The approach to equilibrium in N-body gravitational systems

The evolution of closed gravitational systems is studied by means of $N$-body simulations. This, as well as being interesting in its own right, provides insight into the dynamical and statistical mechanical properties of gravitational systems: the possibility of the existence of stable equilibrium states and the associated relaxation time would provide an ideal situation where relaxation theory can be tested. Indeed, these states are found to exist for single mass $N$-body systems, and the condition condition for this is simply that obtained from elementary thermodynamical considerations applied to self-gravitating ideal gas spheres. However, even when this condition is satisfied, some initial states may not end as isothermal spheres. It is therefore only a necessary condition. Simple considerations also predict that, for fixed total mass, energy and radius, stable isothermal spheres are unique. Therefore, statistically irreversible perturbations to the density profile caused by the accumulation of massive particles near the centre of multimass systems, destroy these equilibria if the aforementioned quantities are kept fixed. The time-scale for this to happen was found to be remarkably short (a few dynamical times when $N= 2500$) in systems undergoing violent relaxation. The time taken to achieve thermal equilibrium depended on the initial conditions and could be comparable to a dynamical time (even when the conditions for violent relaxation were not satisfied) or the two body relaxation time. The relaxation time for velocity anisotropies was intermediate between these two time-scales, being long compared to the dynamical time but much (about four times) shorter than the time-scale of energy relaxation.Comment: To appear in Physical Review

The role of intelligence and feedback in children's strategy competence

A test-intervention-test study was conducted investigating the role of intelligence on four parameters of strategy competence in the context of a numerosity judgment task. Moreover, the effectiveness of two feedback types on these four parameters was tested. In the two test sessions, the choice/no-choice method was used to assess the strategy repertoire, frequency, efficiency, and adaptivity of a group of low-, average-, and high-intelligence children. During the intervention, half of the participants from each intelligence group were given outcome feedback (OFB), whereas the other half received strategy feedback (SFB). The pretest data showed large differences among the three intelligence groups on all four strategy parameters. These differences had disappeared at the posttest due to a particularly strong improvement on all strategy parameters in the low-intelligence group. Furthermore, it was found that SFB was more beneficial than OFB for all parameters involving strategy selection. These results indicate that intelligence plays an important role in children's strategy use and suggest that strategy feedback can be a powerful instructional tool, especially for low-intelligence children. © 2010 Elsevier Inc

Using Segmented Linear Regression Models with Unknown Change Points to Analyze Strategy Shifts in Cognitive Tasks

Some years ago, Beem (1993, 1995) described a program for fitting two regression lines with an unknown change point (Segcurve). He suggested that such models are useful for the analysis of a variety of phenomena and gave an example of an application to the study of strategy shifts in a mental rotation task. This technique has also proven to be very fruitful for investigating strategy use and strategy shifts in other cognitive tasks. Recently, Beem (1999) developed SegcurvN, which fits n regression lines with (n - 1) unknown change points. In the present article we present this new technique and demonstrate the usefulness of a three-phase segmented linear regression model for the identification of strategies and strategy shifts in cognitive tasks by applying it to data from a numerosity judgment experiment. The advantages and shortcomings of this technique are evaluate

The role of verbal and performance intelligence in children's strategy selection and execution

The present study investigated the extent to which verbal intelligence (VIQ) and performance intelligence (PIQ) contribute to strategy selection and execution in the context of a numerosity judgement task. The choice/no-choice method was used to appropriately assess strategy selection (in terms of strategy repertoire, frequency and adaptivity) and strategy execution (in terms of strategy speed and accuracy) in a group of 120 12-year-old children. For each parameter, a regression analysis was carried out with VIQ and PIQ, as independent variables, and with arithmetic proficiency as a control variable. Results showed that VIQ was a significant predictor for all parameters of strategic competence, whereas PIQ only explained a unique portion of the variance above and beyond VIQ in the speed of strategy execution. © 2013 Elsevier Inc