https://www.tau.ac.il/%7Eklafter1/258.pdf

The random walk's guide to anomalous diffusion: a fractional dynamics approach

The mapping of large-scale human movements is important for urban planning, traffic forecasting and epidemic prevention. Work in animals had suggested that their foraging might be explained in terms of a random walk, a mathematical rendition of a series of random steps, or a Levy flight, a random walk punctuated by occasional larger steps. The role of Levy statistics in animal behaviour is much debated — as explained in an accompanying News Feature — but the idea of extending it to human behaviour was boosted by a report in 2006 of Levy flight-like patterns in human movement tracked via dollar bills. A new human study, based on tracking the trajectory of 100,000 cell-phone users for six months, reveals behaviour close to a Levy pattern, but deviating from it as individual trajectories show a high degree of temporal and spatial regularity: work and other commitments mean we are not as free to roam as a foraging animal. But by correcting the data to accommodate individual variation, simple and predictable patterns in human travel begin to emerge. The cover photo (by Cesar Hidalgo) captures human mobility in New York's Grand Central Station. This study used a sample of 100,000 mobile phone users whose trajectory was tracked for six months to study human mobility patterns. Displacements across all users suggest behaviour close to the Levy-flight-like pattern observed previously based on the motion of marked dollar bills, but with a cutoff in the distribution. The origin of the Levy patterns observed in the aggregate data appears to be population heterogeneity and not Levy patterns at the level of the individual. Despite their importance for urban planning1, traffic forecasting2 and the spread of biological3,4,5 and mobile viruses6, our understanding of the basic laws governing human motion remains limited owing to the lack of tools to monitor the time-resolved location of individuals. Here we study the trajectory of 100,000 anonymized mobile phone users whose position is tracked for a six-month period. We find that, in contrast with the random trajectories predicted by the prevailing Levy flight and random walk models7, human trajectories show a high degree of temporal and spatial regularity, each individual being characterized by a time-independent characteristic travel distance and a significant probability to return to a few highly frequented locations. After correcting for differences in travel distances and the inherent anisotropy of each trajectory, the individual travel patterns collapse into a single spatial probability distribution, indicating that, despite the diversity of their travel history, humans follow simple reproducible patterns. This inherent similarity in travel patterns could impact all phenomena driven by human mobility, from epidemic prevention to emergency response, urban planning and agent-based modelling.

/pdf/understanding-individual-human-mobility-patterns-3k1tyob274.pdf

Understanding individual human mobility patterns

Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

Charge Transport in Disordered Organic Photoconductors a Monte Carlo Simulation Study

The website wheresgeorge.com invites its users to enter the serial numbers of their US dollar bills and track them across America and beyond. Why? “For fun and because it had not been done yet”, they say. But the dataset accumulated since December 1998 has provided the ideal raw material to test the mathematical laws underlying human travel, and that has important implications for the epidemiology of infectious diseases. Analysis of the trajectories of over half a million dollar bills shows that human dispersal is described by a ‘two-parameter continuous-time random walk’ model: our travel habits conform to a type of random proliferation known as ‘superdiffusion’. And with that much established, it should soon be possible to develop a new class of models to account for the spread of human disease. The dynamic spatial redistribution of individuals is a key driving force of various spatiotemporal phenomena on geographical scales. It can synchronize populations of interacting species, stabilize them, and diversify gene pools1,2,3. Human travel, for example, is responsible for the geographical spread of human infectious disease4,5,6,7,8,9. In the light of increasing international trade, intensified human mobility and the imminent threat of an influenza A epidemic10, the knowledge of dynamical and statistical properties of human travel is of fundamental importance. Despite its crucial role, a quantitative assessment of these properties on geographical scales remains elusive, and the assumption that humans disperse diffusively still prevails in models. Here we report on a solid and quantitative assessment of human travelling statistics by analysing the circulation of bank notes in the United States. Using a comprehensive data set of over a million individual displacements, we find that dispersal is anomalous in two ways. First, the distribution of travelling distances decays as a power law, indicating that trajectories of bank notes are reminiscent of scale-free random walks known as Levy flights. Second, the probability of remaining in a small, spatially confined region for a time T is dominated by algebraically long tails that attenuate the superdiffusive spread. We show that human travelling behaviour can be described mathematically on many spatiotemporal scales by a two-parameter continuous-time random walk model to a surprising accuracy, and conclude that human travel on geographical scales is an ambivalent and effectively superdiffusive process.

