Averages from Simulations

Molecular dynamics calculations can automatically average and save these values  

The calculation of average energies and their deviations from the mean are useful in several aspects of molecular dynamics simulations, such as these  

Dx = a/(x ) - (x), where Xj = value of EKIN, ETOT, EPOT, TEMP, or another value. 

You choose the values to average in the Molecular Dynamics Averages dialog box. As you run a molecular dynamics simulation, HyperChem stores data in a CSV file. This file has the same name as the HIN file containing the molecular system, plus the extension. CSV. If the molecular system is not yet stored in a HIN file, HyperChem uses the filename chem.csv. 

For more information about the contents of CSV files, see appendix A, HyperChem Files , in the HyperChem Reference Manual. 

---

Ifihe Bath relaxation con start t, t, is greater than 0.1 ps. yon should be able Lo calculate dyriani ic p roperlies, like time correlation fun c-tioris and diffusion constants, from data in the SNP and/or C.SV files (sec "Collecting Averages from Simulations"... 

After initial heating and equilibration, the trajectory may be stable for thousands of time points. During this phase of a simulation, you can collect data. Snapshots and CSV files (see Collecting Averages from Simulations on page 85) store conformational and numeric data that you can later use in thermodynamic calculations. 

Water molecules within 6.0 A from the atomic site are considered. 6Bulk water density taken as average from simulation. 

Statistical errors of dynamic properties could be expressed by breaking a simulation up into multiple blocks, taking the average from each block, and using those values for statistical analysis. In principle, a block analysis of dynamic properties could be carried out in much the same way as that applied to a static average. However, the block lengths would have to be substantial to make a reasonably accurate estimate of the errors. This approach is based on the assumption that each block is an independent sample. 

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

Fig. 19—Shear stress and chain angle as a function of sliding distance, from simulations of alkanethiolates on Au(111) at temperature 1 K (a) results from commensurate sliding show a stick-slip motion with a period of 2.5 A, (b) in incommensurate case both shear stress and chain angle exhibit random fluctuations with a much smaller average friction [45],...

Fig. 38. Variation of polydispersity with average cluster size at short times in a journal bearing flow. The symbols are from simulations and the lines are fits from Eq. (82). The regular flow is the journal bearing flow with only the inner cylinder rotating (Hansen and Ottino, 1996b).

There are two important consequences of this equality for computer simulations of many-body systems. First, it means that statistically averaged properties of these systems are accessible from simulations that are aimed at generating trajectories -e.g., molecular dynamics, or ensemble averages such as Monte Carlo. Furthermore, for sufficiently long trajectories, the time-averaged properties become independent of the initial conditions. Stated differently, it means that for almost all values of qo, Po, the system will pass arbitrarily close to any point x, p, in phase space at some later time. 

Fig. 5.3. Comparison of different free energy estimators. Plotted are distributions of estimated free energies using sample sizes (i.e., number of independent simulation runs) of N = 100 simulations (solid lines), as well as N = 1, 000 (long dashed) and N = 10,000 simulations short dashed lines), (a) Exponential estimator, (5.44). (b) Cumulant estimator using averages from forward and backward paths, (5.47). (c) Cumulant estimator using averages and variances from forward and backward paths, (5.48). (d) Bennett s optimal estimator, (5.50)...

The theory of statistical mechanics provides the formalism to obtain observables as ensemble averages from the microscopic configurations generated by such a simulation. From both the MC and MD trajectories, ensemble averages can be formed as simple averages of the properties over the set of configurations. From the time-ordered properties of the MD trajectory, additional dynamic information can be calculated via the time correlation function formalism. An autocorrelation function Caa( = (a(r) a(t + r)) is the ensemble average of the product of some function a at time r and at a later time t + r. 

How can statistical error be estimated for a single observable from independent simulations There seems little choice but to calculate the standard error in the mean values estimated from each simulation using Equation (1), where the variance is computed among the averages from the independent simulations and NJ L> is set to the number of simulations. In essence, each simulation is treated as a single measurement, and presumed to be totally independent of the other trajectories. Importantly, one can perform a "reality check" on such a calculation because the variance of the observable can also be calculated from all data from all simulations — rather than from the simulation means. The squared ratio of this absolute variance to the variance of the means yields a separate (albeit crude) estimate of the number of independent samples. This latter estimate should be of the same order as, or greater than, the number of... 

Figure 20-30 Effect of signal averaging on a simulated noisy spectrum. Labels refer to number of scans averaged. [From R. Q. Thompson, Experiments in Software Data Handling," J. Chem. 

Figure 7.5 Yields of the best binders as a function of library size. Each data point represents the average from 100 simulated DCLs. Reprinted with permission from Corbett, RT., etal. Org. Lett. 2004, 6, 1825-1827. Copyright (2004) American Chemical Society.

Average SRS simulant was employed for extraction (0 A = 1 3), 50 mM HNO3 for scrubbing (0 A = 5 1), and 1 mM HNO3 for stripping (0 A = 5 1). All contacts were made at 25 °C. Fresh aqueous phase was used for each contact (cross-current contacting). Data from reference [49],... 

Two resins were tested for the removal of succinic acid from simulated medium on a packed column of sorbent to simulate an actual process on a small scale. It is important to test the sorption with medium, because salts and other nutrients can interfere with the sorption. Table 4 presents the results for XUS 40285 MWA-1 was comparable. This indicates that either sorbent can remove succinic acid efficiently from the fermentation broth without direct loss of product. Both columns were then stripped or regenerated with hot water. Stripping with hot water recovered 70-80% of the succinic acid from the XUS 40285 resin whereas less (50-60%) was recovered from the MWA-1. The XUS 40285 column was stripped with 2 column volumes of hot water with eluent concentrations up to 49 g / L. Succinic acid was concentrated on average to 40 g/L in the XUS resin by this operation and to 30 g/L by the MWA-1. The 10-fold concentration factor bodes well for the use of sorbents to purify the fermentation broth. 

The ASEP/MD method, acronym for Averaged Solvent Electrostatic Potential from Molecular Dynamics, is a theoretical method addressed at the study of solvent effects that is half-way between continuum and quantum mechanics/molecular mechanics (QM/MM) methods. As in continuum or Langevin dipole methods, the solvent perturbation is introduced into the molecular Hamiltonian through a continuous distribution function, i.e. the method uses the mean field approximation (MFA). However, this distribution function is obtained from simulations, i.e., as in QM/MM methods, ASEP/MD combines quantum mechanics (QM) in the description of the solute with molecular dynamics (MD) calculations in the description of the solvent. 

The practical advantage of these relations is that, in MD simulations, single molecule properties like the self-diffusion coefficient and rotational relaxation times converge much faster than system properties due to additional averaging over the number of molecules in the ensemble. We applied eqs. 10 and 11 to our MD results using data at 800 K as a reference point in order to predict the viscosity over the entire temperature interval. In Fig. 7 we compare the predicted values with those obtained from simulation. It appears that in the temperature interval 600 K to 800 K predictions of Eq. (10) are more consistent with MD results than are the predictions of Eq. (11). This leads us to conclude that the viscosity temperature dependence in liquid HMX is more correlated... 

---