What is critical about criticality: in praise of the correlation function

The concept of self-organized criticality was introduced to explain the behaviour of the sandpile model. In this model, particles are randomly dropped onto a square grid of boxes. When a box accumulates four particles they are redistributed to the four adjacent boxes or lost off the edge of the grid. Redistributions can lead to further instabilities with the possibility of more particles being lost from the grid, contributing to the size of each avalanche. These model 'avalanches' satisfied a power-law frequency-area distribution with a slope near unity. Other cellular-automata models, including the slider-block and forest-fire models, are also said to exhibit self-organized critical behaviour. It has been argued that earthquakes, landslides, forest fires, and species extinctions are examples of self-organized criticality in nature. In addition, wars and stock market crashes have been associated with this behaviour. The forest-fire model is particularly interesting in terms of its relation to the critical-point behaviour of the site-percolation model. In the basic forest-fire model, trees are randomly planted on a grid of points. Periodically in time, sparks are randomly dropped on the grid. If a spark drops on a tree, that tree and adjacent trees burn in a model fire. The fires are the `avalanches' and they are found to satisfy power-law frequency-area distributions with slopes near unity. This forest-fire model is closely related to the site-percolation model, that exhibits critical behaviour. In the forest-fire model there is an inverse cascade of trees from small clusters to large clusters, trees are lost primarily from model fires that destroy the largest clusters. This quasi steady-state cascade gives a power-law frequency-area distribution for both clusters of trees and smaller fires. The site-percolation model is equivalent to the forest-fire model without fires. In this case there is a transient cascade of trees from small to large clusters and a power-law distribution is found only at a critical density of trees.

Forest fire burn areas in Western Canada modeled as self-similar criticality

We consider the frequency-size statistics of two natural hazards, forest fires and landslides. Both appear to satisfy power-law (fractal) distributions to a good approximation under a wide variety of conditions. Two simple cellular-automata models have been proposed as analogs for this observed behavior, the forest fire model for forest fires and the sand pile model for landslides. The behavior of these models can be understood in terms of a self-similar inverse cascade. For the forest fire model the cascade consists of the coalescence of clusters of trees; for the sand pile model the cascade consists of the coalescence of metastable regions.

Affected by global warming and increased extreme weather, Hunan Province saw a phased and concentrated outbreak of forest fires in 2022, causing significant damage and impact. Predicting the occurrence of forest fires can enhance the ability to make early predictions and strengthen early warning and responses. Currently, fire prevention and extinguishing in China’s forests and grasslands face severe challenges due to the overlapping of natural and social factors. Existing forest fire occurrence prediction models mostly take into account vegetation, topographic, meteorological and human activity factors; however, the occurrence of forest fires is closely related to the forest fuel moisture content. In this study, the traditional driving factors of forest fire such as satellite hotspots, vegetation, meteorology, topography and human activities from 2004 to 2021 were introduced along with forest fuel factors (vegetation canopy water content and evapotranspiration from the top of the vegetation canopy), and a database of factors for predicting forest fire occurrence was constructed. And a forest fire occurrence prediction model was built using machine learning methods such as the Random Forest model (RF), the Gradient Boosting Decision Tree model (GBDT) and the Adaptive Augmentation Model (AdaBoost). The accuracy of the models was verified using Area Under Curve (AUC) and four other metrics. The RF model with an AUC value of 0.981 was more accurate than all other models in predicting the probability of forest fire occurrence, followed by the GBDT (AUC = 0.978) and AdaBoost (AUC = 0.891) models. The RF model, which has the best accuracy, was selected to predict the monthly forest fire probability in Changsha in 2022 and combined with the Inverse Distance Weight Interpolation method to plot the monthly forest fire probability in Changsha. We found that the monthly spatial and temporal distribution of forest fire probability in Changsha varied significantly, with March, April, May, September, October, November and December being the months with higher forest fire probability. The highest probability of forest fires occurred in the central and northern regions. In this study, the core drivers affecting the occurrence of forest fires in Changsha City were found to be vegetation canopy evapotranspiration and vegetation canopy water content. The RF model was identified as a more suitable forest fire occurrence probability prediction model for Changsha City. Meanwhile, this study found that vegetation characteristics and combustible factors have more influence on forest fire occurrence in Changsha City than meteorological factors, and surface temperature has less influence on forest fire occurrence in Changsha City.

Forest fires and the structure of the universe

Applications of statistical mechanics to natural hazards and landforms

Applying genetic algorithms to set the optimal combination of forest fire related variables and model forest fire susceptibility based on data mining models. The case of Dayu County, China

Short description of the Game Method for Modelling (GMM) is given. On GMM base, a model of forest fire is constructed. The model is an illustration of the possible applications of the GMM. It represents the case of wind presence. Intuitionistic fuzzy interpretation of the degree of burnt forest mass.

