Bus bunching is a perennial phenomenon that not only diminishes the efficiency of a bus system, but also prevents transit authorities from keeping buses on schedule. We present a physical theory of buses serving a loop of bus stops as a ring of coupled self-oscillators, analogous to the Kuramoto model. Sustained bunching is a repercussion of the process of phase synchronisation whereby the phases of the oscillators are locked to each other. This emerges when demand exceeds a critical threshold. Buses also bunch at low demand, albeit temporarily, due to frequency detuning arising from different human drivers’ distinct natural speeds. We calculate the critical transition when complete phase locking (full synchronisation) occurs for the bus system, and posit the critical transition to completely no phase locking (zero synchronisation). The intermediate regime is the phase where clusters of partially phase locked buses exist. Intriguingly, these theoretical results are in close correspondence to real buses in a university’s shuttle bus system.
Spatial patterning often occurs in ecosystems as a result of a selforganizing process caused by feedback between organisms and the physical environment. Here, we show that the spatial patterns observable in centuries-old Balinese rice terraces are also created by feedback between farmers' decisions and the ecology of the paddies, which triggers a transition from local to globalscale control of water shortages and rice pests. We propose an evolutionary game, based on local farmers' decisions that predicts specific power laws in spatial patterning that are also seen in a multispectral image analysis of Balinese rice terraces. The model shows how feedbacks between human decisions and ecosystem processes can evolve toward an optimal state in which total harvests are maximized and the system approaches Pareto optimality. It helps explain how multiscale cooperation from the community to the watershed scale could persist for centuries, and why the disruption of this self-organizing system by the Green Revolution caused chaos in irrigation and devastating losses from pests. The model shows that adaptation in a coupled human-natural system can trigger self-organized criticality (SOC). In previous exogenously driven SOC models, adaptation plays no role, and no optimization occurs. In contrast, adaptive SOC is a self-organizing process where local adaptations drive the system toward local and global optima.self-organization | criticality | irrigation | evolutionary games | Pareto optimality T he geometric precision of Balinese rice terraces has inspired generations of postcard photographers. Viewed from above, a changing mosaic of colors appears: green when the rice is young, yellow as it nears harvest, silver when the paddies are flooded, and brown when they are drained. These colors are not uniform across the island, because Bali is an equatorial island with only two seasons, wet and dry. Consequently, farmers can plant their crops at any time, although they avoid harvesting in the rainy season. The colored patches that make up the mosaics are visible in Google Earth. Like many natural phenomena, patches show a characteristic power-law distribution of sizes. However, in this case, the patches are created by the farmer's decisions about when to synchronize irrigation schedules with their neighbors: Each patch displays the outcome of these choices. Adjacent patches tend to be correlated. This correlation weakens with distance, a relationship that also follows a specific power law (Fig. 1). To discover why harvests approach a maximum when both the size distribution of patches and the corresponding correlation functions fit power-law distributions, we need a model that relates the decisions of the farmers to the consequences for irrigation flows and rice growth.In Bali, water is regarded as a public good, the gift of the Goddess of the Lakes. Rice is grown in paddy fields fed by irrigation systems dependent on rainfall. Rainfall varies by season and, in combination with groundwater inflow, determines river flow. By virtue of...
The location and nature of the percolation transition in random networks is a subject of intense interest. Recently, a series of graph evolution processes have been introduced that lead to discontinuous percolation transitions where the addition of a single edge causes the size of the largest component to exhibit a significant macroscopic jump in the thermodynamic limit. These processes can have additional exotic behaviors, such as displaying a "Devil's staircase" of discrete jumps in the supercritical regime. Here we investigate whether the location of the largest jump coincides with the percolation threshold for a range of processes, such as Erdős-Rényipercolation, percolation via edge competition and via growth by overtaking. We find that the largest jump asymptotically occurs at the percolation transition for Erdős-Rényiand other processes exhibiting global continuity, including models exhibiting an "explosive" transition. However, for percolation processes exhibiting genuine discontinuities, the behavior is substantially richer. In percolation models where the order parameter exhibits a staircase, the largest discontinuity generically does not coincide with the percolation transition. For the generalized Bohman-Frieze-Wormald model, it depends on the model parameter. Distinct parameter regimes well in the supercritical regime feature unstable discontinuous transitions-a novel and unexpected phenomenon in percolation. We thus demonstrate that seemingly and genuinely discontinuous percolation transitions can involve a rich behavior in supercriticality, a regime that has been largely ignored in percolation.
In this paper, we have implemented a large-scale agent-based model to study the outbreak of coronavirus infectious diseases (COVID-19) in Singapore, taking into account complex human interaction pattern. In particular, the concept of multiplex network is utilized to differentiate between social interactions that happen in households and workplaces. In addition, weak interactions among crowds, transient interactions within social gatherings, and dense human contact between foreign workers in dormitories are also taken into consideration. Such a categorization in terms of a multiplex of social network connections together with the Susceptible-Exposed-Infectious-Removed (SEIR) epidemic model have enabled a more precise study of the feasibility and efficacy of control measures such as social distancing, work from home, and lockdown, at different moments and stages of the pandemics. Using this model, we study an epidemic outbreak that occurs within densely populated residential areas in Singapore. Our simulations show that residents in densely populated areas could be infected easily, even though they constitute a very small fraction of the whole population. Once infection begins in these areas, disease spreading is uncontrollable if appropriate control measures are not implemented.
We have generalized the two-step approach to the solution of systems of N coupled quantum anharmonic oscillators. By using the squeezed vacuum state of each individual oscillator, we construct the tensor product state, and obtain the optimal squeezed vacuum product state through energy minimization. We then employ this optimal state and its associated bosonic operators to define a basis set to construct the Heisenberg matrix. The diagonalization of the matrix enables us to obtain the energy eigenvalues of the coupled oscillators. In particular, we have applied our formalism to determine the eigenenergies of systems of two coupled quantum anharmonic oscillators perturbed by a general polynomial potential, as well as three and four coupled systems. Furthermore, by performing a first-order perturbation analysis about the optimal squeezed vacuum product state, we have also examined into the squeezing properties of two coupled oscillator systems.
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