Data-driven molecular modeling with the generalized Langevin equation.

The complexity of molecular dynamics simulations necessitates dimension reduction and coarse-graining techniques to enable tractable computation. The generalized Langevin equation (GLE) describes coarse-grained dynamics in reduced dimensions. In spite of playing a crucial role in non-equilibrium dynamics, the memory kernel of the GLE is often ignored because it is difficult to characterize and expensive to solve. To address these issues, we construct a data-driven rational approximation to the GLE. Building upon previous work leveraging the GLE to simulate simple systems, we extend these results to more complex molecules, whose many degrees of freedom and complicated dynamics require approximation methods. We demonstrate the effectiveness of our approximation by testing it against exact methods and comparing observables such as autocorrelation and transition rates.

Reliability assessment for large-scale molecular dynamics approximations.

Molecular dynamics (MD) simulations are used in biochemistry, physics, and other fields to study the motions, thermodynamic properties, and the interactions between molecules. Computational limitations and the complexity of these problems, however, create the need for approximations to the standard MD methods and for uncertainty quantification and reliability assessment of those approximations. In this paper, we exploit the intrinsic two-scale nature of MD to construct a class of large-scale dynamics approximations. The reliability of these methods is evaluated here by measuring the differences between full, classical MD simulations and those based on these large-scale approximations. Molecular dynamics evolutions are non-linear and chaotic, so the complete details of molecular evolutions cannot be accurately predicted even using full, classical MD simulations. This paper provides numerical results that demonstrate the existence of computationally efficient large-scale MD approximations which accurately model certain large-scale properties of the molecules: the energy, the linear and angular momenta, and other macroscopic features of molecular motions.

Difference equations with the Allee effect and the periodic Sigmoid Beverton-Holt equation revisited.

In this paper, we investigate the long-term behaviour of solutions of the periodic Sigmoid Beverton-Holt equation [Formula: see text] where the a ( n ) and δ( n ) are p-periodic positive sequences. Under certain conditions, there are shown to exist an asymptotically stable p-periodic state and a p-periodic Allee state with the property that populations smaller than the Allee state are driven to extinction while populations greater than the Allee state approach the stable state, thus accounting for the long-term behaviour of all initial states. This appears to be the first study of the equation with variable δ. The results are discussed with possible interpretations in Population Dynamics with emphasis on fish populations and smooth cordgrass.