Generalised-Lorentzian Thermodynamics

We extend the recently developed non-gaussian thermodynamic formalism \cite{tre98} of a (presumably strongly turbulent) non-Markovian medium to its most general form that allows for the formulation of a consistent thermodynamic theory. All thermodynamic functions, including the definition of the temperature, are shown to be meaningful. The thermodynamic potential from which all relevant physical information in equilibrium can be extracted, is defined consistently. The most important findings are the following two: (1) The temperature is defined exactly in the same way as in classical statistical mechanics as the derivative of the energy with respect to the entropy at constant volume. (2) Observables are defined in the same way as in Boltzmannian statistics as the linear averages of the new equilibrium distribution function. This lets us conclude that the new state is a real thermodynamic equilibrium in systems capable of strong turbulence with the new distribution function replacing the Boltzmann distribution in such systems. We discuss the ideal gas, find the equation of state, and derive the specific heat and adiabatic exponent for such a gas. We also derive the new Gibbsian distribution of states. Finally we discuss the physical reasons for the development of such states and the observable properties of the new distribution function.Comment: 13 pages, 1 figur

Collective behaviour without collective order in wild swarms of midges

Collective behaviour is a widespread phenomenon in biology, cutting through a huge span of scales, from cell colonies up to bird flocks and fish schools. The most prominent trait of collective behaviour is the emergence of global order: individuals synchronize their states, giving the stunning impression that the group behaves as one. In many biological systems, though, it is unclear whether global order is present. A paradigmatic case is that of insect swarms, whose erratic movements seem to suggest that group formation is a mere epiphenomenon of the independent interaction of each individual with an external landmark. In these cases, whether or not the group behaves truly collectively is debated. Here, we experimentally study swarms of midges in the field and measure how much the change of direction of one midge affects that of other individuals. We discover that, despite the lack of collective order, swarms display very strong correlations, totally incompatible with models of noninteracting particles. We find that correlation increases sharply with the swarm's density, indicating that the interaction between midges is based on a metric perception mechanism. By means of numerical simulations we demonstrate that such growing correlation is typical of a system close to an ordering transition. Our findings suggest that correlation, rather than order, is the true hallmark of collective behaviour in biological systems.Comment: The original version has been split into two parts. This first part focuses on order vs. correlation. The second part, about finite-size scaling, will be included in a separate paper. 15 pages, 6 figures, 1 table, 5 video