simulations. The paper does not report goodness-of-fit statistics, does not justify the scalar, Markovian reduction of the CA lattice, and does not test alternative transition-rate kernels. These gaps leave the headline attribution at the level of a consistency check rather than a demonstrated origin.

major comments (4)
  1. [Section 3, Eq. (14)] The fit of Eq. (14) to the simulated N(E) is not an independent measurement of the transition rate, because Eq. (14) is derived by inserting the assumed power-law form α(E,E') = α0 E^δ (E−E')^−γ (Eq. 13) into the balance equation. The text states explicitly that the transition rate is assumed to have this form. To support the central claim that the power law originates in the transition rates, the simulations should provide a direct histogram of transitions binned by initial lattice energy and released energy, and this histogram should be compared with Eq. (13). At minimum, the authors should test alternative kernels (e.g., exponential or log-normal in released energy) and show that they fail to reproduce N(E) given the measured P(E). Without such a test, the paper demonstrates consistency with the ansatz but not the claimed origin.
  2. [Section 2.2, Eq. (10)] The probability balance equation reduces the full CA lattice state to a scalar energy E with instantaneous Markovian transitions. This reduction is assumed but not justified. Avalanches in CA models have finite durations and consist of many individual redistribution events; treating each avalanche as an instantaneous jump between two energy levels may be invalid if the transition rate depends on internal configurational details that are not captured by the scalar energy. The authors should provide a test of this reduction, for example by comparing the waiting-time distribution predicted from the energy-dependent rate λ(E) in Eq. (15) with the directly measured inter-event statistics, or by verifying that the inferred N(E) is unchanged when conditioning on avalanche duration.
  3. [Section 3, Figures 1-3] The text states that the goodness of fit is evaluated using the χ² test and that a genetic algorithm minimizes the χ² function, but no χ² values, degrees of freedom, or residuals are reported for any of the fits in Figures 1-3. Given that three free parameters are fitted to a smooth power-law-like distribution, reporting the actual χ² values is essential for assessing whether the deviations, particularly in the rollover region, are acceptable. The authors should report χ² (or an equivalent likelihood), the number of data points used, and show residual plots for each fit.
  4. [Section 3, Eqs. (14), (19), (22)] The fitting range used for Eq. (14) is not stated. This matters because the predicted N(E) in Eq. (19) has a rollover near the mean lattice energy due to the finite support of P(E), while the observed rollovers in Figures 1-3 occur at released energies roughly two to five orders of magnitude below the mean and are attributed instead to finite simulation time via Eq. (22). The authors should specify the energy range over which Eq. (14) is fitted and the number of data points in that range, and confirm that the fit is not dominated by the rollover. Without this information, the reported parameters α0, γ, and δ are not reproducible or fully interpretable.
minor comments (5)
  1. [Eq. (24)] Equation (24) is described as the cumulative distribution function, but the expression shown is actually the complementary CDF (survival function). The CDF should include the factor 1 − before the q-exponential term, or the text should be relabeled as the survival function.
  2. [Eq. (23)] In Equation (23), the equality P(Δt) = (2−q)λ e^{−λΔt} ≈ ... is confusing: the ordinary exponential is not exactly equal to the q-exponential. The authors should write the q-exponential definition explicitly and then state that it reduces to the ordinary exponential in the limit q → 1.
  3. [Section 3, Figures 1-3] The fitted curve in the left panels is labeled with the full expression 'α0 (E)^{−γ} ∫ ...' which is difficult to read in the figures. A concise legend entry such as 'Eq. (14) fit' would improve clarity.
  4. [Section 3, Eq. (22)] The notation n(>E) is defined as 'the expected number of events with size greater than E', but the integral in Eq. (22) uses N(E), which is a rate per unit energy per unit time. The authors should make the time interval explicit in the definition to avoid dimensional ambiguity.
  5. [References and reproducibility] The paper does not state the lattice sizes, the number of simulation runs, or the exact driving rates used beyond the inequality in Section 3, and no code or data availability statement is provided. Adding this information would make the numerical results reproducible.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the power-law-origin inference follows from the independently measured quasi-normal P(E) and the general balance equation, not from the fitted transition-rate ansatz.

