Critical Behavior of the Sandpile Model as a Self-Organized Branching Process

Kinetic equations, which explicitly take into account the branching nature of sandpile avalanches, are derived. The dynamics of the sandpile model is described by the generating functions of a branching process. Having used the results obtained the renormalization group approach to the critical behavior of the sandpile model is generalized in order to calculate both critical exponents and height probabilities.Comment: REVTeX, twocolumn, 4 page

Inversion Symmetry and Critical Exponents of Dissipating Waves in the Sandpile Model

Statistics of waves of topplings in the Sandpile model is analysed both analytically and numerically. It is shown that the probability distribution of dissipating waves of topplings that touch the boundary of the system obeys power-law with critical exponent 5/8. This exponent is not indeendent and is related to the well-known exponent of the probability distribution of last waves of topplings by exact inversion symmetry s -> 1/s.Comment: 5 REVTeX pages, 6 figure

Scaling of waves in the Bak-Tang-Wiesenfeld sandpile model

We study probability distributions of waves of topplings in the Bak-Tang-Wiesenfeld model on hypercubic lattices for dimensions D>=2. Waves represent relaxation processes which do not contain multiple toppling events. We investigate bulk and boundary waves by means of their correspondence to spanning trees, and by extensive numerical simulations. While the scaling behavior of avalanches is complex and usually not governed by simple scaling laws, we show that the probability distributions for waves display clear power law asymptotic behavior in perfect agreement with the analytical predictions. Critical exponents are obtained for the distributions of radius, area, and duration, of bulk and boundary waves. Relations between them and fractal dimensions of waves are derived. We confirm that the upper critical dimension D_u of the model is 4, and calculate logarithmic corrections to the scaling behavior of waves in D=4. In addition we present analytical estimates for bulk avalanches in dimensions D>=4 and simulation data for avalanches in D<=3. For D=2 they seem not easy to interpret.Comment: 12 pages, 17 figures, submitted to Phys. Rev.

Self-Organized States in Cellular Automata: Exact Solution

The spatial structure, fluctuations as well as all state probabilities of self-organized (steady) states of cellular automata can be found (almost) exactly and {\em explicitly} from their Markovian dynamics. The method is shown on an example of a natural sand pile model with a gradient threshold.Comment: 4 pages (REVTeX), incl. 2 figures (PostScript

The simulation and optimization of steady state process circuits by means of artificial neural networks

