The role of data in model building and prediction: a survey through examples

The goal of Science is to understand phenomena and systems in order to predict their development and gain control over them. In the scientific process of knowledge elaboration, a crucial role is played by models which, in the language of quantitative sciences, mean abstract mathematical or algorithmical representations. This short review discusses a few key examples from Physics, taken from dynamical systems theory, biophysics, and statistical mechanics, representing three paradigmatic procedures to build models and predictions from available data. In the case of dynamical systems we show how predictions can be obtained in a virtually model-free framework using the methods of analogues, and we briefly discuss other approaches based on machine learning methods. In cases where the complexity of systems is challenging, like in biophysics, we stress the necessity to include part of the empirical knowledge in the models to gain the minimal amount of realism. Finally, we consider many body systems where many (temporal or spatial) scales are at play-and show how to derive from data a dimensional reduction in terms of a Langevin dynamics for their slow components

Thermodynamic assessment of probability distribution divergencies and Bayesian model comparison

Within path sampling framework, we show that probability distribution divergences, such as the Chernoff information, can be estimated via thermodynamic integration. The Boltzmann-Gibbs distribution pertaining to different Hamiltonians is implemented to derive tempered transitions along the path, linking the distributions of interest at the endpoints. Under this perspective, a geometric approach is feasible, which prompts intuition and facilitates tuning the error sources. Additionally, there are direct applications in Bayesian model evaluation. Existing marginal likelihood and Bayes factor estimators are reviewed here along with their stepping-stone sampling analogues. New estimators are presented and the use of compound paths is introduced

Entropy estimates for a class of schemes for the euler equations

In this paper, we derive entropy estimates for a class of schemes for the Euler equations which present the following features: they are based on the internal energy equation (eventually with a positive corrective term at the righ-hand-side so as to ensure consistency) and the possible upwinding is performed with respect to the material velocity only. The implicit-in-time first-order upwind scheme satisfies a local entropy inequality. A generalization of the convection term is then introduced, which allows to limit the scheme diffusion while ensuring a weaker property: the entropy inequality is satisfied up to a remainder term which is shown to tend to zero with the space and time steps, if the discrete solution is controlled in L $\infty$ and BV norms. The explicit upwind variant also satisfies such a weaker property, at the price of an estimate for the velocity which could be derived from the introduction of a new stabilization term in the momentum balance. Still for the explicit scheme, with the above-mentioned generalization of the convection operator, the same result only holds if the ratio of the time to the space step tends to zero

First Principles Study of the Ignition Mechanism for Hypergolic Bipropellants: N,N,N′,N′-Tetramethylethylenediamine (TMEDA) and N,N,N′,N′-Tetramethylmethylenediamine (TMMDA) with Nitric Acid

We report quantum mechanics calculations (B3LYP flavor of density functional theory) to determine the chemical reaction mechanism underlying the hypergolic reaction of pure HNO_3 with N,N,N′,N′-tetramethylethylenediamine (TMEDA) and N,N,N′,N′-tetramethylmethylenediamine (TMMDA). TMEDA and TMMDA are dimethyl amines linked by two CH_2 groups or one CH_2 group, respectively, but ignite very differently with HNO_3. We explain this dramatic difference in terms of the role that N lone-pair electrons play in activating adjacent chemical bonds. We identify two key atomistic level factors that affect the ignition delay: (1) The exothermicity for formation of the dinitrate salt from TMEDA or TMMDA. With only a single CH_2 group between basic amines, the diprotonation of TMMDA results in much stronger electrostatic repulsion, reducing the heat of dinitrate salt formation by 6.3 kcal/mol. (2) The reaction of NO_2 with TMEDA or TMMDA, which is the step that releases the heat and reactive species required to propagate the reaction. Two factors of TMEDA promote the kinetics by providing routes with low barriers to oxidize the C: (a) formation of a stable intermediate with a C–C double bond and (b) the lower bond energy for breaking the C–C single bond (by 18 kcal/mol comparing to alkane) between two amines. Both factors would decrease the ignition delay for TMEDA versus TMMDA. The same factors also explain the shorter ignition delay of 1,4-dimethylpiperazine (DMPipZ) versus 1,3,5-trimethylhexahydro-1,3,5-triazine (TMTZ). These results indicate that TMEDA and DMPipZ are excellent green replacements for hydrazines as the fuel in bipropellants