Local thermodynamics and the generalized Gibbs-Duhem equation in systems with long-range interactions

The local thermodynamics of a system with long-range interactions in d dimensions is studied using the mean-field approximation. Long-range interactions are introduced through pair interaction potentials that decay as a power law in the interparticle distance. We compute the local entropy, Helmholtz free energy, and grand potential per particle in the microcanonical, canonical, and grand canonical ensembles, respectively. From the local entropy per particle we obtain the local equation of state of the system by using the condition of local thermodynamic equilibrium. This local equation of state has the form of the ideal gas equation of state, but with the density depending on the potential characterizing long-range interactions. By volume integration of the relation between the different thermodynamic potentials at the local level, we find the corresponding equation satisfied by the potentials at the global level. It is shown that the potential energy enters as a thermodynamic variable that modifies the global thermodynamic potentials. As a result, we find a generalized Gibbs-Duhem equation that relates the potential energy to the temperature, pressure, and chemical potential. For the marginal case where the power of the decaying interaction potential is equal to the dimension of the space, the usual Gibbs-Duhem equation is recovered. As examples of the application of this equation, we consider spatially uniform interaction potentials and the self-gravitating gas. We also point out a close relationship with the thermodynamics of small systems.

Moths behaving like butterflies. Evolutionary loss of long range attractant pheromones in Castniid Moths: A Paysandisia archon model

Background: In the course of evolution butterflies and moths developed two different reproductive behaviors. Whereas butterflies rely on visual stimuli for mate location, moths use the"female calling plus male seduction" system, in which females release long-range sex pheromones to attract conspecific males. There are few exceptions from this pattern but in all cases known female moths possess sex pheromone glands which apparently have been lost in female butterflies. In the day-flying moth family Castniidae ("butterfly-moths"), which includes some important crop pests, no pheromones have been found so far. Methodology/Principal Findings: Using a multidisciplinary approach we described the steps involved in the courtship of P. archon, showing that visual cues are the only ones used for mate location; showed that the morphology and fine structure of the antennae of this moth are strikingly similar to those of butterflies, with male sensilla apparently not suited to detect female-released long range pheromones; showed that its females lack pheromone-producing glands, and identified three compounds as putative male sex pheromone (MSP) components of P. archon, released from the proximal halves of male forewings and hindwings. Conclusions/Significance: This study provides evidence for the first time in Lepidoptera that females of a moth do not produce any pheromone to attract males, and that mate location is achieved only visually by patrolling males, which may release a pheromone at short distance, putatively a mixture of Z,E-farnesal, E,E-farnesal, and (E,Z)-2,13-octadecadienol. The outlined behavior, long thought to be unique to butterflies, is likely to be widespread in Castniidae implying a novel, unparalleled butterfly-like reproductive behavior in moths. This will also have practical implications in applied entomology since it signifies that the monitoring/control of castniid pests should not be based on the use of female-produced pheromones, as it is usually done in many moths.