/pdf/the-scaling-laws-of-human-travel-49922jssif.pdf

The scaling laws of human travel

Newtonian physics began with an attempt to make precise predictions about natural phenomena, predictions that could be accurately checked by observation and experiment. The goal was to understand nature as a deterministic, “clockwork” universe. The application of probability distributions to physics developed much more slowly. Early uses of probability arguments focused on distributions with well‐defined means and variances. The prime example was the Gaussian law of errors, in which the mean traditionally represented the most probable value from a series of repeated measurements of a fixed quantity, and the variance was related to the uncertainty of those measurements.

Beyond Brownian motion

Quantum information processing systems and related technologies are likely to involve switching and amplification functions in ultrasmall objects such as nanotubes. In today's electronic devices the transistor performs these functions. A 'quantum age' equivalent of the conventional transistor would, ideally, use photons rather than electrons as information carriers because of their speed and robustness against decoherence. But robustness also stops them being easily controlled. Now a team from optETH and ETH in Zurich demonstrates the realization of a single-molecule optical transistor. In it, a single dye molecule coherently attenuates or amplifies a tightly focused laser beam, depending on the power of a second 'gating' beam. The transistor is the most fundamental building block in present-day technologies. For the purpose of quantum information processing schemes and for the development of a 'quantum computer', photons are attractive information carriers because of their speed and robustness against decoherence. However, their robustness also prevents them from being easily controlled; despite this, experiments now show the realization of a quantum optical transistor. The transistor is one of the most influential inventions of modern times and is ubiquitous in present-day technologies. In the continuing development of increasingly powerful computers as well as alternative technologies based on the prospects of quantum information processing, switching and amplification functionalities are being sought in ultrasmall objects, such as nanotubes, molecules or atoms1,2,3,4,5,6,7,8,9. Among the possible choices of signal carriers, photons are particularly attractive because of their robustness against decoherence, but their control at the nanometre scale poses a significant challenge as conventional nonlinear materials become ineffective. To remedy this shortcoming, resonances in optical emitters can be exploited, and atomic ensembles have been successfully used to mediate weak light beams7. However, single-emitter manipulation of photonic signals has remained elusive and has only been studied in high-finesse microcavities10,11,12,13 or waveguides8,14. Here we demonstrate that a single dye molecule can operate as an optical transistor and coherently attenuate or amplify a tightly focused laser beam, depending on the power of a second ‘gating’ beam that controls the degree of population inversion. Such a quantum optical transistor has also the potential for manipulating non-classical light fields down to the single-photon level. We discuss some of the hurdles along the road towards practical implementations, and their possible solutions.

A single-molecule optical transistor

Single dye molecules at cryogenic temperatures exhibit many spectroscopic phenomena known from the study of free atoms and are thus promising candidates for experiments in fundamental quantum optics. However, the existing techniques for their detection have either sacrificed information on the coherence of the excited state or have been inefficient. Here, we show that these problems can be addressed by focusing the excitation light near to the extinction cross-section of a molecule. Our detection scheme enables us to explore resonance fluorescence over nine orders of magnitude of excitation intensity and to separate its coherent and incoherent parts. In the strong excitation regime, we demonstrate the first direct observation of the Mollow fluorescence triplet from a single solid-state emitter. Under weak excitation, we report the detection of a single molecule with an incident power as faint as 600 aW, paving the way for studying nonlinear effects with only a few photons.

Efficient coupling of photons to a single molecule and the observation of its resonance fluorescence

We present experiments where a single subwavelength scatterer is used to examine and control the backscattering induced coupling between counterpropagating high-Q modes of a microsphere resonator. Our measurements reveal the standing wave character of the resulting symmetric and antisymmetric eigenmodes, their unbalanced intensity distributions, and the coherent nature of their coupling. We discuss our findings and the underlying classical physics in the framework common to quantum optics and provide a particularly intuitive explanation of the central processes.

Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light.

We investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous-time random-walk (CTRW) approach. Two different CTRW models are considered: (i) Particles jump between sites (turning points) or (ii) particles move at a constant velocity between sites and choose a new direction at random. For both models we study the mean-squared displacement 〈${\mathit{r}}^{2}$(t)〉 and the propagator P(r,t), the probability to be at location r at time t having started at the origin at t=0. Iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the CTRW framework and scaling arguments. For the case of dispersive motion we discuss the problem of the stationary state and point out its relevance.

Gert Zumofen

Papers

Beyond Brownian motion

A single-molecule optical transistor

Efficient coupling of photons to a single molecule and the observation of its resonance fluorescence

Controlled coupling of counterpropagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light.

Scale-invariant motion in intermittent chaotic systems