Comments on Mathematical Aspects of the Biró–Néda Model

Inácio, Ilda; Velhinho, José M.

doi:10.3390/math10040644

Cited by 3 publications

(3 citation statements)

References 19 publications

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“…Once the needed kernel functions are realistically defined, the dynamics given by the LGGR model should yield the time evolution of the tree-size distribution function. In a general study of the LGGR dynamics it was previously shown 40 , that apart from some pathologic cases, such systems are indeed converging to the stationary distribution. Depending on the starting condition, the mean of the distribution might converge slowly to a stationary value, however, the distribution of x/ x converges quickly to a stationary distribution.…”

Section: Resultsmentioning

confidence: 98%

“…Based on the form of the µ(x) growth-and γ (x) reset rates, the LGGR model is able to reproduce stationary probability distributions, ρ s (x) , that are frequently encountered in complex systems 24,25,32,33 . The LGGR's mathematical apparatus has been comprehensively studied in recent years 24,25,31,39,40 , encompassing aspects of convergence and applicability to various fields of science 24,[32][33][34][35]41 .…”

Section: The Lggr Modeling Frameworkmentioning

confidence: 99%

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Tree size distribution as the stationary limit of an evolutionary master equation

Kelemen,

Józsa,

Hartel

et al. 2024

Sci Rep

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The diameter distribution of a given species of deciduous trees is well approximated by a Gamma distribution. Here we give new experimental evidence for this conjecture by analyzing deciduous tree size data in mature semi-natural forest and ancient, traditionally managed wood-pasture from Central Europe. These distribution functions collapse on a universal shape if the tree sizes are normalized to the mean value in the considered sample. A new evolutionary master equation is used to model the observed distribution. The model incorporates four ecological processes: tree growth, mortality, recruitment, and diversification. Utilizing simple and realistic kernel functions describing the first three, along with an assumed multiplicative dilution due to diversification, the stationary solution of the master equation yields the experimentally observed Gamma distribution. The model as it is formulated allows an analytically compact solution and has only two fitting parameters whose values are consistent with the experimental data related to these processes. We found that the equilibrium size distribution of tree species with different ecology, originating from two contrastingly different semi-natural ecosystem types can be accurately described by a single dynamical mean-field model.

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Section: Resultsmentioning

confidence: 98%

Section: The Lggr Modeling Frameworkmentioning

confidence: 99%

Tree size distribution as the stationary limit of an evolutionary master equation

Kelemen,

Józsa,

Hartel

et al. 2024

Sci Rep

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“…Finally, other generative processes include highly optimized tolerance [ 25 , 26 ], the coherent noise model of biological extinction [ 27 ], the repeated fragmentation model of fixed length elements [ 1 ], the dynamic of times between records in a random process [ 28 ], the Hawkes processes [ 29 ] and out-from-criticality feedback [ 30 ]. A master equation for power laws has been previously obtained in the literature [ 31 – 35 ] but only considers stationary distributions or transient dynamics, while our results give a non-stationary solution [ 36 ]. Furthermore, our choice of transition rates has not been considered before.…”

Section: Introductionmentioning

confidence: 93%

A master equation for power laws

Roman

Bertolotti²

2022

R. Soc. open sci.

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We propose a new mechanism for generating power laws. Starting from a random walk, we first outline a simple derivation of the Fokker–Planck equation. By analogy, starting from a certain Markov chain, we derive a master equation for power laws that describes how the number of cascades changes over time (cascades are consecutive transitions that end when the initial state is reached). The partial differential equation has a closed form solution which gives an explicit dependence of the number of cascades on their size and on time. Furthermore, the power law solution has a natural cut-off, a feature often seen in empirical data. This is due to the finite size a cascade can have in a finite time horizon. The derivation of the equation provides a justification for an exponent equal to 2, which agrees well with several empirical distributions, including Richardson’s Law on the size and frequency of deadly conflicts. Nevertheless, the equation can be solved for any exponent value. In addition, we propose an urn model where the number of consecutive ball extractions follows a power law. In all cases, the power law is manifest over the entire range of cascade sizes, as shown through log–log plots in the frequency and rank distributions.

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Thermodynamical Aspects of the LGGR Approach for Hadron Energy Spectra

Biro

Néda

2022

Symmetry

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The local growth global reset (LGGR) dynamical model is reviewed and its performance in describing the hadron energy spectra in relativistic collisions is demonstrated. It is shown that even for dynamical processes a temperature-like quantity can be defined and distributions resembling statistical equilibrium can be reached. With appropriate growth and reset rates the LGGR model is capable of describing the right energy spectra. These findings draw a certain picture of quark–gluon plasma development with random hadronization and re-swallowing steps and signals the fact that observing an exponential spectrum does not necessarily prove thermal equilibrium in the experiment.

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Comments on Mathematical Aspects of the Biró–Néda Model

Cited by 3 publications

References 19 publications

Tree size distribution as the stationary limit of an evolutionary master equation

Tree size distribution as the stationary limit of an evolutionary master equation

A master equation for power laws

Thermodynamical Aspects of the LGGR Approach for Hadron Energy Spectra

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