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[SSS05-P09] Symmetry breaking that brings self-organized criticality to Olami-Feder-Christensen model of earthquakes
Keywords:self-organized criticality, Olami-Feder-Christensen model
Self-organized criticality (SOC) is an important concept in various natural phenomena, including earthquakes, as pointed out by Bak, Tang, and Wiesenfeld (Bak et al. 1987). SOC refers to systems reaching a critical state without external parameter tuning, with spatio-temporal correlation following a power law. As is well known as the Gutenberg-Richter (GR) law, the size-frequency distribution of earthquakes also follows a power law, and the Olami-Feder-Christensen (OFC) model was proposed as a model that reproduces the GR law (Olami et al. 1992). The OFC model imitates a fault, in which distribution rule and boundary conditions play a crucial role. The distribution rule is generally assumed to be a spatially uniform rule, and SOC is indicated when either open boundary conditions (BCs) or free BCs are imposed around the computational domain as boundary conditions. Interestingly, when the boundary conditions are set to periodic (periodic BCs), the system does not exhibit SOC (Socolar et al. 1993, Grassberger 1994). Models introducing spatial non-uniformity have been proposed (Janosi and Kertesz 1993, Ceva 1995, Bach et al. 2008, Li and Wang 2018, Matin et al. Matin et al. 2020), but a unified understanding of SOC generation remains elusive.
The OFC model operates on a two-dimensional lattice where each site holds an energy value, driven by instantaneous relaxation(toppling) and slow, uniform energy increase(loading). During each toppling, energy is distributed to neighboring sites if a site exceeds the threshold, with assumed energy dissipation. At open or free boundary conditions, energy distribution outside the system is set to zero. Then, clusters are generated from these boundaries (Middleton and Tang 1995), and the whole system spontaneously moves to the state of SOC (Hergarten and Krenn 2011), and it is assumed that the non-uniformity of the distribution rule of artificial boundaries is necessary for SOC to occur (Grassberger 1994).
To explore the fundamental conditions for SOC emergence, we examine the OFC model with a hole, the simplest form of non-uniformity. This system operates under uniform distribution rules and periodic BCs. Introducing a hole disrupts uniformity, acting as an energy sink where dissipation solely occurs. Unlike the conventional OFC model where strong dissipation happens at boundaries, our model delegates this role entirely to the hole, rendering the system spatially uniform except for the hole.
Numerical simulations reveal that SOC only emerges in the presence of at least one hole, contrasting with no SOC exhibited in hole-less scenarios. Increasing the number of holes introduces global non-uniformity, shifting the size frequency distribution from a power law to exponential, thus halting SOC behavior. This aligns with findings in the OFC model with lattice defects (Ceva 1995). From the above, local non-uniformity (holes) proves vital for SOC emergence, while perfect uniformity or global non-uniformity disrupts it. Previous studies suggested energy clustering starts from system boundaries, but we discovered that even slight non-uniformity within the system is the key to SOC emergence, unintentionally overlooked by conventional boundary assumptions.
In the present study, the non-uniformity is not assigned to the outer boundary of the computational domain, but to the hole in the inner domain, from which the cluster structure is formed. Then, the next question is, what does a hole correspond to in the real world? If we can give a physical interpretation to the hole, it will be a great clue to understand the power law of earthquakes. In addition, this study shows that the state of SOC appears by itself when the symmetry of space is broken. The clustering of energy prior to the SOC is reminiscent of symmetry breaking (e.g., the phenomenon in which spins of many-body systems align simultaneously and acquire magnetization in macroscopic magnetism).
The OFC model operates on a two-dimensional lattice where each site holds an energy value, driven by instantaneous relaxation(toppling) and slow, uniform energy increase(loading). During each toppling, energy is distributed to neighboring sites if a site exceeds the threshold, with assumed energy dissipation. At open or free boundary conditions, energy distribution outside the system is set to zero. Then, clusters are generated from these boundaries (Middleton and Tang 1995), and the whole system spontaneously moves to the state of SOC (Hergarten and Krenn 2011), and it is assumed that the non-uniformity of the distribution rule of artificial boundaries is necessary for SOC to occur (Grassberger 1994).
To explore the fundamental conditions for SOC emergence, we examine the OFC model with a hole, the simplest form of non-uniformity. This system operates under uniform distribution rules and periodic BCs. Introducing a hole disrupts uniformity, acting as an energy sink where dissipation solely occurs. Unlike the conventional OFC model where strong dissipation happens at boundaries, our model delegates this role entirely to the hole, rendering the system spatially uniform except for the hole.
Numerical simulations reveal that SOC only emerges in the presence of at least one hole, contrasting with no SOC exhibited in hole-less scenarios. Increasing the number of holes introduces global non-uniformity, shifting the size frequency distribution from a power law to exponential, thus halting SOC behavior. This aligns with findings in the OFC model with lattice defects (Ceva 1995). From the above, local non-uniformity (holes) proves vital for SOC emergence, while perfect uniformity or global non-uniformity disrupts it. Previous studies suggested energy clustering starts from system boundaries, but we discovered that even slight non-uniformity within the system is the key to SOC emergence, unintentionally overlooked by conventional boundary assumptions.
In the present study, the non-uniformity is not assigned to the outer boundary of the computational domain, but to the hole in the inner domain, from which the cluster structure is formed. Then, the next question is, what does a hole correspond to in the real world? If we can give a physical interpretation to the hole, it will be a great clue to understand the power law of earthquakes. In addition, this study shows that the state of SOC appears by itself when the symmetry of space is broken. The clustering of energy prior to the SOC is reminiscent of symmetry breaking (e.g., the phenomenon in which spins of many-body systems align simultaneously and acquire magnetization in macroscopic magnetism).