Ikuo Fukuda

Numerical examination of the extended phase‐space volume‐preserving integrator by the Nosé‐Hoover molecular dynamics equations

This article illustrates practical applications to molecular dynamics simulations of the recently developed numerical integrators [Phys Rev E 2006, 73, 026703] for ordinary differential equations. This method consists of extending any set of ordinary differential equations in order to define a time invariant function, and then use the techniques of divergence‐free solvable decomposition and symmetric composition to obtain volume‐preserving integrators in the extended phase space. Here, we have developed the technique by constructing multiple extended‐variable formalism in order to enhance the handling in actual simulation, and by constituting higher order integrators to obtain further accuracies. Using these integrators, we perform constant temperature molecular dynamics simulations of liquid water, liquid argon and peptide in liquid water droplet. The temperature control is obtained through an extended version of the Nosé‐Hoover equations. Analyzing the effects of the simulation conditions including time step length, initial values, boundary conditions, and equation parameters, we investigate local accuracy, global accuracy, computational cost, and sensitivity along with the sampling validity. According to the results of these simulations, we show that the volume‐preserving integrators developed by the current method are more effective than traditional integrators that lack the volume‐preserving property.

Numerical Integration Techniques Based on a Geometric View and Application to Molecular Dynamics Simulations

In this chapter we address numerical integration techniques of ordinary differential equation (ODE), especially that for molecular dynamics (MD) simulation. Since most of the fundamental equations of motion in MD are represented by nonlinear ODEs with many degrees of freedom, numerical integration becomes essential to solve the equations for analyzing the properties of a target physical system. To enhance the molecular simulation performance, we demonstrate two techniques for numerically integrating the ODE. The first object we present is an invariant function, viz., a conserved quantity along a solution, of a given ODE. The second one is a numerical integrator itself, which numerically solves the ODE by capturing certain geometric properties of the ODE.

Molecular dynamics equation designed for realizing arbitrary density: Application to sampling method utilizing the Tsallis generalized distribution

Several molecular dynamics techniques applying the Tsallis generalized distribution are presented. We have developed a deterministic dynamics to generate an arbitrary smooth density function ρ. It creates a measure-preserving flow with respect to the measure ρdω and realizes the density ρ under the assumption of the ergodicity. It can thus be used to investigate physical systems that obey such distribution density. Using this technique, the Tsallis distribution density based on a full energy function form along with the Tsallis index q ≥ 1 can be created. From the fact that an effective support of the Tsallis distribution in the phase space is broad, compared with that of the conventional Boltzmann-Gibbs (BG) distribution, and the fact that the corresponding energy-surface deformation does not change energy minimum points, the dynamics enhances the physical state sampling, in particular for a rugged energy surface spanned by a complicated system. Other feature of the Tsallis distribution is that it provides more degree of the nonlinearity, compared with the case of the BG distribution, in the deterministic dynamics equation, which is very useful to effectively gain the ergodicity of the dynamical system constructed according to the scheme. Combining such methods with the reconstruction technique of the BG distribution, we can obtain the information consistent with the BG ensemble and create the corresponding free energy surface. We demonstrate several sampling results obtained from the systems typical for benchmark tests in MD and from biomolecular systems.

A Simple Efficient Molecular Dynamics Scheme for Evaluating Electrostatic Interaction of Particle Systems

We present results from a recently developed new molecular dynamics method to calculate electrostatic interaction of point particle systems. This method is very simple, which is based on the charge neutralized pairwise summation developed by Wolf et al., while solves problems presented in previous similar approaches and consistently gives the force, potential, and the total energy of the system.

Application of the Tsallis Statistics to the Molecular Dynamics Simulation

We have developed a deterministic algorithm that produces the Tsallis distribution in continuous systems. Using this, efficient sampling of the states is performed to construct the Boltzmann‐Gibbs distributions.