Structural Properties of the Disordered Spherical and other Mean Field Spin Models

We extend the approach of Aizenman, Sims and Starr for the SK-type models to their spherical versions. Such an extension has already been performed for diluted spin glasses. The factorization property of the optimal structures found by Guerra for the SK model, which holds for diluted models as well, is verified also in the case of spherical systems, with the due modifications. Hence we show that there are some common structural features in various mean field spin models. These similarities seem to be quite paradigmatic, and we summarize the various techniques typically used to prove the structural analogies and to tackle the computation of the free energy per spin in the thermodynamic limit.Comment: 24 page

The activity of Main Belt comets

Main Belt comets represent a recently discovered class of objects. They are quite intriguing because, while having a Tisserand invariant value higher than 3, are showing cometary activity. We study the activity of the Main Belt comets making the assumption that they are icy-bodies and that the activity has been triggered by an impact. We determine the characteristics of this activity and if the nowadays impact rate in the Main Asteroid Belt is compatible with the hypothesis of an activity triggered by a recent impact. Due to the fact that the Main Belt comets can be considered as a kind of comets, we apply a thermal evolution model developed for icy bodies in order to simulate their activity. We also apply a model to derive the impact rate, with respect to the size of the impactor, in the Main Belt. We demonstrate that a stable activity can result from a recent impact, able to expose ice-rich layers, and that the impact rate in the Main Belt is compatible with this explanation.Comment: 9 pages, 7 figure

On the transition to efficiency in Minority Games

The existence of a phase transition with diverging susceptibility in batch Minority Games (MGs) is the mark of informationally efficient regimes and is linked to the specifics of the agents' learning rules. Here we study how the standard scenario is affected in a mixed population game in which agents with the `optimal' learning rule (i.e. the one leading to efficiency) coexist with ones whose adaptive dynamics is sub-optimal. Our generic finding is that any non-vanishing intensive fraction of optimal agents guarantees the existence of an efficient phase. Specifically, we calculate the dependence of the critical point on the fraction $q$ of `optimal' agents focusing our analysis on three cases: MGs with market impact correction, grand-canonical MGs and MGs with heterogeneous comfort levels.Comment: 12 pages, 3 figures; contribution to the special issue "Viewing the World through Spin Glasses" in honour of David Sherrington on the occasion of his 65th birthda

The contamination of the surface of Vesta by impacts and the delivery of the dark material

The Dawn spacecraft observed the presence of dark material, which in turn proved to be associated with OH and H-rich material, on the surface of Vesta. The source of this dark material has been identified with the low albedo asteroids, but it is still a matter of debate whether the delivery of the dark material is associated with a few large impact events, to micrometeorites or to the continuous, secular flux of impactors on Vesta. The continuous flux scenario predicts that a significant fraction of the exogenous material accreted by Vesta should be due to non-dark impactors likely analogous to ordinary chondrites, which instead represent only a minor contaminant in the HED meteorites. We explored the continuous flux scenario and its implications for the composition of the vestan regolith, taking advantage of the data from the Dawn mission and the HED meteorites. We used our model to show that the stochastic events scenario and the micrometeoritic flux scenario are natural consequences of the continuous flux scenario. We then used the model to estimate the amounts of dark and hydroxylate materials delivered on Vesta since the LHB and we showed how our results match well with the values estimated by the Dawn mission. We used our model to assess the amount of Fe and siderophile elements that the continuous flux of impactors would mix in the vestan regolith: concerning the siderophile elements, we focused our attention on the role of Ni. The results are in agreement with the data available on the Fe and Ni content of the HED meteorites and can be used as a reference frame in future studies of the data from the Dawn mission and of the HED meteorites. Our model cannot yet provide an answer to the fate of the missing non-carbonaceous contaminants, but we discuss possible reasons for this discrepancy.Comment: 31 pages, 7 figures, 4 tables. Accepted for publication on the journal ICARUS, "Dark and Bright Materials on Vesta" special issu

Small crater populations on Vesta

The NASA Dawn mission has extensively examined the surface of asteroid Vesta, the second most massive body in the main belt. The high quality of the gathered data provides us with an unique opportunity to determine the surface and internal properties of one of the most important and intriguing main belt asteroids (MBAs). In this paper, we focus on the size frequency distributions (SFDs) of sub-kilometer impact craters observed at high spatial resolution on several selected young terrains on Vesta. These small crater populations offer an excellent opportunity to determine the nature of their asteroidal precursors (namely MBAs) at sizes that are not directly observable from ground-based telescopes (i.e., below ~100 m diameter). Moreover, unlike many other MBA surfaces observed by spacecraft thus far, the young terrains examined had crater spatial densities that were far from empirical saturation. Overall, we find that the cumulative power-law index (slope) of small crater SFDs on Vesta is fairly consistent with predictions derived from current collisional and dynamical models down to a projectile size of ~10 m diameter (Bottke et al., 2005a,b). The shape of the impactor SFD for small projectile sizes does not appear to have changed over the last several billions of years, and an argument can be made that the absolute number of small MBAs has remained roughly constant (within a factor of 2) over the same time period. The apparent steady state nature of the main belt population potentially provides us with a set of intriguing constraints that can be used to glean insights into the physical evolution of individual MBAs as well as the main belt as an ensemble.Comment: Accepted by PSS, to appear on Vesta cratering special issu

Statistical mechanics and stability of a model eco-system

We study a model ecosystem by means of dynamical techniques from disordered systems theory. The model describes a set of species subject to competitive interactions through a background of resources, which they feed upon. Additionally direct competitive or co-operative interaction between species may occur through a random coupling matrix. We compute the order parameters of the system in a fixed point regime, and identify the onset of instability and compute the phase diagram. We focus on the effects of variability of resources, direct interaction between species, co-operation pressure and dilution on the stability and the diversity of the ecosystem. It is shown that resources can be exploited optimally only in absence of co-operation pressure or direct interaction between species.Comment: 23 pages, 13 figures; text of paper modified, discussion extended, references adde