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The science of diffusion had its beginnings in the nineteenth century, although the blacksmiths and metal artisans of antiquity already, used diffusion phenomena to make such objects as hardened iron swords and gilded bronze wares. Diffusion as a scientific discipline is based on several cornerstones. The most important ones are: (i) The continuum theory of diffusion originating from the work of the German physiologist Adolf Fick, who was inspired by elegant experiments on diffusion in gases and of salt in water performed by the Scotsman Thomas Graham. (ii) The Brownian motion, observed for the first time by the Scotish botanist Robert Brown, was interpreted decades later by the famous German-Jewish physicist Albert Einstein and almost at the same time by the Polish physicist Marian von Smoluchowski. Their theory related the mean square displacement of atoms to the diffusion coefficient. This provided the statistical cornerstone of diffusion and bridged the gap between mechanics and thermodynamics. The Einstein-Smoluchowski relation was verified in tedious experiments by the French Nobel laureate Jean Baptiste Perrin and his coworkers. (iii) Solid-state diffusion was first studied systematically on the example of gold in lead by the British metallurgist Roberts-Austen in 1896. Using a natural radioisotope of lead the Austro-Hungarian Georg von Hevesy and his coworkers performed for the first time studies of self-diffusion in liquid and solid lead around 1920. (iv) The atomistics of diffusion in materials had to wait for the birthday of solid-state physics, heralded by the experiments of the German Nobel laureate Max von Laue. Equally important was the perception of the Russian and German scientists Jakov Frenkel and Walter Schottky, reinforced by the experiments of the American metallurgist Ernest Kirkendall, that point defects play an important role for the properties of crystalline substances, most notably for those controlling diffusion and the many properties that stem from it. (v) The American physicist and twofold Nobel laureate John Bardeen was the first who pointed out the role of correlation in defect-mediated diffusion in solids, an aspect, which was treated in great detail by the American physicist John Manning. (vi) The first systematic studies of grain-boundary diffusion, a transport phenomenon of fundamental as well as technological importance, were initiated by the American materials scientist David Turnbull.

The understanding of human mobility and the development of qualitative models as well as quantitative theories for it is of key importance to the research of human infectious disease dynamics on large geographical scales. In our globalized world, mobility and traffic have reached a complexity and volume of unprecedented degree. Long range human mobility is now responsible for the rapid geographical spread of emergent infectious diseases. Multiscale human mobility networks exhibit two prominent features: (1) Networks exhibit a strong heterogeneity, the distribution of weights, traffic fluxes and populations sizes of communities range over many orders of magnitude. (2) Although the interaction magnitude in terms of traffic intensities decreases with distance, the observed power-laws indicate that long range interactions play a significant role in spatial disease dynamics. We will review how the topological features of traffic networks can be incorporated in models for disease dynamics and show, that the way topology is translated into dynamics can have a profound impact on the overall disease dynamics. We will also introduce a class of spatially extended models in which the impact and interplay of both spatial heterogeneity as well as long range spatial interactions can be investigated in a systematic fashion. Our analysis of multiscale human mobility networks is based on a proxy network of dispersing US dollar bills, which we incorporated in a model to produce real-time epidemic forecasts that projected the spatial spread of the recent outbreak of Influenza A(H1N1).

In this paper we consider the spread of modern technological innovations. We contrast social learning and threshold heterogeneity models of innovation diffusion, and show how the typical temporal evolution of the distribution of adopters may be consistent with either explanation. Noting the likelihood that each model contains some useful independent explanatory power, we introduce a combined model. We also consider a spatially-structured population in which the spread of an innovation by social influence is modelled as a reaction-diffusion system, and show that the typical spatiotemporal evolution of the distribution is also consistent with a heterogeneity explanation. Additional contextual information is required to estimate the relative importance of social learning and of economic inequalities in observed adoption lags.

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