Ensemble inequivalence in systems with long-range interactions

Ensemble inequivalence has been observed in several systems. In particular it has been recently shown that negative specific heat can arise in the microcanonical ensemble in the thermodynamic limit for systems with long-range interactions. We display a connection between such behaviour and a mean-field like structure of the partition function. Since short-range models cannot display this kind of behaviour, this strongly suggests that such systems are necessarily non-mean field in the sense indicated here. We illustrate our results showing an application to the Blume-Emery-Griffiths model. We further show that a broad class of systems with non-integrable interactions are indeed of mean-field type in the sense specified, so that they are expected to display ensemble inequivalence as well as the peculiar behaviour described above in the microcanonical ensemble.Comment: 12 pages, no figure

The Earliest Phases of Galaxy Evolution

In this paper we study the very early phases of the evolution of our Galaxy by means of a chemical evolution model which reproduces most of the observational constraints in the solar vicinity and in the disk. We have restricted our analysis to the solar neighborhood and present the predicted abundances of several elements (C, N, O, Mg, Si, S, Ca, Fe) over an extended range of metallicities $[Fe/H] = -4.0$ to $[Fe/H] = 0.0$ compared to previous models. We adopted the most recent yield calculations for massive stars taken from different authors (Woosley & Weaver 1995 and Thielemann et al. 1996) and compared the results with a very large sample of data, one of the largest ever used to this purpose. These data have been analysed with a new and powerful statistical method which allows us to quantify the observational spread in measured elemental abundances and obtain a more meaningful comparison with the predictions from our chemical evolution model. Our analysis shows that the ``plateau'' observed for the [$\alpha$/Fe] ratios at low metallicities ($-3.0< [Fe/H] <-1.0$) is not perfectly constant but it shows a slope, especially for oxygen. This slope is very well reproduced by our model with both sets of yields. This is not surprising since realistic chemical evolution models, taking into account in detail stellar lifetimes, never predicted a completely flat plateau. This is due either to the fact that massive stars of different mass produce a slightly different O/Fe ratio or to the often forgotten fact that supernovae of type Ia, originating from white dwarfs, start appearing already at a galactic age of 30 million years and reach their maximum at 1 Gyr.Comment: 32 pages, 9 figures, to be published in Ap

Grid of theoretical NLTE equivalent widths of four Ba II lines and barium abundance in cool stars

We present a grid of computed non-local thermodynamic equilibrium (NLTE) equivalent widths (EW) and NLTE abundance corrections for four Ba II lines: 4554, 5853, 6141, and 6496 A. The grid can be useful in deriving the NLTE barium abundance in stars having parameters in the following ranges: effective temperature from 4000 K to 6500 K, surface gravity log g from 0 to 5, microturbulent velocity 0 km s^-1 to 3 km s^-1, metallicity [Fe/H] from -2 to +0.5, and [Ba/Fe] from -0.4 to +0.6. The NLTE abundance can be either derived by EW interpolation (using the observed Ba II line EW) or by using the NLTE correction applied to a previously determined LTE abundance. Ba II line equivalent widths and the NLTE corrections were calculated using the updated MULTI code and the Ba II atomic model that was previously applied to determine the NLTE barium abundance in different types of stars. The grid is available on-line through the web, and we find that the grid Ba NLTE corrections are almost as accurate as direct NLTE profile fitting (to within 0.05-0.08 dex). For the weakest Ba II line (5853 A) the LTE abundances almost agree with the NLTE abundances, whereas the other three Ba II lines, 4554, 6141, and 6496 A, need NLTE corrections even at the highest metallicities tested here. The 4554 A line is extremely strong and should not be used for abundance analysis above [Fe/H]= -1. Furthermore, we tested the impact of different model atmospheres and spectrum synthesis codes and found average differences of 0.06 dex and 0.09 dex, respectively, for all four lines. At these metallicities we find an average Delta NLTE of +/-0.1 dex for the three useful Ba lines for subsolar cool dwarfs.Comment: 9 pages 8 figures submitted to A&