Finite-size system

Fig. 2.2. Average electrostatic potential mc at the position of the methane-like Lennard-Jones particle Me as a function of its charge q. mc contains corrections for the finite system size. Results are shown from Monte Carlo simulations using Ewald summation with N = 256 (plus) and N = 128 (cross) as well as GRF calculations with N = 256 water molecules (square). Statistical errors are smaller than the size of the symbols. Also included are linear tits to the data with q < 0 and q > 0 (solid lines). The fit to the tanh-weighted model of two Gaussian distributions is shown with a dashed line. Reproduced with permission of the American Chemical Society...

E. Energetics, Thermodynamics, Response, and Dynamics of Ultracold Finite Systems Size Effects on the Superfluid Transition in ( He) Finite Systems... 

To address the problem of finite system size, the EFP method has also been combined with continuum models in order to model the effects of the neglected bulk solvent [125], The Onsager equation was used to obtain the dipole polarization of the solute molecule (modeled quantum mechanically) and explicit water molecules (modeled by effective fragment potentials) due to the dielectric continuum. Thus the energy becomes... 

To overcome problems arising fi-om the finite system size used in MC or MD simulation, boundary conditions are imposed using periodic-stochastic approximations or continuum models. In particular, in stochastic boundary conditions the finite system is not duplicated but a boundary force is applied to interact with atoms of the system. This force is set as to reproduce the solvent regions that have been neglected. Anyway, in general any of the methods used to impose boundary conditions in MC or MD can be used in the QM/MM approach. 

Space does not permit us to give here a detailed discussion of the effects of finite-system size on molecular dynamic calculations of time correlations functions. We have given elsewhere a discussion of such effects on the velocity autocorrelation function from a hydrodynamic point of view. This reference can also be consulted for a more extensive discussion of results for both the velocity autocorrelation function and the super-Burnett self-diffusion coefficient, including comparisons with theoretical predictions. 

E) Compare results for the different system sizes. If the effects of the finite system size are not acceptable, return to (D). 

A milestone in the simulation of crystallisation was performed by Streitz and co-workers, who simulated the crystallisation (solidification) of a metal without size effects. In particular, they simulated the spontaneous nucleation and growth of a solid from the liquid phase. The authors found that 16 million atoms were sufficient to simulate metal solidification from the melt with no approximations due to finite system size. Equipped with an IBM BlueGene/L computer at LLNL they were able to simulate a billion atoms. On the other hand those of us without such resources can still simulate nanoparticles as one can easily consider all the atoms comprising the nanoparticle explicitly. 

Starting with a deterministic initial condition P X, Y, Z, 0) = x,20000 K, isooo 2,250 one observes a rapid growth followed by a final saturation to a constant value, very close to the plateau found in Figure 14, up to a small correction arising from the finite system size as before. One notes (see insert to Fig-... 

L of a periodic system, in agreement with previous studies [68,91,112]. Simulations for various system sizes for polymers of lengths Nm = 10, 20, and 40 allow an extrapolation to infinite system size, which yields Do/y/k Ta /mRi 1.7 x 10 , in good agreement with the diffusion coefficient of a monomer in the same solvent. The values of are about 30% larger than the finite-system-size values presented in Fig. 10. Similarly the diffusion coefficient for a polymer chain with excluded vol-lune interactions displays the dependence Dh l/Rn [73]. 

Applying eqn (2.91) to our GCMC results for the 0-Pt(321) system, we obtain the results in Figure 2.17, where we plot v5. J at a simulation chemical potential of -0.7 eV/O. This chemical potential is shown because it most clearly exhibits a sharp maximum arormd 800 K, whereas most other results not shown exhibit broad maxima rather than sharp peaks. The presence of a gradual increase and decrease of the heat capacity around the maximum rather than a sharp delta function singularity is a consequence of the finite system size being unable to capture the true behavior at phase transitions. ... 

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