Dissertation (Ph.D.) -- University of Stellenbosch, 1993.ENGLISH ABASTRACT: Since the advent of modern process industries engineers engaged in the modelling and simulation of chemical and metallurgical processes have had to contend with two important dilemmas. The first concerns the ill-defined nature of the processes they have to describe, while the second relates to the limitations of prevailing computational resources. Current process simulation procedures are based on explicit process models in one form or another. Many chemical and metallurgical processes are not amenable to this kind of modelling however, and can not be incorporated effectively into current commercial process simulators. As a result many process operations do not benefit from the use of predictive models and simulation routines and plants are often poorly designed and run, ultimately leading to considerable losses in revenue. In addition to this dilemma, process simulation is in a very real way constrained by available computing resources. The construction of adequate process models is essentially meaningless if these models can not be solved efficiently - a situation occurring all too often. In the light of these problems, it is thus not surprising that connectionist systems or neural network methods are singularly attractive to process engineers, since they provide a powerful means of addressing both these dilemmas. These nets can form implicit process models through learning by example, and also serve as a vehicle for parallel supercomputing devices. In this dissertation the use of artificial neural networks for the steady state modelling and optimization of chemical and metallurgical process circuits is consequently investigated. The first chapter is devoted to a brief overview of the simulation of chemical and metallurgical plants by conventional methods, as well as the evolution and impact of computer technology and artificial intelligence on the process industries. Knowledge of the variance covariance matrices of process data is of paramount importance to data reconciliation and gross error detection problems, and although various methods can be employed to estimate these often unknown variances, it is shown in the second chapter that the use of feedforward neural nets can be more efficient than conventional strategies. In the following chapter the important problem of gross error detection in process data is addressed. Existing procedures are statistical and work well for systems subject to linear constraints. Non-linear constraints are not handled well by these methods and it is shown that back propagation neural nets can be trained to detect errors in process systems, regardless of the nature of the constraints. In the fourth chapter the exploitation of the massively parallel information processing structures of feedback neural nets in the optimization of process data reconciliation problems is investigated. Although effective and sophisticated algorithms are available for these procedures, there is an ever present demand for computational devices or routines that can accommodate progressively larger or more complex problems. Simulations indicate that neural nets can be efficient instruments for the implementation of parallel strategies for the optimization of such problems. In the penultimate chapter a gold reduction plant and a leach plant are modelled with neural nets and the models shown to be considerably better than the linear regression models used in practice. The same technique is also demonstrated with the modelling of an apatite flotation plant. Neural nets can also be used in conjunction with other methods and in the same chapter the steady state simulation and optimization of a gravity separation circuit with the use of two linear programming models and a neural net are described.AFRIKAANSE OPSOMMING: Sedert die ontstaan van prosesingenieurswese, het ingenieurs gemoeid met die modellering en simulasie van chemiese en metallurgiese prosesse met twee belangrike dilemmas te kampe gehad. Die eerste het te make met die swakgedefinieerde aard van chemiese prosesse, wat die beskrywing en dus ook die beheer daarvan kompliseer, terwyl die tweede verband hou met die beperkinge van huidige berekeningsmiddele. Die prosesse wat tans gebruik word om chemiese prosesse te simuleer is gebaseer op eksplisiete prosesmodelle van een of ander aard. Baie chemiese en metallurgiese prosesse kan egter nie op 'n eksplisiete wyse gemodelleer word nie, en kan gevolglik ook nie doeltreffendheid deur kommersiële prosessimulators beskryf word nie. Die bedryf van baie prosesse vind derhalwe nie baat by die gebruik van voorspellende modelle en simulasie-algoritmes nie en aanlegte word dikwels suboptimaal ontwerp en bedryf, wat uiteindelik tot aansienlike geldelike verliese kan lei. Prosessimulasie word op die koop toe ook beperk deur die beskikbaarheid van berekeningsfasiliteite. Die konstruksie van geskikte prosesmodelle hou geen voordeel in as hierdie modelle nie doeltreffendheid opgelos kan word nie. Teen die agtergrond van hierdie probleme is dit nie verrassend dat neurale netwerke 'n besondere bekoring vir prosesingenieurs inhou nie, aangesien hulle beide hierdie dilemmas aanspreek. Hierdie nette kan implisiete prosesmodelle konstrueer deur te leer van voorbeelde en dien ook as 'n raamwerk vir parallelle superrekenaars. In hierdie proefskrif word die gebruik van kunsmatige neurale netwerke vir gestadigde toestandsmodellering en optimering van chemiese en metallurgiese prosesse gevolglik ondersoek. Die eerste hoofstuk word gewy aan 'n kort oorsig oor die simulasie van chemiese en metallurgiese aanlegte met konvensionele tegnieke, asook die ontwikkeling en impak van rekenaartegnologie en skynintelligensie in die prosesnywerhede. Kennis van die variansie-kovariansie-matrikse van prosesdata is van kardinale belang vir datarekonsiliasie en die identifikasie en eliminasie van sistematiese foute en alhoewel verskeie metodes aangewend kan word om hierdie onbekende variansies te beraam, word daar in die tweede hoofstuk getoon dat die gebruik van neurale netwerke meer doeltreffend is as konvensionele strategieë. In die volgende hoofstuk word die belangrike probleem van sistematiese foutopsporing in prosesdata ondersoek. Bestaande prosedures is statisties van aard en werk goed vir stelsels onderworpe aan lineêre beperkinge. Nie-lineêre beperkinge kan nie doeltreffend deur hierdie prosedures hanteer word nie en daar word gewys dat terugwaarts-propagerende nette geleer kan word om sulke foute in prosessisteme op te spoor, ongeag die aard van die beperkinge. In die vierde hoofstuk word die rekonsiliasie van prosesdata met behulp van massiewe parallelle dataverwerkingstrukture soos verteenwoordig deur terugvoerende neurale nette, ondersoek. Alhoewel doeltreffende en gesofistikeerde algoritmes beskikbaar is vir die optimering van die tipe probleme, is daar 'n onversadigbare aanvraag na rekenaars wat groter en meer komplekse stelsels kan akkommodeer. Simulasie dui aan dat neurale nette effektief aangewend kan word vir die implementering van parallelle strategieë vir dié tipe optimeringsprobleme. In die voorlaaste hoof stuk word die konneksionistiese modellering van 'n goudreduksie- en 'n logingsaanleg beskryf en daar word aangetoon dat die neurale netwerk-modelle aansienlik beter resultate lewer as die linneêre regressie modelle wat in die praktyk gebruik word. Dieselfde tegnieke vir die modellering van 'n flottasie-aanleg vir apatiet word ook bespreek. Neural nette kan ook saam met ander metodes aangewend word en in dieselfde hoofstuk word die gebruik van twee lineêre programmeringsmodelle en 'n neural net om 'n gravitasieskeidingsbaan onder gestadigde toestande te simuleer en te optimeer, beskryf

Dynamically Driven Renormalization Group Applied to Sandpile Models

The general framework for the renormalization group analysis of self-organized critical sandpile models is formulated. The usual real space renormalization scheme for lattice models when applied to nonequilibrium dynamical models must be supplemented by feedback relations coming from the stationarity conditions. On the basis of these ideas the Dynamically Driven Renormalization Group is applied to describe the boundary and bulk critical behavior of sandpile models. A detailed description of the branching nature of sandpile avalanches is given in terms of the generating functions of the underlying branching process.Comment: 18 RevTeX pages, 5 figure

Critical Behaviour of the Drossel-Schwabl Forest Fire Model

We present high statistics Monte Carlo results for the Drossel-Schwabl forest fire model in 2 dimensions. They extend to much larger lattices (up to $65536\times 65536$) than previous simulations and reach much closer to the critical point (up to $\theta \equiv p/f = 256000$). They are incompatible with all previous conjectures for the (extrapolated) critical behaviour, although they in general agree well with previous simulations wherever they can be directly compared. Instead, they suggest that scaling laws observed in previous simulations are spurious, and that the density $\rho$ of trees in the critical state was grossly underestimated. While previous simulations gave $\rho\approx 0.408$, we conjecture that $\rho$ actually is equal to the critical threshold $p_c = 0.592...$ for site percolation in $d=2$. This is however still far from the densities reachable with present day computers, and we estimate that we would need many orders of magnitude higher CPU times and storage capacities to reach the true critical behaviour -- which might or might not be that of ordinary percolation.Comment: 8 pages, including 9 figures, RevTe