epl draft Multi-scale time-stepping in molecular dynamics

We introduce a modified molecular dynamics algorithm that allows one to freeze the dynamics of parts of a physical system, and thus concentrate the simulation effort on selected, central degrees of freedom. This freezing, in contrast to other multi-scale methods, introduces no approximations in the thermodynamic behaviour of the non-central variables while conserving the Newtonian dynamics of the region of physical interest. Molecular dynamics simulation is the backbone of much numerical work in condensed matter physics and chemistry. It is formulated in terms of the underlying forces of the physical model and allows one to study thermodynamic and dynamic processes in microscopic detail. However, it is also clear that the method is, in some geometries, extravagantly wasteful. An important and obvious example occurs in biophysical applications where a single interesting complex is studied in the presence of a solvent: A solvated protein requires detailed simulation of large number of water molecules—in order to correctly hydrate the molecule, and also to separate periodic images in the simulation cell. Such systems seem to be obvious candidates for a multi-scale approach. Close to the protein all the dynamics are to be followed, but at larger distances from the protein the water molecules are spectators, which should give rise to self-averaging in large systems. Nevertheless, this “distant spectator” must be well modeled in order to limit finite size artefacts. Such ideas have led to a number of interesting papers for the formulation of multiscale simulation [1, 2] where the spatial resolution of the solvent model changes as a function of position in the simulation cell. An alternative to this molecular coarse-graining is to use different integration methods in different regions using a simple, robust large-step integrator in the far field, coupled to more precise near field stepper, [3]. Another physical system where multi-scale ideas are important is in the modelling of deformations in solid materials. Here short-range atomistic modeling is coupled to larger scale continuum descriptions in order to reduce artefacts from the tiny samples that can be comfortably simulated [4,5]. It is clear that careful matching is required to best interface the continuum and atomistic models. The present paper introduces a complementary view on how to treat the far-field degrees of freedom. Rather than spatially coarse-graining or introducing continuum couplings we introduce a hierarchy in the speed of simulation. Central regions in the physical system advance with a full, molecular dynamic algorithm. Regions further away are regularly frozen so that on average time advances more slowly. However, the whole dynamic system is assembled in such a way that detailed-balance is obeyed; the entire simulated system thus samples the proper equilibrated thermodynamic ensemble. Thus there is no sharp boundary between interesting and spectator regions. All degrees of freedom are free to relax and fluctuate. Physical intuition tells us that this can be potentially useful: In the simulation of water the dominant local process is Debye dielectric relaxation of the orientational dipole occurring on the scale of 1ps. In a long (microsecond simulation) simulation of a bio-molecule a single, distant water molecule influences, above all, due to the averaged, static dielectric properties which are fully developed at this tiny time scale. Putting more computational effort into this rotational process than is needed is clearly inefficient. The same can be said of a distant region in a simulation of mechanical deformation of a solid: full high statistics simulation of distant regions do not substantially increase the accuracy of the simulation of the near field. We now consider the phase space dynamics of a system which has been partially frozen: To begin, consider the motion of a single particle with a reduced Hamiltonian Hi which contains just that part of the full Hamiltonian H p-1 ar X iv :1 70 3. 07 99 0v 1 [ co nd -m at . s ta tm ec h] 2 3 M ar 2 01 7