We present an exhaustive study of the rank-distribution of city-population and population-dynamics of the 50 Spanish provinces (more than 8000 municipalities) in a time-window of 15 years (1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010). We exhibit compelling evidence regarding how well the MaxEnt principle describes the equilibrium distributions. We show that the microscopic dynamics that governs population growth is the deciding factor that originates the observed macroscopic distributions. The connection between microscopic dynamics and macroscopic distributions is unravelled via MaxEnt.

Understanding demographic and migrational patterns constitutes a great challenge. Millions of individual decisions, motivated by economic, political, demographic, rational and/or emotional reasons underlie the high complexity of demographic dynamics. Significant advances in quantitatively understanding such complexity have been registered in recent years, as those involving the growth of cities but many fundamental issues still defy comprehension. We present here compelling empirical evidence of a high level of regularity regarding time and spatial correlations in urban sprawl, unravelling patterns about the inertia in the growth of cities and their interaction with each other. By using one of the world's most exhaustive extant demographic data basisthat of the Spanish Government's Institute INE, with records covering 111 years and (in 2011) 45 million people, distributed among more than 8000 population nuclei-we show that the inertia of city growth has a characteristic time of 15 years, and its interaction with the growth of other cities has a characteristic distance of 80 km. Distance is shown to be the main factor that entangles two cities (60% of total correlations). The power of our current social theories is thereby enhanced.

A quantitative understanding of cities' demographic dynamics is becoming a potentially useful tool for planning sustainable growth. The concomitant theory should reveal details of the cities' past and also of its interaction with nearby urban conglomerates for providing a reasonably complete picture. Using the exhaustive database of the Census Bureau in a time window of 170 years, we exhibit here empirical evidence for time and space correlations in the demographic dynamics of US counties, with a characteristic memory time of 25 years and typical distances of interaction of 200 km. These correlations are much larger than those observed in a European country (Spain), indicating more coherent evolution in US cities. We also measure the resilience of US cities to historical events, finding a demographical post-traumatic amnesia after wars (such as the American Civil War) or economic crisis (such as the 1929 Stock Market Crash).

The distribution of firms' growth and firms' sizes is a topic under intense scrutiny. In this paper, we show that a thermodynamic model based on the maximum entropy principle, with dynamical prior information, can be constructed that adequately describes the dynamics and distribution of firms' growth. Our theoretical framework is tested against a comprehensive database of Spanish firms, which covers, to a very large extent, Spain's economic activity, with a total of 1 155 142 firms evolving along a full decade. We show that the empirical exponent of Pareto's law, a rule often observed in the rank distribution of large-size firms, is explained by the capacity of economic system for creating/destroying firms, and that can be used to measure the health of a capitalist-based economy. Indeed, our model predicts that when the exponent is larger than 1, creation of firms is favoured; when it is smaller than 1, destruction of firms is favoured instead; and when it equals 1 (matching Zipf's law), the system is in a full macroeconomic equilibrium, entailing ‘free’ creation and/or destruction of firms. For medium and smaller firm sizes, the dynamical regime changes, the whole distribution can no longer be fitted to a single simple analytical form and numerical prediction is required. Our model constitutes the basis for a full predictive framework regarding the economic evolution of an ensemble of firms. Such a structure can be potentially used to develop simulations and test hypothetical scenarios, such as economic crisis or the response to specific policy measures.

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