Self-organization without conservation: true or just apparent scale-invariance?

Abstract

The existence of true scale-invariance in slowly driven models of self-organized criticality without a conservation law, as forest-fires or earthquake automata, is scrutinized in this paper. By using three different levels of description - (i) a simple mean field, (ii) a more detailed mean-field description in terms of a (self-organized) branching processes, and (iii) a full stochastic representation in terms of a Langevin equation-, it is shown on general grounds that non-conserving dynamics does not lead to bona fide criticality. Contrarily to conserving systems, a parameter, which we term "re-charging" rate (e.g. the tree-growth rate in forest-fire models), needs to be fine-tuned in non-conserving systems to obtain criticality. In the infinite size limit, such a fine-tuning of the loading rate is easy to achieve, as it emerges by imposing a second separation of time-scales but, for any finite size, a precise tuning is required to achieve criticality and a coherent finite-size scaling picture. Using the approaches above, we shed light on the common mechanisms by which "apparent criticality" is observed in non-conserving systems, and explain in detail (both qualitatively and quantitatively) the difference with respect to true criticality obtained in conserving systems. We propose to call this self-organized quasi-criticality (SOqC). Some of the reported results are already known and some of them are new. We hope the unified framework presented here helps to elucidate the confusing and contradictory literature in this field. In a second accompanying paper, we shall discuss the implications of the general results obtained here for models of neural avalanches in Neuroscience for which self-organized scale-invariance in the absence of conservation has been claimed.

40 pages, 7 figures.

References in corpus (97)

