Basis functions defined using a set of grid points with nonuniform spacings. Vertical lines show the positions of the grid points, the first grid point being at . The top panel shows functions for even , and the bottom panel shows functions for odd . Each basis function is nonzero in two adjacent intervals and zero elsewhere. Solid curves are the basis functions (for ) associated with first interior grid point (0.9 for the present case) and dashed curves are examples of basis functions associated with other interior grid points.
The CG force function for large distances. The solid line is the average of the results of five variational calculations using statistically independent atomistic data. The error bars are estimated from the variance of the five results in the usual way. Note the expanded vertical scale.
Dependence of the calculated CG force and potential functions on the parameter for solute mole fraction of 0.25. Results for values of 5.2, 8.0, and 10.5 were obtained from the variational calculation using data from atomistic simulations of systems of 1000, 4000, and 8000 particles, respectively. The main figures show the results for and the insets give the difference between the functions for and those for a smaller .
Test of the pairwise additivity approximation of the CG potential for a system with solute mole fraction of 0.25. The top panel shows the atomistic and CG functions, and the bottom panel shows the difference between the two.
Test of the pairwise additivity approximation for various solute mole fractions for the model system. The top panel shows the CG functions for various mole fractions of solute as well as the atomistic . The bottom panel shows the difference between each CG and the atomistic , Numbers shown in the legends of both the figures give the mole fractions of solute ( particles) in the atomistic systems.
Comparison of CG potentials obtained from force matching calculations of systems with different compositions. Numbers shown in the legends give the mole fractions of solute particles in the atomistic systems. The thinner solid line in the inset is the truncated and shifted Lennard-Jones potential, which is the correct CG potential in the limit of solute mole fraction of 1.
Test of the first strategy for the construction of short-ranged CG potentials (for details, see the text). Comparison of radial distribution functions from atomistic and CG simulations with short-ranged potentials constructed using the first strategy for a system with solute mole fraction of 0.25.
Test of the second strategy for the construction of short-ranged CG potentials (for details, see the text). Comparison of radial distribution functions from atomistic and CG simulations with short-ranged potentials constructed using the second strategy for a system with solute mole fraction of 0.25. is the position of the last grid point or the distance beyond which the basis function part of the total CG force [i.e., , see Eq. (1)] is zero.
Effects of grid spacing on the quality of fitting in the force matching calculation. The same atomistic data set is used with three different choices of grid spacings. The top panel shows the overall fit. The middle and bottom panels show the effects on the large and short distance parts of the CG force. Neither of the uniform grid spacings led to an adequate representation of the force function both in the first peaks, where the function varies rapidly, and at large distances where there is more statistical error in the force information. (In the middle panel, with its expanded scale, the dashed curve is almost completely coincident with the solid curve. In the bottom panel, the dotted curve is almost completely coincident with the solid curve.)
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