The application of Statistical Physics to social systems is mainly related to the search for macroscopic laws, that can be derived from experimental data averaged in time or space,assuming the system in a steady state. One of the major goals would be to find a connection between the statistical laws to the microscopic properties: for example to understand the nature of the microscopic interactions or to point out the existence of interaction networks. The probability theory suggests the existence of few classes of stationary distributions in the thermodynamics limit, so that the question is if a statistical physics approach could be able to enroll the complex nature of the social systems. We have analyzed a large GPS data base for single vehicle mobility in the Florence urban area, obtaining statistical laws for path lengths, for activity downtimes and for activity degrees. We show also that simple generic assumptions on the microscopic behavior could explain the existence of stationary macroscopic laws, with an universal function describing the distribution. Our conclusion is that understanding the system complexity requires dynamical data-base for the microscopic evolution, that allow to solve both small space and time scales in order to study the transients.

Recent studies of human mobility largely focus on displacements patterns and power law fits of empirical long-tailed distributions of distances are usually associated to scale-free superdiffusive random walks called Lévy flights. However, drawing conclusions about a complex system from a fit, without any further knowledge of the underlying dynamics, might lead to erroneous interpretations. Here we show, on the basis of a data set describing the trajectories of 780,000 private vehicles in Italy, that the Lévy flight model cannot explain the behaviour of travel times and speeds. We therefore introduce a class of accelerated random walks, validated by empirical observations, where the velocity changes due to acceleration kicks at random times. Combining this mechanism with an exponentially decaying distribution of travel times leads to a short-tailed distribution of distances which could indeed be mistaken with a truncated power law. These results illustrate the limits of purely descriptive models and provide a mechanistic view of mobility.

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