• Statistical mechanics of complex networks
• Power laws, Pareto distributions and Zipf's law
• Nonequilibrium Critical Phenomena and Phase Transitions into Absorbing States
• Universality classes in nonequilibrium lattice systems
• Dynamical synapses causing self-organized criticality in neural networks
• Dynamics of a ferromagnetic domain wall: avalanches, depinning transition and the Barkhausen effect
• Universal scaling behavior of non-equilibrium phase transitions
• Paths to Self-Organized Criticality
• Self-Organized Branching Processes: A Mean-Field Theory for Avalanches
• The abelian sandpile and related models
• Influence of Generic Scale Invariance on Classical and Quantum Phase Transitions
• How self-organized criticality works: A unified mean-field picture
• Experiments in vortex avalanches
• Renormalization of pinned elastic systems: how does it work beyond one loop ?
• The universality class of absorbing phase transitions with a conserved field
• 2-loop Functional Renormalization Group Theory of the Depinning Transition
• Self-organized criticality as an absorbing-state phase transition
• Avalanche and spreading exponents in systems with absorbing states
• Tracer Dispersion in a Self-Organized Critical System
• Transport in Sand Piles, Interface Depinning, and Earthquake Models
• Absorbing-state phase transitions in fixed-energy sandpiles
• Driving, conservation and absorbing states in sandpiles
• Driven Interface Depinning in a Disordered Medium
• Multifractal scaling in the Bak-Tang-Wiesenfeld Sandpile and edge events
• Acoustic Emission from Paper Fracture
• Integration of Langevin Equations with Multiplicative Noise and Viability of Field Theories for Absorbing Phase Transitions
• Interfacial Velocity Corrections due to Multiplicative Noise
• Order Parameter and Scaling Fields in Self-Organized Criticality
• Self-Organized Criticality in Non-conserved Systems
• Critical Behaviour of the Drossel-Schwabl Forest Fire Model
• Self-organized criticality and synchronization in a lattice model of integrate-and-fire oscillators
• Forest fires and other examples of self-organized criticality
• Rare events and breakdown of simple scaling in the Abelian sandpile
• Self-organized branching processes: Avalanche models with dissipation
• Mapping Self-Organized Criticality onto Criticality
• Avalanche dynamics, surface roughening and self-organized criticality - experiments on a 3 dimensional pile of rice
• Pulse-coupled relaxation oscillators: from biological synchronization to Self-Organized Criticality
• Scaling of waves in the Bak-Tang-Wiesenfeld sandpile model
• Avalanches and the Renormalization Group for Pinned Charge-Density Waves
• Universal 1/f Noise from Dissipative SOC Models
• Properties of Foreshocks and Aftershocks of the Non-Conservative SOC Olami-Feder-Christensen Model: Triggered or Critical Earthquakes?
• Vortex avalanches and self organized criticality in superconducting niobium
• Exact Results for the One-Dimensional Self-Organized Critical Forest-Fire Model
• Self-Organized Criticality and Universality in a Nonconservative Earthquake Model
• Field theory of absorbing phase transitions with a non-diffusive conserved field
• Finite-size effects in the self-organized critical forest-fire model
• Power spectra of self-organized critical sandpiles
• A Nonconservative Earthquake Model of Self-Organized Criticality on a Random Graph
• On Self-Organized Criticality and Synchronization in Lattice Models of Coupled Dynamical Systems
• Self-organized criticality in a rice-pile model
• Percolation in real Wildfires
• Scaling and correlations in the dynamics of forest-fire occurrence
• Broken scaling in the Forest Fire Model
• Random Neighbor Theory of the Olami-Feder-Christensen Earthquake Model
• Numerical schemes for continuum models of reaction-diffusion systems subject to internal noise
• Transitions in non-conserving models of Self-Organized Criticality
• Quasiperiodic events in an earthquake model
• Quenched noise and over-active sites in sandpile dynamics
• From waves to avalanches: two different mechanisms of sandpile dynamics
• Avalanche Merging and Continuous Flow in a Sandpile Model
• On the robustness of scale invariance in SOC models
• Critical Behavior of the Sandpile Model as a Self-Organized Branching Process
• Confirming and extending the hypothesis of universality in sandpiles
• Olami-Feder-Christensen Model on different Networks
• Analysis of a dissipative model of self-organized criticality with random neighbors
• Transient and stationary behavior of the Olami-Feder-Christensen earthquake model
• The Oslo rice pile model is a quenched Edwards-Wilkinson equation
• Scaling in a Nonconservative Earthquake Model of Self-Organised Criticality
• Absorbing states and elastic interfaces in random media: two equivalent descriptions of self-organized criticality
• Corrections to scaling in the forest-fire model
• The complex scaling behavior of non--conserved self--organized critical systems
• Cusps, self-organization, and absorbing states
• Avalanches and Self-Organized Criticality in Superconductors
• Short period attractors and non-ergodic behavior in the deterministic fixed energy sandpile model
• Interface Scaling in the Contact Process
• Self-Organized Criticality Induced by Diversity
• Length Scales and Power Laws in the Two-Dimensional Forest-Fire Model
• Self-organized Criticality and Absorbing States: Lessons from the Ising Model
• Sandpiles and absorbing-state phase transitions: recent results and open problems
• On the random neighbor Olami-Feder-Christensen slip-stick model
• SOC in a population model with global control
• How to discriminate easily between Directed-percolation and Manna scaling
• Periodicity and criticality in the Olami-Feder-Christensen model of earthquakes
• Algorithmic Mapping Criticality into Self Organized Criticality
• Field theory of self-organized fractal etching
• Sticky grains do not change the universality class of isotropic sandpiles
• A solvable non-conservative model of Self-Organized Criticality
• Self-organized criticality as a phase transition
• Avalanche Behavior in an Absorbing State Oslo Model
• Comment on ``Self-organized criticality and absorbing states: Lessons from the Ising model"
• Self-organisation to criticality in a system without conservation law
• Dissipative Abelian Sandpiles and Random Walks
• Dynamic roughening of the magnetic flux landscape in YBa${}_2$Cu${}_3$O${}_{7-x}$
• Self-organized critical earthquake model with moving boundary
• Reply to "Comment on `Self-organized Criticality and Absorbing States: Lessons from the Ising Model'"
• Non conservative Abelian sandpile model with BTW toppling rule
• Boundary-induced heterogeneous absorbing states

