Simulation algorithms

To speed up the process of attainment of the temperature steady value one can use special operations calculation without a kiln rotation, using large time intervals and calculation in two-dimensional R-tp geometry without regard for heat and mass transfer along an axis The program for realization of discussed simulation algorithms enables to calculate temperature in cells, a total number of which can not exceed 130 thousands A circular kiln structure can contain up to three layers. 

P., Campbell, T. J., Ogata, S., Shimojo, F., Saini, S., Scalable atomistic simulation algorithms for materials research,... 

Cemy, V., "Thermodynamic Approach to the Travelling Salesman Problem An Efficient Simulation Algorithm", J. Opt. Theory Applic., 45,41-51 (1985). 

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. 

Lamoureux G, Roux B (2003) Modeling induced polarization with classical Drude oscillators theory and molecular dynamics simulation algorithm. J Chem Phys 119(6) 3025-3039... 

Wang, L. Hermans, J., Change of bond length in free-energy simulations algorithmic improvements, but when is it necessary , J. Chem. Phys. 1994,100, 9129-9139... 

Computer simulations of confined polymers have been popular for several reasons. For one, they provide exact results for the given model. In addition, computer simulations provide molecular information that is not available from either theory or experiment. Finally, advances in computers and simulation algorithms have made reasonably large-scale simulations of polymers possible in the last decade. In this section I describe computer simulations of polymers at surfaces with an emphasis on the density profiles and conformational properties of polymers at single flat surfaces. 

The first MC (16) and MD (17) studies were used to simulate the properties of single particle fluids. Although the basic MC (11,12) and MD (12,13) methods have changed little since the earliest simulations, the systems simulated have continually increased in complexity. The ability to simulate complex interfacial systems has resulted partly from improvements in simulation algorithms (15,18) or in the interaction potentials used to model solid surfaces (19). The major reason, however, for this ability has resulted from the increasing sophistication of the interaction potentials used to model liquid-liquid interactions. These advances have involved the use of the following potentials Lennard-Jones 12-6 (20), Rowlinson (21), BNS... 

LaBerge, L. J. and Tully, J.C., Arigorous procedure for combining molecular dynamics and Monte Carlo simulation algorithms, Chem. Phys., 260, 183, 2000. 

On the other hand, it is well known that there is a relationship between Lyapunov exponents and the divergence of the vector field deduced from the differential equations describing a dynamical system. This relation provides a test on the numerical values obtained from the simulation algorithm. This relationship is, according to the definition of Lyapunov exponents ... 

However, in other types of molecular simulation, any sampled configuration may be correlated with configurations not sequential in the ultimate "trajectory7 produced. That is, the final result of some simulation algorithms is really a list of configurations, with unknown correlations, and not a true trajectory in the sense of a time series. 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

The discrete Markov process used to define a kinetic SDE in Eq. (2.315) or (2.318) can be directly implemented as a numerical algorithm for the integration of a set of SDEs. The resulting simulation algorithm would require the evaluation of neither derivatives of the mobility nor any corrective pseudoforce. It would, however, require an efficient method of calculating the elements of the mobility tensor and derivatives of U and in in the chosen system of generalized coordinates. 

In this section, we have considered four possible ways of formulating and interpreting a set of SDE to describe Brownian motion, and tried to clarify the relationships among them. Because each interpretation may be defined as the At 0 limit of a discrete Markov processes, this discussion of SDEs provides a useful starting point for the discussion of possible simulation algorithms. 

H.-Ch. Ottinger, Stochastic Processes in Polymer Fluids Tools and Examples for Developing Simulation Algorithms, (Springer-Verlag, Berlin, 1996). 

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