Velocity-Verlet method

This actually translates into a fairly simple algorithm, based closely on the standard velocity Verlet method. Written in a Fortran-like pseudo-code, it is as follows. At tire start of the run we calculate both rapidly-varying (1) and slowly-varying (F) forces, then, in the main loop ... 

The velocity Verlet method [Swope et al. 1982] gives positions, velocities and accelerations at the same time and does not compromise precision ... 

The velocity Verlet method is actually implemented as a three-stage procedure because, as can be seen from Equation (7.15), to calculate the new velocities requires the accelerations at both t and t + 8t. Thus in the first step the positions at f I- are calculated according to Equation (7.14) using the velocities and the accelerations at time t. The velocities at time t + 6t are then determined using ... 

The dWi are Gaussian white noise processes, and their strength a is related to the kinetic friction y through the fluctuation-dissipation relation.72 When deriving integrators for these methods, one has to be careful to take into account the special character of the random forces employed in these simulations.73 A variant of the velocity Verlet method, including a stochastic dynamics treatment of constraints, can be found in Ref. 74. The stochastic... 

The classical equations of motion are solved with the velocity Verlet method [66]. The simulation is performed in the microcanonical ensemble (N,V,E) in a cubic box with length 15 A. Periodic boundary conditions are applied. The system is equilibrated by performing a constant temperature simulation [67] for 10 ps to achieve a mean temperature of 1000 K. The temperature control is then turned off to gain constant energy conditions. 

In this scheme, v and r refer to one of the 3N velocities or positions, respectively. Note that the different types of force are calculated throughout the algorithm. It can be readily seen that the method reduces to the standard velocity Verlet method if Mi, M2 and M3 are set equal to 1. 

The r-RESPA method has been applied to a variety of systems, including simple model systems [Tuckerman et al. 1992] but also organic molecules [Watanabe and Karplus 1993], fullerene crystals [Procacci and Berne 1994] and also proteins [Humphreys et al. 1994, 1996]. In these studies the reduction in computational time compared with the standard velocity Verlet method varied between 4-5 and 20-40, depending upon the size of the system, without any noticeable loss in accuracy. Other developments of the r-RESPA algorithm include its coupling to the fast multipole method (see Section 6.8.3) [Zhou and Berne 1995]. 

We can think of the velocity Verlet method as being defined by a splitting into three parts ... 

A natural simplification is to truncate the Taylor series after the constant term, resulting in a constant extrapolation of the slow force. The velocity Verlet method [6] can be easily modified to implement this constant extrapolation multiple time-step method ... 

A new era in multiple time-step methods arrived in the early 1990s when Grubmiiller et al. and Tuckerman et al. independently published multiple time-step methods that appeared to overcome the energy instabihty of extrapolation methods. Their idea is to mimic the kick-drift nature of the velocity Verlet method itself. In Eq. [6] the force supplies a kick, or impulse, in the first line, and the system then drifts as the updated half-step velocity contributes to the position at the new step. The velocity Verlet method can be modified so that the slow force is also applied as an impulse ... 

The velocity Verlet method is a three-stage algorithm because the calculation of the new velocities (O Eq. 7.28) requires both the acceleration at time t and at time t+8t. Therefore, first, the positions att+St re calculated using O Eq. 7.27 and the velocities and accelerations at time t. The velocities at time t + i d i are then calculated using... 

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