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Diffusion maps approximate the generator of Langevin dynamics from simulation data. They afford a means of identifying the slowly-evolving principal modes of high-dimensional molecular systems. When combined with a biasing mechanism, diffusion maps can accelerate the sampling of the stationary Boltzmann-Gibbs distribution. In this work, we contrast the local and global perspectives on diffusion maps, based on whether or not the data distribution has been fully explored. In the global setting, we use diffusion maps to identify metastable sets and to approximate the corresponding committor functions of transitions between them. We also discuss the use of diffusion maps within the metastable sets, formalising the locality via the concept of the quasi-stationary distribution and justifying the convergence of diffusion maps within a local equilibrium. This perspective allows us to propose an enhanced sampling algorithm. We demonstrate the practical relevance of these approaches both for simple models and for molecular dynamics problems (alanine dipeptide and deca-alanine).

The implementation and validation of the adaptive buffered force QM/MM method in two popular packages, CP2K and AMBER are presented. The implementations build on the existing QM/MM functionality in each code, extending it to allow for redefinition of the QM and MM regions during the simulation and reducing QM-MM interface errors by discarding forces near the boundary according to the buffered force-mixing approach. New adaptive thermostats, needed by force-mixing methods, are also implemented. Different variants of the method are benchmarked by simulating the structure of bulk water, water autoprotolysis in the presence of zinc and dimethyl-phosphate hydrolysis using various semiempirical Hamiltonians and density functional theory as the QM model. It is shown that with suitable parameters, based on force convergence tests, the adaptive buffered-force QM/MM scheme can provide an accurate approximation of the structure in the dynamical QM region matching the corresponding fully QM simulations, as well as reproducing the correct energetics in all cases. Adaptive unbuffered force-mixing and adaptive conventional QM/MM methods also provide reasonable results for some systems, but are more likely to suffer from instabilities and inaccuracies.

Molecular dynamics is one of the most commonly used approaches for studying the dynamics and statistical distributions of many physical, chemical, and biological systems using atomistic or coarse-grained models. It is often the case, however, that the interparticle forces drive motion on many time scales, and the efficiency of a calculation is limited by the choice of time step, which must be sufficiently small that the fastest force components are accurately integrated. Multiple time-stepping algorithms partially alleviate this inefficiency by assigning to each time scale an appropriately chosen step-size. However, such approaches are limited by resonance phenomena, wherein motion on the fastest time scales limits the step sizes associated with slower time scales. In atomistic models of biomolecular systems, for example, resonances limit the largest time step to around 5-6 fs. In this paper, we introduce a set of stochastic isokinetic equations of motion that are shown to be rigorously ergodic and that can be integrated using a multiple time-stepping algorithm that can be easily implemented in existing molecular dynamics codes. The technique is applied to a simple, illustrative problem and then to a more realistic system, namely, a flexible water model. Using this approach outer time steps as large as 100 fs are shown to be possible.

We consider dimensional reduction techniques for the Liouville-von Neumann equation for the evaluation of the expectation values in a mixed quantum system. In applications such as nuclear spin dynamics the main goal for simulations is being able to simulate a system with as many spins as possible, for this reason it is very important to have an efficient method that scales well with respect to particle numbers. We describe several existing methods that have appeared in the literature, pointing out their limitations particularly in the setting of large systems. We introduce a method for direct computation of expectations via Chebyshev polynomials (DEC) based on evaluation of a trace formula combined with expansion in modified Chebyshev polynomials. This reduction is highly efficient and does not destroy any information. We demonstrate the practical application of the scheme for a nuclear spin system and compare with several alternatives, focusing on the performance of the various methods with increasing system dimension. Our method may be applied to autonomous quantum problems where the desired outcome of quantum simulation, rather than being a full description of the system dynamics, is only the expectation value of some given observable., Comment: 8 pages, 4 figures, submitted to SIAM J. Sci.Comp.

We consider a natural dimension reduction technique for the Liouville-von Neumann equation for a mixed quantum system based on evaluation of a trace formula combined with a direct expansion in modified Chebyshev polynomials. This reduction is highly efficient and does not destroy any information. We demonstrate the practical application of the scheme with a model problem and compare with popular alternatives. This method can be applied to autonomous quantum problems where the desired outcome of quantum simulation is the expectation of an observable., Comment: 6 pages, 4 figures

(Anti)-/ferromagnetic Heisenberg spin models arise from discretization of Landau-Lifshitz models in micromagnetic modelling. In many applications it is essential to study the behavior of the system at a fixed temperature. A formulation for thermostatted spin dynamics was given by Bulgac and Kusnetsov which incorporates a complicated nonlinear dissipation/driving term while preserving spin length. It is essential to properly model this term in simulation, and simplified schemes give poor numerical performance, e.g. requiring an excessively small timestep for stable integration. In this paper we present an efficient, structure-preserving method for thermostatted spin dynamics., Comment: 22 pages, 9 figures, 34 ps-files. Submitted to BIT