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.

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The scaling laws of human travel

Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.

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The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

Brownian motors: noisy transport far from equilibrium

We survey the newly developed Hilbert spectral analysis method and its applications to Stokes waves, nonlinear wave evolution processes, the spectral form of the random wave field, and turbulence. Our emphasis is on the inadequacy of presently available methods in nonlinear and nonstationary data analysis. Hilbert spectral analysis is here proposed as an alternative. This new method provides not only a more precise definition of particular events in time-frequency space than wavelet analysis, but also more physically meaningful interpretations of the underlying dynamic processes.

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A new view of nonlinear water waves: the Hilbert spectrum

Chaos, fractional kinetics, and anomalous transport

Lévy Flights and Related Topics in Physics

Part I. General Concepts: Hamiltonian dynamics Stability and chaos Part II. Dymanic Order and Choas: The stochastic layer Stochastic layer - stochastic sea transition The stochastic web Uniform web Part III. Spatial Patterns: Two-dimensional patterns with quasi-symmetry Two-dimensional hydrodynamic patterns with symmetry and quasi-symmetry Chaos and streamlines Part IV. Miscellanea: Patterns in art and nature References Index.

Weak chaos and quasi-regular patterns

Part 1 Particles: the elements of dynamics - phase space, systems with one degree of freedom, an example - the nonlinear pendulum, two more examples of nonlinear oscillations, Poincare's integral invariants, multidimensional integrable systems, mappings, some remarks in conclusion approximate methods - perturbation theory, the averaging method, adiabatic invariants, charged particles in a magnetic field, linear analogues of adiabatic invariance special methods - nonlinear resonance, the Kolmogorov-Arnold-Moser (KAM), structural properties of phase trajectories, simple bifurcations ergodic theory and chaos - ergodicity and mixing, K-systems, examples, recurrences and periodic orbits chaos in detail - a universal mapping for nonlinear oscillations, overlapping of resonances, formation of a stochastic layer, destruction of the integrals of motion, stochastic attractors, examples of stochastic attractors, general notes on the onset of chaos elements of kinetics - the Fokker-Planck-Kolmogorov equation, kinetics in dissipative mappings, stochastic acceleraton and "heating" of particles fractal properties of chaos - fractals, fractals and chaos. Part 2 Waves: nonlinear stationary waves - steepening of waves, stationary waves, examples of stationary waves, collision-free shock waves Hamiltonian description of waves - variational principles, resonance interaction of waves, nonlinear wave resonances, interaction of nonlinear waves chaos in wave fields - weakly nonlinear fields, the fermi-pasta-ulam (FPU) problem, turbulence of a weekly nonlinear field, stochastic instability of a nonlinear wave strong turbulence - Lorenz model, convective cells, features of the onset of turbulence, Langmuir turbulence, soliton turbulence exactly integrable wave equations - integration of the KdV equation, integrable equations. Part 3 Examples: motion of particles in wave fields - regular and stochastic dynamics of particles, motion in a magnetic field and the field of a wave packet, the paradox of the disappearance of Landau damping, stochastic web billiards - mixing billiards, nonlinear-ray dynamics nonlinear optics - nonlinear geometrical optics, nonlinear co-operative phenomena structural properties of one dimensional chains - atom chains, spin chains, excitation in chains of molecules perturbations in Kepler's problem - nonlinear dynamics in a coulomb field, excitation and ionization of a hydrogen atom, diffusion of the eccentricity of orbits in the gravitational field of planets, diffusion of comets from the oort cloud. Part 4 Numerical simulation: nonlinear physics in colour - general notes on the pictures diskettes the ATRS program.

Nonlinear physics : from the pendulum to turbulence and chaos

Using wavelets we re-examine the U.S. stock market price index for any evidence of self-similarity or order that might be revealed at different scales. The wavelet transform localized in time can be used to indicate how the power of the projection of the signal onto the kernel varies with the scale of observation. By comparing how the local power scales vary over time much information about the structure of the data can be obtained. Such evidence is not at all evident from standard analyses of untransformed data, including projections onto a Fourier basis. Wavelets can detect structures in data that are highly localized in time and therefore non-detectable by Fourier transforms. The main conclusion is that while the data are clearly complex, there seems to be some evidence of non-randomness in the data. There is also some limited evidence of quasi-periodicity in the occurrence of large amplitude shocks to the system.

An analysis of u.s. stock price behavior using wavelets

The problem of the foundation of statistical physics emerged with the derivation by Ludwig Boltzmann of a kinetic equation for a gas of molecules that required monotonic growth of entropy. Boltzmann's theory leads to modern thermodynamics, and, for example, to the impossibility of gas spontaneously gathering in one part of a container in the absence of external forces. This result, known as the H‐theorem, met with strong contemporary opposition, especially from mathematician Ernst Zermelo.

George M. Zaslavsky

Papers

Lévy Flights and Related Topics in Physics

Weak chaos and quasi-regular patterns

Nonlinear physics : from the pendulum to turbulence and chaos

An analysis of u.s. stock price behavior using wavelets

Chaotic Dynamics and the Origin of Statistical Laws