Coarse-graining procedure

Table 5.1 Parameters of the united atom force field for polyethylene used as the atomistic input for the coarse-graining procedure. The Lennard-Jones parameters pertain to CH2-group interaction, since chain ends were not considered in the coarse-graining. 

Fig. 5.5. Illustration of the coarse-graining procedure for a united atom chain. The chain is a segment of PE at 509 K from molecular dynamics simulations with the united atom model [Eqs. (5.7)—(5.11)]. One coarse-grained bond represents the end-to-end distance of n = 5 consecutive united atom bonds. From [32]...

Figure 8-1. Coarse-graining procedure, (a) A snapshot of an atomistic system, (b) Groups of atoms of the atomistic system are combined into CG sites in order to reduce the number of degrees of freedom, (c) The atomistic system of (a) as represented by CG sites...

Figure 25 Multi-scale modeling of HDI (a) ab initio (atomic), (b) MC/MD (molecular), (c) coarse-graining procedure (molecular/meso-scale), (d) diffusion characteristics and surface topography, and (e) LBM (meso-scale) and integration HDI (system design).

The coarse-graining procedure simplifies the picture and provides the justification for various nonequilibrium approximations. One example is the Landau-Lifshitz equation... 

Unlike the density of bulk fluids, which is a function of pressure and temperature only (and composition for a mixture), the average density across the interface between a liquid and its vapor, as well as at the liquid/liquid interface, varies as a function of the distance along the interface normal p(z). Like other local thermodynamic quantities[30], it is defined by a coarse-graining procedure The volume of the system is divided into slabs perpendicular to the interface normal, and the density of each slab is computed in the usual way. The thickness of the slabs is chosen to be small enough so that the density does not vary much... 

This is a coarse-graining procedure that is valid if the spacing between frequencies a>j is much smaller than the inverse time resolution of any conceivable observation. See Section 1.4.4 and Eq. (1.182). 

In many practical situations the random process under observation is continuous in the sense that (1) the space of possible states is continuous (or it can be transformed to a continuous-like representation by a coarse-graining procedure), and (2) the change in the system state during a small time interval is small, that is, if the system is found in state x at time t then the probability to find it in state y x at time t + St vanishes when St 0. When these, and some other conditions detailed below, are satisfied, we can derive a partial differential equation for the probability distribution, the Fokker-Planck equation, which is discussed in this Section. 

Milano, G., MiiUta--Plathe, F. Mapping atomistic simulations to mesoscopic models a systematic coarse-graining procedure for vinyl polymer chains. J. Phys. Chem. B 109, 18609-18619 (2005)... 

Constructing of the coarse-graining procedures in which microscopic fluctuations are eliminated or replaced with simpler stochastic models. 

In DPD [42], the two-body interactions fl between two fluid particles i and j are assumed to be central and short-ranged. This can be defined as a sum of a conservative force Fc, dissipative component Fq, and the Brownian force Fb. The Brownian factor represents the thermal fluctuations averaged out due to coarse-graining procedure. The following equations show the basic formula describing the interparticle forces. 

The bottom-up atomistic approach — coarse-graining procedure replacing atoms interacting with conservative and central forces with clusters of atoms interacting with dissipative forces [85]. 

The foundations of DPD have been considered in a number of publica-tions. The rules of dissipative particle dynamics were derived from the underlying molecular interactions by a systematic coarse graining procedure. Evans derived expressions for the self-diffusion coefficient and shear viscosity of the DPD particles in the form of the Green-Kubo time correlation functions. DPD can be used to model arbitrarily shaped objects made up of fused spheres by... 

Thus, the next approximation level makes it necessary to take account of the interactions as well as the V2 ones. The idea of a coarse-graining procedure of the structural elements remains as in the case considered above. I he coanse-grained propagator in its diagrammatic form is shown in Figure 5.13, and the interaction line is in Figure 5.14. 

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