The universal visitation law of human mobility.

Human mobility impacts many aspects of a city, from its spatial structure1-3 to its response to an epidemic4-7. It is also ultimately key to social interactions8, innovation9,10 and productivity11. However, our quantitative understanding of the aggregate movements of individuals remains incomplete. Existing models-such as the gravity law12,13 or the radiation model14-concentrate on the purely spatial dependence of mobility flows and do not capture the varying frequencies of recurrent visits to the same locations. Here we reveal a simple and robust scaling law that captures the temporal and spatial spectrum of population movement on the basis of large-scale mobility data from diverse cities around the globe. According to this law, the number of visitors to any location decreases as the inverse square of the product of their visiting frequency and travel distance. We further show that the spatio-temporal flows to different locations give rise to prominent spatial clusters with an area distribution that follows Zipf's law15. Finally, we build an individual mobility model based on exploration and preferential return to provide a mechanistic explanation for the discovered scaling law and the emerging spatial structure. Our findings corroborate long-standing conjectures in human geography (such as central place theory16 and Weber's theory of emergent optimality10) and allow for predictions of recurrent flows, providing a basis for applications in urban planning, traffic engineering and the mitigation of epidemic diseases.

Swarmalators on a ring with uncorrelated pinning.

We present a case study of swarmalators (mobile oscillators) that move on a 1D ring and are subject to pinning. Previous work considered the special case where the pinning in space and the pinning in the phase dimension were correlated. Here, we study the general case where the space and phase pinning are uncorrelated, both being chosen uniformly at random. This induces several new effects, such as pinned async, mixed states, and a first-order phase transition. These phenomena may be found in real world swarmalators, such as systems of vinegar eels, Janus matchsticks, electrorotated Quincke rollers, or Japanese tree frogs.

Modeling the Interplay Between Seasonal Flu Outcomes and Individual Vaccination Decisions.

Seasonal influenza presents an ongoing challenge to public health. The rapid evolution of the flu virus necessitates annual vaccination campaigns, but the decision to get vaccinated or not in a given year is largely voluntary, at least in the USA, and many people decide against it. In some early attempts to model these yearly flu vaccine decisions, it was often assumed that individuals behave rationally, and do so with perfect information-assumptions that allowed the techniques of classical economics and game theory to be applied. However, these assumptions are not fully supported by the emerging empirical evidence about human decision-making behavior in this context. We develop a simple model of coupled disease spread and vaccination dynamics that instead incorporates experimental observations from social psychology to model annual vaccine decision-making more realistically. We investigate population-level effects of these new decision-making assumptions, with the goal of understanding whether the population can self-organize into a state of herd immunity, and if so, under what conditions. Our model agrees with the established results while also revealing more subtle population-level behavior, including biennial oscillations about the herd immunity threshold.

Quantifying the sensing power of vehicle fleets.

Sensors can measure air quality, traffic congestion, and other aspects of urban environments. The fine-grained diagnostic information they provide could help urban managers to monitor a city's health. Recently, a "drive-by" paradigm has been proposed in which sensors are deployed on third-party vehicles, enabling wide coverage at low cost. Research on drive-by sensing has mostly focused on sensor engineering, but a key question remains unexplored: How many vehicles would be required to adequately scan a city? Here, we address this question by analyzing the sensing power of a taxi fleet. Taxis, being numerous in cities, are natural hosts for the sensors. Using a ball-in-bin model in tandem with a simple model of taxi movements, we analytically determine the fraction of a city's street network sensed by a fleet of taxis during a day. Our results agree with taxi data obtained from nine major cities and reveal that a remarkably small number of taxis can scan a large number of streets. This finding appears to be universal, indicating its applicability to cities beyond those analyzed here. Moreover, because taxis' motion combines randomness and regularity (passengers' destinations being random, but the routes to them being deterministic), the spreading properties of taxi fleets are unusual; in stark contrast to random walks, the stationary densities of our taxi model obey Zipf's law, consistent with empirical taxi data. Our results have direct utility for town councilors, smart-city designers, and other urban decision makers.

Comparison of Strong-field Ionization Models in the Wavelength-scaling of High Harmonic Generation.

We report the use of wavelength-tuneable laser pulses from an optical parametric amplifier to generate high-order harmonics in a range of noble gases. The variation of the harmonic cut-off wavelength and phasematching pressure with gas species and fundamental wavelength were recorded. The experimental results are compared to a phenomenological model of the harmonic generation process, incorporating two separate models of photo-ionization. While the calculated phasematching pressure is generally insensitive to the ionization model, for the harmonic cut-off we obtain superior agreement between experiment and theory when the Yudin-Ivanov (YI) ionization model is used, compared to the commonly utilised Ammosov-Delone-Krainov (ADK) model.

