Sampling Different Ensembles

In the following we are interested in the classical approximation of pv, which means 

Here W is the Hamilton function of the system and = (fcgr) . The factory arises from the translation of Xu to a corresponding integration over classical phase space, i.e. 

This formula applies to particles completely characterizable through their position in space r = where is a relative coordinate independent of the size of 

Here U is the potential energy of the system. All in all this probability describes classical systems with variable volume, and particle number. 

How do we apply this First we must decide which ensemble to use. Is it sufiicient to just translate the particles at constant temperature, volume and particle number This would be the canonical ensemble. Or do we model an open system with variable particle number at constant chemical potential, volume, and temperature This would be the grand-canonical ensemble. Remember our discussion of the isosteric heat of adsorption for methane on graphite on p. 206. In this example methane is well represented as a point particle. Here step (i) of a MC procedure consists in a random change of ( , V, N). We can select a methane molecule at random and move it a random distance in a random direction. Volume and particle number would be constant. But we can also decide to just change the particle number. We must decide whether to insert or remove a particle from the system. The following algorithm, used to generate the simulation results in the aforementioned example, alternates between these two MC moves . The volume is kept constant all the time. Insertion and removal of particles makes additional translation of existent particles obsolete in this case.  

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Monte Carlo Sampling from Different Ensembles... 

An ensemble of 16 nonadiabatic surface hopping trajectories have been calculated sampling different initial conditions from a 300 K ground state simulation. A monoexponential fit to the Sj population gives a lifetime of 1.0 ps. Estimating the lifetime using the relation (10-13) yields the interval [0.9...1.6. ..6.0] ps, which indicates that the result from the exponential fit probably underestimates the lifetime [41,42], The 7H-keto tautomer is thus considerably longer lived than the 9H-keto form. 

The molecular dynamics and Monte Carlo simulation methods differ in a variety of ways. The most obvious difference is that molecular dynamics provides information about the time dependence of the properties of the system whereas there is no temporal relationship between successive Monte Carlo configurations. In a Monte Carlo simulation the outcome of each trial move depends only upon its immediate predecessor, whereas in molecular dynamics it is possible to predict the configuration of the system at any time in the future - or indeed at any time in the past. Molecular dynamics has a kinetic energy contribution to the total energy whereas in a Monte Carlo simulation the total energy is determined directly from the potential energy function. The two simulation methods also sample from different ensembles. Molecular dynamics is traditionally performed under conditions of constant number of particles (N), volume (V) and energy (E) (the microcanonical or constant NVE ensemble) whereas a traditional Monte Carlo simulation samples from the canonical ensemble (constant N, V and temperature, T). Both the molecular dynamics and Monte Carlo techniques can be modified to sample from other ensembles for example, molecular dynamics can be adapted to simulate from the canonical ensemble. Two other ensembles are common ... 

A Monte Carlo simulation traditionally samples from the constant NVT (canonical) ensemble, but the teclmique can also be used to sample from different ensembles. A common alternative is the isothermal-isobaric, or constant NPT, ensemble. To simulate from this ensemble, it is necessary to have a scheme for changing the volume of the simulation cell in order to keep the pressure constant. This is done by combining random displacements of the particles with random changes in the volume of the simulation cell. The size of each volume change is governed by the maximum volume change, V ax-Thus a new volume is generated from the old volume as follows ... 

The importance of treating the NMR observables and from them derived parameters as ensembles over relevant time scales rather than individual conformers is highlighted by Angyan and Gaspari who discuss several MD simulation techniques to interpret the structural and dynamical data from NMR. Internal dynamics in proteins take place at several different time scales. It is therefore important to sample the ensembles covering the specific time scales. 

Standard molecular dynamics calculations, i.e., those that solve Hamilton s equation, are performed on NVE ensembles, i.e., samples with a constant number of atoms N), fixed volume (V), and constant energy ( ). In standard Monte Carlo simulations the more widely applicable NVT ensembles are used, i.e., constant temperature (T) rather than energy, although both schemes can be modified to work in different ensembles. In particular, free energies can be directly evaluated using Monte Carlo methods in the Grand Canonical ensemble, although technical difficulties involved... 

Although in principle shooting and shifting moves sample the path ensemble ergodically, it can be difficult to sample different classes of pathways if they are not connected in path space. This can be solved by parallel tempering [237], which is discussed for trcinsition path sampling in appendix C. 

A related algorithm can be written also for the Brownian trajectory [10]. However, the essential difference between an algorithm for a Brownian trajectory and equation (4) is that the Brownian algorithm is not deterministic. Due to the existence of the random force, we cannot be satisfied with a single trajectory, even with pre-specified coordinates (and velocities, if relevant). It is necessary to generate an ensemble of trajectories (sampled with different values of the random force) to obtain a complete picture. Instead of working with an ensemble of trajectories we prefer to work with the conditional probability. I.e., we ask what is the probability that a trajectory being at... 

Umbrella sampling can give free-energy differences, but not absolute free energies Usually done in NPT—isothermal/isobaric ensembles, including a water box... 

To extract the conformational properties of the molecule that is being studied, the conformational ensemble that was sampled and optimized must be analyzed. The analysis may focus on global properties, attempting to characterize features such as overall flexibility or to identify common trends in the conformation set. Alternatively, it may be used to identify a smaller subset of characteristic low energy conformations, which may be used to direct future drug development efforts. It should be stressed that the different conformational analysis tools can be applied to any collection of molecular conformations. These... 

Realistic samples contain CNTs with different layer numbers, circumferences, and orientations. If effects of small interlayer interactions are neglected, the magnetic properties of a multi-walled CNT (MWCNT) are given by those of an ensemble of single-walled CNTs (SWCNTs). The distribution function for the circumference, p(L), is not known and therefore we shall consider following two different kinds. The first is the rectangular distribution, p(L) = mn)... 

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