full rationale

The paper's central inference is not circular. Section 2.2 defines the flare frequency-size distribution N(E) via the general convolution N(E) = ∫ P(E') α(E', E'-E) dE' (Eq. 11), which does not require the power-law ansatz of Eq. (13). The simulations independently yield a sharply peaked, quasi-normal P(E) (right panels of Figures 1-3). For E well below the mean lattice energy, Eq. (11) makes N(E) approximately proportional to α(Ebar, Ebar-E), so the observed power-law N(E) directly forces a power-law release-energy dependence in the transition rate. Thus the conclusion that the power law originates in the transition rates, rather than in P(E), follows from measured quantities plus the general balance equation. The subsequent use of Eq. (13) is a parametric fitting form: fitting Eq. (14) estimates α0, γ, and δ and tests consistency; it is not presented as an independent prediction of N(E) from a previously determined α. The self-citations (Wheatland 2008; Farhang et al. 2018) provide the balance-equation framework and the optimized CA model as inputs; neither is invoked as an unverified uniqueness theorem or as the sole support for the origin claim. The scalar Markovian reduction behind Eq. (10) is an assumption whose validity could be questioned, and the missing chi-squared values are an incompleteness issue, but these are correctness concerns rather than circularity. I find no step in which a central claim reduces by construction to its inputs.

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

The central claim relies on a scalar-energy Markovian description of the cellular automaton and on assuming a power-law form for transition rates. The parameters of that form and of the waiting-time distribution are fitted to simulated data, and no new physical entities are introduced. The fitted parameters are the main empirical content linking the simulations to the master-equation interpretation.

free parameters (6)
  • alpha_0 = 0.06, 0.085, 2.72 (per model)
    Scaling factor in the assumed power-law transition rate, fitted to simulated N(E) via Equation 14.
  • gamma = 1.40, 1.65, 1.65
    Power-law index of the transition rate, fitted to the simulated flare frequency-size distribution.
  • delta = 0.07, 0.06, 0.01
    Excess-energy exponent in the transition rate, fitted as part of Equation 14.
  • q = 1.00001, 1.28, 1.00010
    Shape parameter of the q-exponential waiting-time distribution, fitted to the simulated waiting-time CDFs.
  • lambda = not quoted
    Rate parameter in the q-exponential WTD fits; values are not reported in the text.
  • E0 = not specified
    Low-energy cutoff in the assumed transition rate (Equation 13); its value is not stated.
assumptions (4)
  • domain assumption The active region energy evolution is a Markov process described by the master equation (Equation 10).
    The probability balance equation is taken from Van Kampen and from Wheatland and Glukhov; the paper does not justify Markovianity for cellular automaton dynamics.
  • domain assumption A flare corresponds to an instantaneous transition from energy E to a lower energy E' with rate alpha(E,E').
    Used to derive Equation 11; the finite duration of an avalanche is ignored.
  • ad hoc to paper The transition rate has the power-law form alpha = alpha0 E^delta (E-E')^(-gamma), Equation 13.
    Assumed for fitting and never derived from the cellular automaton rules.
  • domain assumption The system reaches a steady state so that the time-independent probability balance equation applies.
    The paper states that a stationary state is achieved after a sufficient number of driving steps, but no convergence diagnostics are provided.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Energy Balance in Avalanche Models for Solar Flares." pith.science (2026). https://pith.science/paper/4JLGPRMM

@misc{pith2026190900195,
  author       = {Pith},
  title        = {Pith review of: Energy Balance in Avalanche Models for Solar Flares},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4JLGPRMM}},
  note         = {Machine review of arXiv:1909.00195}
}
read the original abstract

The distributions of solar flare energies and waiting times have not been described simultaneously by a single physical model, yet. In this research, we investigate whether recent avalanche models can describe the distributions for both the released energies and waiting times of flares in an active region. Flaring events are simulated using the modified Lu and Hamilton model (Lu and Hamilton (1991), ApJ, 380, 89) and also the optimized model (Farhang et al. (2018), ApJ, 859, 41). Applying a probability balance equation approach, we study the statistics of the simulated flaring events and investigate the origin of the observed power law in the flare frequency-size distribution. The results indicate that the power law originates in the distribution of transition rates (the distribution of the probabilities of transitions between different energies) rather than the distribution of the energy of the active region. It is also observed that the waiting-time distribution of simulated flaring events follows a q-exponential function which approximates a simple Poisson distribution.

Figures

Figures reproduced from arXiv: 1909.00195 by the authors.

Figure 1
Figure 1. The statistics of the simulated flaring events for the reduced LH model. (I) The frequency-size distribution of flaring events together with the result of fitting the Equation (14) on the data. (II) The probability distribution of lattice energies. The location of the peak of P(E) is shown with the red line. 6 6.1 6.2 6.3 6.4 6.5 x 1013 10−16 10−15 10−14 10−13 10−12 Lattice energy P(E) 10−2 100 102 104 10−8 10−6 10−… view at source ↗
Figure 2
Figure 2. The statistics of the simulated flaring events for the anisotropic optimized model. (I) The frequency-size distribution of flaring events together with the result of fitting the Equation (14) on the data. (II) The probability distribution of lattice energies. The location of the peak of P(E) is shown with the red line [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. The statistics of the simulated flaring events for the isotropic optimized model. (I) The frequency-size distribution of flaring events together with the result of fitting the Equation (14) on the data. (II) The probability distribution of lattice energies. The location of the peak of P(E) is shown with the red line. 0 5000 10000 10−4 10−3 10−2 10−1 100 Waiting time CDF Reduced LH model 0 2 4 x 104 10−3 10−2 10−1 10… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The cumulative waiting-time distributions for the reduced LH model (I), the anisotropic optimized model (II), and the isotropic optimized model (III), respectively. The red line in each panel represents the cumulative q-exponential distribution of Equation (24). The ob…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Correlation in the Waiting-time Distributions of Solar Flares