Correlated disorder in the Kuramoto model: Effects on phase coherence, finite-size scaling, and dynamic fluctuations.

We consider a mean-field model of coupled phase oscillators with quenched disorder in the natural frequencies and coupling strengths. A fraction p of oscillators are positively coupled, attracting all others, while the remaining fraction 1-p are negatively coupled, repelling all others. The frequencies and couplings are deterministically chosen in a manner which correlates them, thereby correlating the two types of disorder in the model. We first explore the effect of this correlation on the system's phase coherence. We find that there is a critical width γc in the frequency distribution below which the system spontaneously synchronizes. Moreover, this γc is independent of p. Hence, our model and the traditional Kuramoto model (recovered when p = 1) have the same critical width γc. We next explore the critical behavior of the system by examining the finite-size scaling and the dynamic fluctuation of the traditional order parameter. We find that the model belongs to the same universality class as the Kuramoto model with deterministically (not randomly) chosen natural frequencies for the case of p < 1.

Synchronization as Aggregation: Cluster Kinetics of Pulse-Coupled Oscillators.

We consider models of identical pulse-coupled oscillators with global interactions. Previous work showed that under certain conditions such systems always end up in sync, but did not quantify how small clusters of synchronized oscillators progressively coalesce into larger ones. Using tools from the study of aggregation phenomena, we obtain exact results for the time-dependent distribution of cluster sizes as the system evolves from disorder to synchrony.

Quasi-phase-matched high-order harmonic generation using tunable pulse trains.

A simple technique for generating trains of ultrafast pulses is demonstrated in which the linear separation between pulses can be varied continuously over a wide range. These pulse trains are used to achieve tunable quasi-phase-matching of high harmonic generation over a range of harmonic orders up to the harmonic cut-off, resulting in enhancements of the harmonic intensity in excess of an order of magnitude. The peak enhancement of the harmonics is clearly shown to depend on the separation between pulses, as well as the number of pulses in the train, representing an easily tunable source of quasi-phase-matched high harmonic generation.

General analytic solution for far-field phase and amplitude control, with a phase-only spatial light modulator.

We present an analytical solution for the phase introduced by a phase-only spatial light modulator to generate far-field phase and amplitude distributions within a domain of interest. The solution is demonstrated experimentally and shown to enable excellent control of the far-field amplitude and phase.

Solvable model of driven matter with pinning.

We present a simple model of driven matter in a 1D medium with pinning impurities, applicable to magnetic domains walls, confined colloids, and other systems. We find rich dynamics, including hysteresis, reentrance, quasiperiodicity, and two distinct routes to chaos. In contrast to other minimal models of driven matter, the model is solvable: we derive the full phase diagram for small N, and for large N, we derive expressions for order parameters and several bifurcation curves. The model is also realistic. Its collective states match those seen in the experiments of magnetic domain walls.

Swarmalators with delayed interactions.

We investigate the effects of delayed interactions in a population of "swarmalators," generalizations of phase oscillators that both synchronize in time and swarm through space. We discover two steady collective states: a state in which swarmalators are essentially motionless in a disk arranged in a pseudocrystalline order, and a boiling state in which the swarmalators again form a disk, but now the swarmalators near the boundary perform boiling-like convective motions. These states are reminiscent of the beating clusters seen in photoactivated colloids and the living crystals of starfish embryos.

Order, chaos, and dimensionality transition in a system of swarmalators.

Similarly to sperm, where individuals self-organize in space while also striving for coherence in their tail swinging, several natural and engineered systems exhibit the emergence of swarming and synchronization. The arising and interplay of these phenomena have been captured by collectives of hypothetical particles named swarmalators, each possessing a position and a phase whose dynamics are affected reciprocally and also by the space-phase states of their neighbors. In this work, we introduce a solvable model of swarmalators able to move in two-dimensional spaces. We show that several static and active collective states can emerge and derive necessary conditions for each to show up as the model parameters are varied. These conditions elucidate, in some cases, the displaying of multistability among states. Notably, in the active regime, the system exhibits hyperchaos, maintaining spatial correlation under certain conditions and breaking it under others on what we interpret as a dimensionality transition.

Complete spatial characterization of an optical wavefront using a variable-separation pinhole pair.