Cited by in corpus (53)

• Colloquium: Criticality and dynamical scaling in living systems
• Power laws and Self-Organized Criticality in Theory and Nature
• Dragon-kings: mechanisms, statistical methods and empirical evidence
• 25 Years of Self-Organized Criticality: Concepts and Controversies
• Landau-Ginzburg theory of cortex dynamics: Scale-free avalanches emerge at the edge of synchronization
• Signatures of Self-Organised Criticality in an Ultracold Atomic Gas
• Self-organization without conservation: Are neuronal avalanches generically critical?
• A Brief History of Excitable Map-Based Neurons and Neural Networks
• Self-Organized Bistability Associated With First-Order Phase Transitions
• Feedback mechanisms for self-organization to the edge of a phase transition
• Continuously sheared granular matter reproduces in detail seismicity laws
• Neutral theory and scale-free neural dynamics
• Universality in driven open quantum matter
• A simple unified view of branching process statistics: random walks in balanced logarithmic potentials
• Growing Critical: Self-Organized Criticality in a Developing Neural System
• Hybrid-type synchronization transitions: where marginal coherence, scale-free avalanches, and bistability live together
• Synaptic balance due to homeostatically self-organized quasicritical dynamics
• Self-organized critical balanced networks: a unified framework
• Theoretical foundations of studying criticality in the brain
• Quantitative Analysis of Bloggers Collective Behavior Powered by Emotions
• Self-organized bistability and its possible relevance for brain dynamics
• Can dynamical synapses produce true self-organized criticality?
• Steady-State Dynamics of the Forest Fire Model on Complex Networks
• Critical neural networks with short and long term plasticity
• Quasi-Neutral theory of epidemic outbreaks
• Correlations Induced by Depressing Synapses in Critically Self-Organized Networks with Quenched Dynamics
• Critical neuronal models with relaxed timescales separation
• Conway's game of life is a near-critical metastable state in the multiverse of cellular automata
• Chaos and correlated avalanches in excitatory neural networks with synaptic plasticity
• Hydrodynamic stabilization of self-organized criticality in a driven Rydberg gas
• Enhancing (quasi-)long-range order in a two-dimensional driven crystal
• Identification of criticality in neuronal avalanches: II. A theoretical and empirical investigation of the driven case
• Cooperation, competition and the emergence of criticality in communities of adaptive systems
• Homeostatic Criticality in Neuronal Networks
• Hyperuniformity and conservation laws in non-equilibrium systems
• Phase transitions and self-organized criticality in networks of stochastic spiking neurons
• The Emergence of Weak Criticality in SOC systems
• Self-Organized Criticality as Witten-type Topological Field Theory with Spontaneously Broken Becchi-Rouet-Stora-Tyutin Symmetry
• Coexistence of scale invariant and rhythmic behavior in self-organized criticality
• Controlling excitation avalanches in driven Rydberg gases
• Physics of Psychophysics: two coupled square lattices of spiking neurons have huge dynamic range at criticality
• Dynamical and structural properties of an absorbing phase transition: a case study from granular systems
• Memory in neural activity: long-range order without criticality
• Memory-induced long-range order in dynamical systems
• Mechanisms of self-organized quasicriticality in neuronal networks models
• Locally self-organized quasi-critical percolation in a multiple disease model
• Discrete scaling and criticality in a chain of adaptive excitable integrators
• Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience
• Brain, Rain and Forest Fires -- What is critical about criticality: In praise of the correlation function
• Critical Density of the Abelian Manna Model via a Multi-type Branching Process
• A Spiking Neural Dynamical Drift-Diffusion Model on Collective Decision Making with Self-Organized Criticality
• A neuronal dynamics study on a neuromorphic chip
• System-size independence of a large deviation function for frequency of events in a one-dimensional forest-fire model with a single ignition site