The temperature programmed molecular dynamics (TPMD) method is a recent addition to the list of rare-event simulation techniques for materials. Study of thermally-activated events that are rare at molecular dynamics (MD) timescales is possible with TPMD by employing a temperature program that raises the temperature in stages to a point where the transitions happen frequently. Analysis of the observed waiting time distribution yields a wealth of information including kinetic mechanisms in the material, their rate constants and Arrhenius parameters. The first part of this review covers the foundations of the TPMD method. Recent applications of TPMD are discussed to highlight its main advantages. These advantages offer the possibility for rapid construction of kinetic Monte Carlo (KMC) models of a chosen accuracy using TPMD. In this regards, the second part focuses on the latest developments on uncertainty measures for KMC models. The third part focuses on current challenges for the TPMD method and ways of resolving them.

The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary. In this case we even obtain pointwise error estimates for the strain.

The primary objective of this work is to develop coarse-grainingschemes for stochastic many-body microscopic models and quantify theireffectiveness in terms of a priori and a posteriori error analysis. Inthis paper we focus on stochastic lattice systems ofinteracting particles at equilibrium. The proposed algorithms are derived from an initial coarse-grainedapproximation that is directly computable by Monte Carlo simulations, and the corresponding numerical error is calculated using the specific relative entropy between the exact and approximate coarse-grained equilibrium measures. Subsequently we carry out a cluster expansion around this first – and often inadequate – approximation and obtain more accurate coarse-graining schemes.The cluster expansions yield also sharp a posteriori error estimates forthe coarse-grained approximations that can be used for the construction ofadaptive coarse-graining methods.
We present a number of numerical examples that demonstrate that thecoarse-graining schemes developed here allow for accurate predictions of critical behavior and hysteresis in systems with intermediate and long-range interactions. We also present examples where they substantially improvepredictions of earlier coarse-graining schemes for short-range interactions.