Langevin dynamics discretization scheme

Table 7.1 The expected long-time computed average using the given second-order Langevin dynamics discretization scheme...

G. Zhang and T. Schlick. Implicit discretization schemes for Langevin dynamics. Mol. Phys., 84 1077-1098, 1995. 

This gives the necessary Lyapunov condition for the Nosd-Hoover-Langevin equations, and hence the dynamics are geometrically ergodic. Solution trajectories will sample the extended canonical distribution, and it remains to compute solutions using an appropriate discretization scheme. 

The Langevin equation is discretized temporally by a set of equally spaced time intervals. At predetermined times, the ion dynamics is frozen, and the spatial distribution of the force is calculated from the vector sum of all its components, including both the long-range and the short-range contributions. The components of the force are then kept constant, while the dynamics resumes under the effect of the updated field distribution. Self-consistency between the force field and the ionic motion in the phase space is obtained by iterating this procedure for a desired amount of simulation time. The choice of the spatial and temporal discretization schemes plays a crucial role in computational performance and model accuracy. 

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