Algorithmic methods particle dynamics

Carmesin, I. and Kremer, K. The bond fluctuation method a new effective algorithm for the dynamics of polymers in all spatial dimensions. Macromolecules, 21,2819-23 (1988). Hoogerbrugge, P. J. and Koelman, J. M. V. A. Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhys Lett., 19,155-60 (1992). 

The CG method not only provides general models for studying a class of block copolymers but also conducts efficient algorithms for simulation. In this chapter, we overview the theoretical and computational approaches toward the simulations of dynamics of microphase separation of block copolymers with the focus on the recent contributions applying Monte Carlo (MC), dissipation particle dynamics... 

For the numerical simulation of flowing polymers, several mesoscopic models have been proposed in the last few years that describe polymer (hydro-)dynamics on a mesoscopic scale of several micrometers, typically. Among these methods, we like to mention dissipative particle dynamics (DPD) [168], stochastic rotation dynamics (sometimes also called multipartide collision dynamics) [33], and lattice Boltzmann algorithms [30]. Hybrid simulation schemes for polymer solutions have been developed recenfly, combining these methods for solvent dynamics with standard particle simulations of polymer beads (see Refs [32, 169, 170]). Extending the mesoscopic fluid models to nonideal fluids including polymer melts is currently in progress [30, 159,160,171]. 

One important feature of MFC algorithms is that the dynamics is well-defined for an arbitrary time step. At. In contrast to methods such as molecular dynamics simulations (MD) or dissipative particle dynamics (DFD), which approximate the continuous-time dynamics of a system, the time step does not have to be small. MFC defines a discrete-time dynamics which has been shown to yield the correct longtime hydrodynamics one consequence of the discrete dynamics is that the transport coefficients depend exphdtly on At. In fact, this freedom can be used to tune the Schmidt number. Sc [15] keeping all other parameters fixed, decreasing At leads to... 

The main difference between the force-bias and the smart Monte Carlo methods is that the latter does not impose any limit on the displacement that m atom may undergo. The displacement in the force-bias method is limited to a cube of the appropriate size centred on the atom. However, in practice the two methods are very similar and there is often little to choose between them. In suitable cases they can be much more efficient at covering phase space and are better able to avoid bottlenecks in phase space than the conventional Metropolis Monte Carlo algorithm. The methods significantly enhance the acceptance rate of trial moves, thereby enabling Icirger moves to be made as well as simultaneous moves of more than one particle. However, the need to calculate the forces makes the methods much more elaborate, and comparable in complexity to molecular dynamics. 

Molecular dynamics, in contrast to MC simulations, is a typical model in which hydrodynamic effects are incorporated in the behavior of polymer solutions and may be properly accounted for. In the so-called nonequilibrium molecular dynamics method [54], Newton s equations of a (classical) many-particle problem are iteratively solved whereby quantities of both macroscopic and microscopic interest are expressed in terms of the configurational quantities such as the space coordinates or velocities of all particles. In addition, shear flow may be imposed by the homogeneous shear flow algorithm of Evans [56]. 

Since MPC dynamics yields the hydrodynamic equations on long distance and time scales, it provides a mesoscopic simulation algorithm for investigation of fluid flow that complements other mesoscopic methods. Since it is a particle-based scheme it incorporates fluctuations, which are essential in many applications. For macroscopic fluid flow averaging is required to obtain the deterministic flow fields. In spite of the additional averaging that is required the method has the advantage that it is numerically stable, does not suffer from lattice artifacts in the structure of the Navier-Stokes equations, and boundary conditions are easily implemented. 

Another development that we will undertake in the near fumre is development of algorithms for non-BO calculations of molecules with 7i-electrons (the CH radical is an example of such a system). We also contemplate development of methods for describing systems where only the light nuclei (apart from electrons) are treated as quantum particles, and the other heavier nuclei are described either classically or by using a low-level approximation. This development would move us closer to cosidering the quantum dynamics of such reactions as inter- and intramolecular proton transfer. 

The implementation of molecular dynamics simulations on parallel computers needs a method that distributes over the processors both the evaluation of pair interactions and the integration of particle motions. The force terms involved in integrating the set of coupled differential equations (Newton s equations) characteristic of any MD simulation are typically nonlinear functions of the distance between pairs of atoms and may be either long-range or short-range. We use this attribute of the force terms in detailing the parallel algorithmic work conducted to date. 

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