Canonical ensemble Force

An alternative method, proposed by Andersen [23], shows that the coupling to the heat bath is represented by stochastic impulsive forces that act occasionally on randomly selected particles. Between stochastic collisions, the system evolves at constant energy according to the normal Newtonian laws of motion. The stochastic collisions ensure that all accessible constant-energy shells are visited according to their Boltzmann weight and therefore yield a canonical ensemble. 

The theory presented in this section is based on the grand canonical ensemble formulation, which is perfectly well-suited for the description of confined systems. Undoubtedly, in the case of attractive-repulsive interparticle forces unexpected structural and thermodynamic behavior in partly... 

In the canonical ensemble (P2) = 3kBTM and p M. In the microcanonical ensemble (P2) = 3kgT i = 3kBTMNm/(M + Nm) [49]. If the limit M —> oo is first taken in the calculation of the force autocorrelation function, then p = Nm and the projected and unprojected force correlations are the same in the thermodynamic limit. Since MD simulations are carried out at finite N, the study of the N (and M) dependence of (u(t) and the estimate of the friction coefficient from either the decay of the momentum or force correlation functions is of interest. Molecular dynamics simulations of the momentum and force autocorrelation functions as a function of N have been carried out [49, 50]. 

There is considerable interest in the use of discretized path-integral simulations to calculate free energy differences or potentials of mean force using quantum statistical mechanics for many-body systems [140], The reader has already become familiar with this approach to simulating with classical systems in Chap. 7. The theoretical basis of such methods is the Feynmann path-integral representation [141], from which is derived the isomorphism between the equilibrium canonical ensemble of a... 

The pairwise nature of the bond-boost makes this task easier since such traps would show up as a non-convexity of some of the biased effective pair potentials, which in the canonical ensemble can be taken to be the pairwise potential of mean force (PMF, denoted as V). Thus, assuming that V is approximately quadratic for lei safety condition can be enforced by setting Sa[Pg.92]

This statistical mechanical expression for surface tension depends explicitly on the potentials of inteimolecular force and molecular distribution functions. Upon recognition that the two-phase system under consideration is thermodynamically open, it follows that the distribution functions must be represented in the grand canonical ensemble. Thus, the dependence of y on temperature, T, and chemical potentials, enters through the implicit dependence of the distribution func-... 

One also finds that fixing the director generates a new equilibrium ensemble where the Green-Kubo relations for the viscosities are considerably simpler compared to the conventional canonical ensemble. They become linear functions of time correlation function integrals instead of rational functions. The reason for this is that all the thermodynamic forces are constants of motion and all the thermodynamic fluxes are zero mean fluctuating phase functions in the constrained ensemble. 

To determine the free-energy profile (rather then the potential energy profile as above), one performs a (typically short) constant temperature (canonical ensemble) FPMD simulation of the system (while constraining the reaction coordinate to the prescribed value) and records the mean force of constraint along the chosen reaction coordinate. This process is then repeated for a number of points along the reaction coordinate connecting the reactant and... 

Fa(1 0) and v-0> are the mean intermolecular forces in singlet space and pair space in this equilibrium state. Equations IV.7 and IV.8 are exact since the averages are performed in a canonical ensemble. 

Most of the work done to date is based on classical effective potentials that describe with sufficient accuracy the relevant interatomic and intermolecular forces, in particular the vdW couplings between hydrogen molecules and the various carbon structures considered (mainly graphite and nanotubes). An important point to note is that, since the electrons are not explicitly treated in such models, the computational cost of the simulations is relatively small, which makes it possible to perform, for realistic model systems composed of hundreds of atoms, sophisticated statistical calculations in the grand-canonical ensemble, and thus compute the amount of hydrogen that can be stored at particular temperature and pressure conditions, and even consider the quantum effects associated to the hydrogen molecules. Useful reviews of the most important results and literature can be found in Hirscher and Becher, Meregalh and Parrinello and Simonyan and Johnson. ... 

FI. Resat, M. Mezei, and J. A. McCammon,/. Phys. Ghent., 100, 1426 (1996). Use of the Grand Canonical Ensemble in Potential of Mean Force Calculations. 

The Metropolis MC method in the canonical ensemble is the most frequently used simulation approach to solve the primitive model of electrolytes. Averages of static properties are taken from a large set of Boltzmann-weighted configurations. Molecular dynamics and Brownian dynamics constitute two other methods to determine static and dynamic properties of molecular systems. Their implementations are, however, comphcated for systems possessing impulsive forces in combination with other forces. Hence, a soft-sphere repulsion is frequently used instead of the hard-sphere one when simulating such systems with these methods. 

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