A Hamiltonian mean field model, where the potential is inspired by dipole-dipole interactions, is proposed to characterize the behavior of systems with long-range interactions. The dynamics of the system remains in quasistationary states before arriving at equilibrium. The equilibrium is analytically derived from the canonical ensemble and coincides with that obtained from molecular dynamics simulations (microcanonical ensemble) at only long time scales. The dynamics of the system is characterized by the behavior of the mean value of the kinetic energy. The significance of the results, compared to others in the recent literature, is that two plateaus sequentially emerge in the evolution of the model under the special considerations of the initial conditions and systems of finite size. The first plateau decays to a different second one before the system reaches equilibrium, but the dynamics of the system is expected to have only one plateau when the thermodynamics limit is reached because the difference between them tends to disappear as N tends to infinity. Hence, the first plateau is a type of quasistationary state the lifetime of which depends on a power law of N and the second seems to be a true quasistationary state as reported in the literature. We characterize the general behavior of the model according to its dynamics and thermodynamics.
The present study regards the zeroth order mean field approximation of a dipole-type interaction model, which is analytically solved in the canonical and microcanonical ensembles. After writing the canonical partition function, the free and internal energies, magnetization and the specific heat are derived and graphically represented. A crucial derivation is the calculation of the free energy, which is variationally evaluated, and it is shown that the exact result coincides with the approximate trend when N tends to infinity. In the microcanonical ensemble, the entropy as other thermodynamic properties are calculated. We notice that both schemes coincide in equilibrium.
Previously, we observed that the student workload follows an inverse relation with the learning rate (an application of the kinematic notion of speed contextualized to the learning process). Motivated by this finding, we propose a quantitative estimation of the learning rate using a different source of information: the historical records of final grades of a given course. According to empirical data analyzed in other similar studies, the distribution functions of final grades exhibit a regular pattern: a Gaussian behavior for the approval region and a homogeneous distribution for the failed one. This fact is combined with the incidence of student elimination–desertion rules for introducing two simple agent-based models. Our analysis is complemented by revisiting the performance indicators typically employed to characterize the student promotion and progression. We discuss some other performance indicators to characterize the learning advancement of students: the group learning rate and the learning curve. We compare the results of Monte Carlo simulations with empirical data, observing a good agreement in the behavior of performance indicators derived from these sources. This analysis suggests an adaptive method for the readjustment of the student workload (the number of academic credits) considering the group learning rates during a follow-up period, which resembles the readjustment of prices of goods (and services) in correspondence with the evolution of supply and demand.
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