We present a technique for measuring the transverse spatial properties of an optical wavefront. Intensity and phase profiles are recovered by analysis of a series of interference patterns produced by the combination of a scanning X-shaped slit and a static horizontal slit; the spatial coherence may be found from the same data. We demonstrate the technique by characterizing high harmonic radiation generated in a gas cell, however the method could be extended to a wide variety of light sources.

Pinning in a system of swarmalators.

We study a population of swarmalators (swarming/mobile oscillators) which run on a ring and are subject to random pinning. The pinning represents the tendency of particles to stick to defects in the underlying medium which competes with the tendency to sync and swarm. The result is rich collective behavior. A highlight is low dimensional chaos which in systems of ordinary, Kuramoto-type oscillators is uncommon. Some of the states (the phase wave and split phase wave) resemble those seen in systems of Janus matchsticks or Japanese tree frogs. The others (such as the sync and unsteady states) may be observable in systems of vinegar eels, electrorotated Quincke rollers, or other swarmalators moving in disordered environments.

Diverse behaviors in non-uniform chiral and non-chiral swarmalators.

We study the emergent behaviors of a population of swarming coupled oscillators, dubbed swarmalators. Previous work considered the simplest, idealized case: identical swarmalators with global coupling. Here we expand this work by adding more realistic features: local coupling, non-identical natural frequencies, and chirality. This more realistic model generates a variety of new behaviors including lattices of vortices, beating clusters, and interacting phase waves. Similar behaviors are found across natural and artificial micro-scale collective systems, including social slime mold, spermatozoa vortex arrays, and Quincke rollers. Our results indicate a wide range of future use cases, both to aid characterization and understanding of natural swarms, and to design complex interactions in collective systems from soft and active matter to micro-robotics.

Attractive and repulsive interactions in the one-dimensional swarmalator model.

We study a population of swarmalators, mobile variants of phase oscillators, which run on a ring and have both attractive and repulsive interactions. This one-dimensional (1D) swarmalator model produces several of collective states: the standard sync and async states as well as a splaylike "polarized" state and several unsteady states such as active bands or swirling. The model's simplicity allows us to describe some of the states analytically. The model can be considered as a toy model for real-world swarmalators such as vinegar eels and sperm which swarm in quasi-1D geometries.

Travel distance, frequency of return, and the spread of disease.

Human mobility is a key driver of infectious disease spread. Recent literature has uncovered a clear pattern underlying the complexity of human mobility in cities: [Formula: see text], the product of distance traveled r and frequency of return f per user to a given location, is invariant across space. This paper asks whether the invariant [Formula: see text] also serves as a driver for epidemic spread, so that the risk associated with human movement can be modeled by a unifying variable [Formula: see text]. We use two large-scale datasets of individual human mobility to show that there is in fact a simple relation between r and f and both speed and spatial dispersion of disease spread. This discovery could assist in modeling spread of disease and inform travel policies in future epidemics-based not only on travel distance r but also on frequency of return f.

Quasi-phase-matching high harmonic generation using trains of pulses produced using an array of birefringent plates.

Quasi-phase-matched high harmonic generation using trains of up to 8 counter-propagating pulses is explored. For trains of up to 4 pulses the measured enhancement of the harmonic signal scales with the number of pulses N as (N + 1)², as expected. However, for trains with N > 4, no further enhancement of the harmonic signal is observed. This effect is ascribed to changes of the coherence length Lc within the generating medium. Techniques for overcoming the variation of Lc are discussed. The pressure dependence of quasi-phase-matching is investigated and the switch from true-phase-matching to quasi-phase-matching is observed.

Optical rotation quasi-phase-matching for circularly polarized high harmonic generation.

A scheme for quasi-phase-matching high harmonic generation of circularly polarized radiation is proposed: optical rotation quasi-phase-matching (ORQPM). In ORQPM, propagation of the driving radiation in a system exhibiting circular birefringence causes its plane of polarization to rotate; by appropriately matching the period of rotation to the coherence length, it is possible to avoid destructive interference of the generated radiation. It is shown that ORQPM is approximately five times more efficient than conventional QPM, and half as efficient as true phase-matching.

Swarmalators on a ring with distributed couplings.

We study a simple model of identical "swarmalators," generalizations of phase oscillators that swarm through space. We confine the movements to a one-dimensional (1D) ring and consider distributed (nonidentical) couplings; the combination of these two effects captures an aspect of the more realistic two-dimensional swarmalator model. We discover several collective states which we describe analytically. These states imitate the behavior of vinegar eels, catalytic microswimmers, and other swarmalators which move on quasi-1D rings.