    astro-ph.SR 2019-08 reject novelty 7.0 of 10

    A new analysis of GOES soft X-ray data finds that larger flares in isolated active regions tend to be followed by longer waiting times until the next flare, a 'saturation' pattern that supports the build-up/release mo...

Reference graph

Works this paper leans on

43 extracted references · 37 canonical work pages · cited by 1 Pith paper

  1. [1]

    2011, Self-Organized Criticality in Astrophysics (Springer-Verlag Berlin Heidelberg), XIV, 416

    Aschwanden, M. 2011, Self-Organized Criticality in Astrophysics (Springer-Verlag Berlin Heidelberg), XIV, 416

    work page 2011

  2. [2]

    Aschwanden, M., & van Ballegooijen, A. A. 2018, in Catalyzing Solar Connections

    work page 2018

  3. [3]

    Crosby, N., Dimitropoulou, M., et al

    Aschwanden, M., B. Crosby, N., Dimitropoulou, M., et al. 2014, Space Science Reviews, 198

    2014

  4. [4]

    J., & McTiernan, J

    Aschwanden, M. J., & McTiernan, J. M. 2010, The Astrophysical Journal, 717, 683

    work page 2010

  5. [5]

    J., Tarbell, T

    Aschwanden, M. J., Tarbell, T. D., Nightingale, R. W., et al. 2000, The Astrophysical Journal, 535, 1047

    work page 2000

  6. [6]

    J., Caspi, A., Cohen, C

    Aschwanden, M. J., Caspi, A., Cohen, C. M. S., et al. 2017, The Astrophysical Journal, 836, 17

    work page 2017

  7. [7]

    1987, Physical Review Letters, 59, 381

    Bak, P., Tang, C., & Wiesenfeld, K. 1987, Physical Review Letters, 59, 381

    work page 1987

  8. [8]

    Barpi, F., Borri-Brunetto, M., & Veneri, L. D. 2007, Journal of Cold Regions Engineering, 21, 121 Boffetta, G., Carbone, V., Giuliani, P., Veltri, P., &

    work page 2007

Show all 43 references
  1. [9]

    1999, Physical Review Letters, 83, 4662

    Vulpiani, A. 1999, Physical Review Letters, 83, 4662

    1999

  2. [10]

    2003, Astronomy and Astrophysics, 406, 1061

    Buchlin, E., Aletti, V., Galtier, S., et al. 2003, Astronomy and Astrophysics, 406, 1061

    2003

  3. [11]

    2005, Astronomy and astrophysics, 436, 355 Cant´ o, J., & Mart ´ ınez-G´ omez, S

    Buchlin, E., Galtier, S., & Velli, M. 2005, Astronomy and astrophysics, 436, 355 Cant´ o, J., & Mart ´ ınez-G´ omez, S. C. . E. 2009, Astronomy and Astrophysics, 501, 1259

    2005

  4. [12]

    W., Liu, H.-L., & Bogdan, T

    Charbonneau, P., McIntosh, S. W., Liu, H.-L., & Bogdan, T. J. 2001, Solar Physics, 203, 321

    2001

  5. [13]

    B., Aschwanden, M

    Crosby, N. B., Aschwanden, M. J., & Dennis, B. R. 1993, Solar Physics, 143, 275

    1993

  6. [14]

    Farhang, N., Safari, H., & Wheatland, M. S. 2018, The Astrophysical Journal, 859, 41

    2018

  7. [15]

    R., Hudson, H

    Fletcher, L., Dennis, B. R., Hudson, H. S., et al. 2011, Space Sience Review, 159, 19

    2011

  8. [16]

    2017, The Astrophysical Journal, 847, 115

    Gheibi, A., Safari, H., & Javaherian, M. 2017, The Astrophysical Journal, 847, 115

    2017

  9. [17]

    McClements, K. G. 2003, Physical Review Letters, 90, 131101

    2003

  10. [18]

    1998, Astronomy and Astrophysics, 335, 1085

    Isliker, H., Anastasiadis, A., Vassiliadis, D., & Vlahos, L . 1998, Astronomy and Astrophysics, 335, 1085