We examine the cosmological evolution of ultralight axionlike (ULA) scalar fields with potentials of the form $V(\ensuremath{\phi})={m}^{2}{f}^{2}[1\ensuremath{-}\mathrm{cos}(\ensuremath{\phi}/f){]}^{n}$, with particular emphasis on the deviation in their behavior from the corresponding small--$\ensuremath{\phi}$ power-law approximations to these potentials: $V(\ensuremath{\phi})\ensuremath{\propto}{\ensuremath{\phi}}^{2n}$. We show that in the slow-roll regime, when ${\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\phi}}}^{2}/2\ensuremath{\ll}V(\ensuremath{\phi})$, the full ULA potentials yield a more interesting range of possibilities for quintessence than do the corresponding power law approximations. For rapidly oscillating scalar fields, we derive the equation of state parameter and oscillation frequency for the ULA potentials and show how they deviate from the corresponding power-law values. We derive an analytic expression for the equation of state parameter that better approximates the ULA value than does the pure power-law approximation.

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

The forest fire occurrence prediction model is a very useful tool for preventing and extinguishing forest fires. The determination of forest fire drivers is important for establishing a high‐precision forest fire prediction model. In this paper, we studied the relative influence of different types of factors on forest fire occurrence in forest areas of Jiangxi Province. Several models, i.e., Multilayer perceptron (MLP), Logistic, and Support vector machine (SVM), are used to predict the occurrence of forest fires. Through modeling and analysis of forest fire data from 2010 to 2016 years, we found that climatic and topographic are influential factors in the model of forest fire occurrence in Jiangxi Province. Subsequently, we established the MLP occurrence model based on the significant factors after the variable screening. Using ROC plots to compare the effects of the three models, MLP scored 0.984, which was higher than Logistic of 0.933 and SVM of 0.974. For the independent validation set of 2017‐2018, an accuracy of 91.73% was also achieved. Therefore, the multilayer perceptron is well suited for the prediction of forest fires in Jiangxi Province. Based on the prediction results, a fire risk level map of Jiangxi Province was produced. Finally, we analyzed the changes in forest fire quantity under climate change, which can be helpful for fire prevention and suppression of forest fires.

Neighbourhood rules make or break spatial scale invariance in a classic model of contagious disturbance

A modified version of compact directed percolation on a square lattice is examined in the context of a model problem for the spread of forest fires. The modification relates to conditioning the extent of the fire spread along the forest boundaries. Exact expressions are given for the mean perimeter length and the mean size of the damaged forest under such conditioning.

Recent Kerr relaxation experiments by Gottke et al. have revealed the existence of a pronounced temporal power law decay in the orientational relaxation near the isotropic-nematic phase transition (INPT) of nematogens of rather small aspect ratio, kappa (kappa approximately 3-4). We have carried out very long (50 ns) molecular dynamics simulations of model (Gay-Berne) prolate ellipsoids with aspect ratio 3 in order to investigate the origin of this power law. The model chosen is known to undergo an isotropic to nematic phase transition for a range of density and temperature. The distance dependence of the calculated angular pair correlation function correctly shows the emergence of a long range correlation as the INPT is approached along the density axis. In the vicinity of INPT, the single particle second rank orientational time correlation function exhibits power law decay, (t(-alpha)) with exponent alpha approximately 2/3. More importantly, we find the sudden appearance of a pronounced power-law decay in the collective part of the second rank orientational time correlation function at short times when the density is very close to the transition density. The power law has an exponent close to unity, that is, the correlation function decays almost linearly with time. At long times, the decay is exponential-like, as predicted by Landau-de Gennes mean field theory. Since Kerr relaxation experiments measure the time derivative of the collective second rank orientational pair correlation function, the simulations recover the near independence of the signal on time observed in experiments. In order to capture the microscopic essence of the dynamics of pseudonematic domains inside the isotropic phase, we introduce and calculate a dynamic orientational pair correlation function (DOPCF) obtained from the coefficients in the expansion of the distinct part of orientational van Hove time correlation function in terms of spherical harmonics. The DOPCF exhibits power law relaxation when the pair separation length is below certain critical length. The orientational relaxation of a local director, defined in terms of the sum of unit vectors of all the ellipsoidal molecules, is also found to show slow power law relaxation over a long time scale. These results have been interpreted in terms of a newly developed mode coupling theory of orientational dynamics near the INPT. In the present case, the difference between the single particle and the collective orientational relaxation is huge which can be explained by the frequency dependence of the memory kernel, calculated from the mode coupling theory. The relationship of this power law with the one observed in a supercooled liquid near its glass transition temperature is explored.

In this paper the assignment and theoretical investigations of the problems of crown forest fire initiation and spread in windy condition were carried out. Mathematical model of forest fire was based on an analysis of known experimental data and using concept and methods from reactive media mechanics. The investigation takes in to account the mutual interaction of the forest fires and three dimensional atmosphere flows. The research is done by means of mathematical modeling of physical processes. It is based on numerical solution of Reynolds equations for chemical components and equations of energy conservation for gaseous and condensed phases. It is assumed that the forest during a forest fire can be modeled as a two-temperature multiphase non-deformable porous reactive medium. A discrete analog for the system of equations was obtained by means of the finite volume method. The developed model of forest fire initiation and spreading would make it possible to obtain a detailed picture of the variation in the velocity, temperature and chemical species concentration fields with time. Mathematical model and the result of the calculation give an opportunity to evaluate critical conditions of the forest fire initiation and spread which allows applying the given model for of means for preventing fires.