### Thermodynamics of firms’ growth

E. Zambrano, A. Hernando, A.F. Bariviera, R. Hernando, A. Plastino, J. R. Soc. Interface **12**, 20150789 (2015)

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 econ- omic 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 econ- omic 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.

### Memory-endowed US cities and their demographic interactions

A. Hernando, R. Hernando, A. Plastino, E. Zambrano, J. R. Soc. Interface **12**, 20141185 (2015)

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).

### Space–time correlations in urban sprawl

A. Hernando, R. Hernando, A. Plastino, J. R. Soc. Interface **11**, 20130930 (2014)

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 basis—that 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.

### The workings of the maximum entropy principle in collective human behaviour

A. Hernando, R. Hernando, A. Plastino, A. R. Plastino, J. R. Soc. Interface **10**, 20120758 (2013)

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 (1996–2010). 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.

### Scale-invariance underlying the logistic equation and its social applications

A. Hernando and A. Plastino, Phys. Lett. A **377**, 176 (2013)

On the basis of dynamical principles we i) advance a derivation of the Logistic Equation (LE), widely employed (among multiple applications) in the simulation of population growth, and ii) demonstrate that scale-invariance and a mean-value constraint are sufficient and necessary conditions for obtaining it. We also generalize the LE to multi-component systems and show that the above dynamical mechanisms underlie a large number of scale-free processes. Examples are presented regarding city-populations, diffusion in complex networks, and popularity of technological products, all of them obeying the multi-component logistic equation in an either stochastic or deterministic way.

### Thermodynamics of urban population flows

A. Hernando and A. Plastino, Phys. Rev. E **86**, 066105 (2012)

Orderliness, reflected via mathematical laws, is encountered in different frameworks involving social groups. Here we show that a thermodynamics can be constructed that macroscopically describes urban population flows. Microscopic dynamic equations and simulations with random walkers underlie the macroscopic approach. Our results might be regarded, via suitable analogies, as a step towards building an explicit social thermodynamics.

### Variational principle underlying scale invariant social systems

A. Hernando and A. Plastino, Eur. Phys. J. B **85**, 293 (2012)

MaxEnt’s variational principle, in conjunction with Shannon’s logarithmic information measure, yields only exponential functional forms in straightforward fashion. In this communication we show how to overcome this limitation via the incorporation, into the variational process, of suitable dynamical information. As a consequence, we are able to formulate a somewhat generalized Shannonian maximum entropy approach which provides a unifying “thermodynamic-like” explanation for the scale-invariant phenomena observed in social contexts, as city-population distributions. We confirm the MaxEnt predictions by means of numerical experiments with random walkers, and compare them with some empirical data.

### MaxEnt and dynamical information

A. Hernando, A. Plastino, A. R. Plastino, Eur. Phys. J. B **85**, 147 (2012)

The MaxEnt solutions are shown to display a variety of behaviors (beyond the traditional and customary exponential one) if adequate dynamical information is inserted into the concomitant entropic-variational principle. In particular, we show both theoretically and numerically that power laws and power laws with exponential cut-offs emerge as equilibrium densities in proportional and other dynamics.

### Unravelling the size distribution of social groups with information theory in complex networks

A. Hernando, D. Villuendas, C. Vesperinas, M. Abad and A. Plastino, Eur. Phys. J. B **76**, 87 (2010)

The minimization of Fisher’s information (MFI) approach of Frieden et al. Phys. Rev. E **60**, 48 (1999) is applied to the study of size distributions in social groups on the basis of a recently established analogy between scale invariant systems and classical gases Phys. A **389**, 490 (2010). Going beyond the ideal gas scenario is seen to be tantamount to simulating the interactions taking place, for a competitive cluster growth process, in a *scale-free ideal network* – a non-correlated network with a connection-degree’s distribution that mimics the scale-free ideal gas density distribution. We use a scaling rule that allows one to classify the final cluster-size distributions using only one parameter that we call the *competitiveness*, which can be seen as a measure of the strength of the interactions. We find that both empirical city-size distributions and electoral results can be thus reproduced and classified according to this competitiveness-parameter, that also allow us to infer the maximum number of stable social relationships that one person can maintain, known as the Dunbar number, together with its standard deviation. We discuss the importance of this number in connection with the empirical phenomenon known as “six-degrees of separation”. Finally, we show that scaled city-size distributions of large countries follow, in general, the same universal distribution.

### Fisher information and the thermodynamics of scale-invariant systems

A. Hernando, C. Vesperinas and A. Plastino, Physica A **389**, 490 (2010)

We present a thermodynamic formulation for scale-invariant systems based on the minimization with constraints of the Fisher information measure. In such a way a clear analogy between these systems’ thermal properties and those of gases and fluids is seen to emerge in a natural fashion. We focus our attention on the non-interacting scenario, speaking thus of *scale-free ideal gases* (SFIGs) and present some empirical evidences regarding such disparate systems as electoral results, city populations and total citations in Physics journals, that seem to indicate that SFIGs do exist. We also illustrate the way in which Zipf’s law can be understood in a thermodynamical context as the surface of a finite system. Finally, we derive an equivalent microscopic description of our systems which totally agrees with previous numerical simulations found in the literature.

### Zipf’s law from a Fisher variational-principle

A. Hernando, D. Puigdoménech, D. Villuendas, C. Vesperinas and A. Plastino, Phys. Lett. A **374**, 18 (2009)

Zipf’s law is shown to arise as the variational solution of a problem formulated in Fisher’s terms. An appropriate minimization process involving Fisher information and scale-invariance yields this universal rank distribution. As an example we show that the number of citations found in the most referenced physics journals follows this law.