Monte Carlo averaging

Quirke, N. Jacucci, G., Energy difference functions in Monte Carlo simulations application to the calculation of free energy of liquid nitrogen. II. The calculation of fluctuation in Monte Carlo averages, Mol. Phys. 1982, 45, 823-838... 

The choice of other variables R, r, h, 0, and r appropriate for Monte-Carlo averaging is made by pseudo random numbers generated on computer. The reactive cross section can be found by averaging the reaction probability over the impact parameter and rotational state... 

From this dynamic interpretation of the Monte Carlo averaging we can obtain a formal estimate of the number of steps Mq that have to be omitted at the beginning of the averaging. Usually, the order parameter ijf is the slowest relaxing quantity and then... 

These quantum mechanical (QM) results have been compared (Diestler and Karplus, 1971) with classical mechanical (CM) ones obtained from Monte Carlo averages of classical trajectories with the same potential. The QM threshold is about 007 eV lower than the CM one, and QM total probabilities of reactions are consistently larger. The CM results did not show the sharp QM dip present just above threshold for vibrational excitation of products. 

Since the number of chains n is kept constant and only the order parameter M, Eq. (2), can fluctuate, the first factor on the right hand side of Eq. (140) is constant and hence cancels out from the Monte Carlo averages. The transition probability for the grandcanonical moves where one goes from an old configuration (c) to a new configuration (c ) via an identity switch of a chain (A B), Fig. 16 (lower part), hence becomes,... 

In this form it becomes evident that the transmission coefficient is just the plateau value of the average of a unidirectional thermal flux. To obtain the dynamical corrections to TST there is no need to know the TST rate itself. Numerically it may be actually easier to compute the transmission coefficient Ka(f) than the magnitude of the one-dimensional TST rate. The ID TST rate may be obtained only after extensive Monte Carlo averaging. A small error in the activation energy can easily mask the correction term, which is just a prefactor. 

N. Quirke and G. Jacucci, Mol. Phys., 45, 823 (1982). Energy Difference Functions in Monte Carlo Simulations. Application to (1) The Calculation of the Free Energy of Liquid Nitrogen, (2) The Fluctuation of Monte Carlo Averages. 

Binder and Stauffer have calculated Monte Carlo average surface energies of cubic clusters of a lattice gas. When their data for clusters of up to 256 atoms are plotted in the same form as we did before for argon and water (Fig. 13), we see the surface energy of these lattice gas clusters is well described by Eq. (50). 

To obtain Monte Carlo averages in the microcanonical ensemble, one can radially project the velocities Vp onto the hypersphere of constant energy. 

In the Monte Carlo approximation the integrals in these equations are replaced by sums. When the collisions involve polyatomic molecules, the additional degrees of freedom obviously result in additional integrals. The reader will find comprehensive, explicit descriptions of how to compute various Monte Carlo averages in previous reviews. ... 

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