    1998

  11. [19]

    2000, Astronomy and Astrophysics, 363, 1134

    Isliker, H., Anastasiadis, A., & Vlahos, L. 2000, Astronomy and Astrophysics, 363, 1134

    2000

  12. [20]

    2013, Acta Geophysica, 61

    Jimnez, A. 2013, Acta Geophysica, 61

    2013

  13. [21]

    2017, Genetic Algorithm Essentials, Studies in Computational Intelligence (Springer International Publishing)

    Kramer, O. 2017, Genetic Algorithm Essentials, Studies in Computational Intelligence (Springer International Publishing)

    2017

  14. [22]

    F., & Uzdensky, D

    Loureiro, N. F., & Uzdensky, D. A. 2016, Plasma Physics and Controlled Fusion, 58, 014021

    2016

  15. [23]

    T., & Hamilton, R

    Lu, E. T., & Hamilton, R. J. 1991, The Astrophysical Journal, 380, L89

    1991

  16. [24]

    T., Hamilton, R

    Lu, E. T., Hamilton, R. J., McTiernan, J. M., & Bromund, K. R. 1993, The Astrophysical Journal, 412, 841

    1993

  17. [25]

    2015, Earth, Planets and Space, 67, 59

    Maehara, H., Shibayama, T., Notsu, Y., et al. 2015, Earth, Planets and Space, 67, 59

    2015

  18. [26]

    Moore, E. F. 1962, Machine Models of Self-Reproduction, ed. R. Bellman, ”Mathematical problems in the biological sciences (American Mathematical Society)

    1962

  19. [27]

    2008, The Astrophysical Journal, 682, 654 Picoli Jr., S., Mendes, R

    Morales, L., & Charbonneau, P. 2008, The Astrophysical Journal, 682, 654 Picoli Jr., S., Mendes, R. S., Malacarne, L. C., & Santos, R. P. B. 2009, Brazilian Journal of Physics, 39, 468

    2008

  20. [28]

    Robinson, P. A. 1994, Physical Review E, 49, 1984

    1994

  21. [29]

    Rosner, R., & Vaiana, G. S. 1978, The Astrophysical Journal, 222, 1104

    1978

  22. [30]

    2011, Living Reviews in Solar Physics, 8, 6

    Shibata, K., & Magara, T. 2011, Living Reviews in Solar Physics, 8, 6

    2011

  23. [31]

    1995, Publications of the Astronomical Society of Japan, 47, 251

    Shimizu, T. 1995, Publications of the Astronomical Society of Japan, 47, 251

    1995

  24. [32]

    2014, Solar Physics, 289, 2993

    Strugarek, A., Charbonneau, P., Joseph, R., & Pirot, D. 2014, Solar Physics, 289, 2993

    2014

  25. [33]

    2004, Continuum Mechanics and Thermodynamics, 16, 223

    Tsallis, C., & Brigatti, E. 2004, Continuum Mechanics and Thermodynamics, 16, 223

    2004

  26. [34]

    Upper, D. R. 1997, PhD thesis, University of California,

    1997

  27. [35]

    Berkeley, USA Van Kampen, N. G. 1992, Stochastic Processes in Physics and Chemistry (Amsterdam: Elsevier Science) Von Neumann, J. 1966, Theory of Self-Reproducing Automata (Champain, IL: University of Illionois Press)

    1992

  28. [36]

    Weimar, J. R. 1998, Simulation with Cellular Automata (Berlin: Technical University Braunschweig)

    1998

  29. [37]

    2001, Solar Physics, 203, 87

    Wheatland, M. 2001, Solar Physics, 203, 87

    2001

  30. [38]

    Wheatland, M. S. 2008, The Astrophysical Journal, 679, 1621

    2008

  31. [39]

    S., & Glukhov, S

    Wheatland, M. S., & Glukhov, S. 1998, The Astrophysical Journal, 494, 858

    1998

  32. [40]

    S., & Litvinenko, Y

    Wheatland, M. S., & Litvinenko, Y. E. 2001, The Astrophysical Journal, 557, 332

    2001

  33. [41]

    S., & Litvinenko, Y

    Wheatland, M. S., & Litvinenko, Y. E. 2002, Solar Physics, 211, 255

    2002

  34. [42]

    C., Rabassa, P., & Beck, C

    Yalcin, G. C., Rabassa, P., & Beck, C. 2016, Journal of Physics A: Mathematical and Theoretical, 49, 154001

    2016

  35. [43]

    B., & Cleveland, F

    Zirker, J. B., & Cleveland, F. M. 1993, Solar Physics, 144, 341

    1993

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.