Molecular dynamics is a popular method for exploring the molecular conformations accessible at given conditions of state.  It is widely used for applications in Chemistry, Physics, Biology and Engineering, with the number of atoms ranging from a few dozen to a few billion.   In order to ensure sampling of the canonical ensemble some sort of thermal regulation device is needed, such as the Nose-Hoover thermostat or Langevin dynamics.  The latter uses stochastic dynamics to provide a complete (ergodic) sampling of the distribution.   

Numerical methods rely on splitting up the equations of motion within a timestep of fixed length.   Due to the presence of stiff components in the dynamics, small timesteps are generally needed to integrate the equations of motion. Given the metastable character of the free energy landscape (with many local minima which may 'trap' sampling trajectories), the challenge is to find numerical methods that (a) allow stable simulations to be performed with the largest possible timestep, while (b) providing sufficient accuracy with respect to thermal averages so that the algorithms retain their predictive power.   In order to enhanced sampling, many schemes have been proposed, including the removal of fast modes by use of rigid constraints, and multiple timestepping methods which use different timesteps in a hierarchical fashion to resolve different components of the dynamics.

In this talk I will discuss two recent articles on the design of integrators for molecular sampling.  In the first article [1] (with C. Matthews) we have studied the accuracy of Langevin dynamics splitting methods, demonstrating that the choice of ordering of the steps of the algorithm can have profound effect on the attainable resolution of configurational averages. The ideal method allows simulation of a solvated biomolecule to be performed with stepsizes up to very near 3fs which is the upper limit for schemes that accurately resolve all dynamical modes.  In joint work with D. Margul and M. Tuckerman (NYU) [2] we have developed new stochastic-dynamical methods that incorporate isokinetic constraints in order to tame resonance-induced instabilities in multiple timestepping.  These methods enable stable ergodic simulation to be performed with 'outer' timesteps of up to around 100fs in biomolecular simulation.

[1] B. Leimkuhler and C. Matthews, Robust and efficient configurational molecular sampling via Langevin dynamics, J. Chem. Phys. 138, 174102 (2013)

[2] B. Leimkuhler, D. Margul and M. Tuckerman, Stochastic resonance-free multiple time-step algorithm for molecular dynamics with very large time steps, Mol. Phys., to appear (2013).