/ #brownian motion #diffusion

# Humans are superdiffusive

When tea is poured in a cup of hot water, we observe a phenomenon called diffusion: in the end particles of tea spread evenly throughout the mass of water and we enjoy our cup of tea. Diffusion occurs as a result of the second law of thermodynamics (increase of entropy) and can be modeled quantitatively using the diffusion equation (or heat equation). This is a funny equation, since it establishes that the velocity of spreading is infinite while the mean root square fluctuations of the position of the particles grows in time as

$$\langle x^2 \rangle = 2 D t$$

Specifically, this means that the typical volume covered by the particles of tea in the cup grows like the square root of time, while there is always a chance to find a particle anywhere in the cup. The coefficient D is called the diffusion constant and depends on thermodynamical properties of the liquid. It was Einstein in his miraculous year (1905) who found the relationship with temperature and mobility of particles in the liquid which is named after him. There are numerous examples of diffusion processes (also known as Brownian motions) from the erratic motion of particles in water found by the botanist Brown in 1827 to the description of price fluctuations in stock markets made by Bachelier in 1900.

Of course, not everything in nature is diffusive. Actually, diffusive behavior is related to the Central Limit Theory and the fact that the mean root square fluctuations is growing linearly in time is telling us that the sum of the kicks than a particle of coffee suffers in the cup add like random numbers to get the CLT result: the variance grows linearly in time. The ubiquity of diffusive behavior is related to the fact that convergence in CLT does not depend on the microscopic details of the random numbers that are summed.

In a recent study published in Nature and made by D. Brockmann, L. Hufnagel and T. Geisel, these researchers have obtained quantitative assessment of the displacements of humans by analyzing the circulation of bank notes in the United States obtained in the Where is George? website. They observe that human travel is somehow a random process governed by super-diffusive jumps to get

$$\langle x^2 \rangle = 2 D_1 t$$

which says that the typical area covered by humans when traveling is linear in time. This behavior is nothing unexpected, since humans tend to mix short travels around their neighborhood with business or holiday travels. The authors study several statistical properties of the travel of humans to find that it can be described by a Continuous Time Random Walk (CTRW). Their study can be relevant to any other thing carried by humans, like viruses or diseases and thus it pertains to epidemiology.