Boltzmann distribution space configuration

When calculating free energies, one generates, either by molecular dynamics or MC, configuration space samples distributed according to a probability distribution function (e.g., the Boltzmann distribution in the case of the Helmholtz free energy). 

Random search of configurational space followed by calculation of probabilities based on a Boltzmann distribution (see energy surface, conformational search, stochastic search). 

Similar in spirit is the Milestoning [90] method by Fiber and coworkers, who assume that the diffusion of interest occurs through a tube in configuration space, and translate the rare process into a non-Markovian hopping between configuration space hyperplanes, the so-called milestones (which are in fact rather similar to the TIS interfaces, except that they do not form a foliation). The kinetics is obtained from starting an equilibrium ensemble on a milestone, and measuring the time distribution needed to reach the next milestone. The distribution can subsequently be used to construct the kinetics. The assumption is that there is an equilibrium (Boltzmann) distribution on each milestone. 

No special equilibrium between activated complexes and reactants is assumed. It is supposed however, that within the space between qi and qi+Aqi, q2 and q2+Aq2, configurations (activated complexes) have impulses (motions) between pi and pi+Api,p2 and P2+AP2 respectively. These configurations are computed in accordance with Maxwell-Boltzmann distribution. Recall from the gas laws that an energy profile for molecules can be describe by the Maxwell-Boltzman distribution diagram (Figure 3.3). As the temperature goes up, the population of molecules with more energy also increases. 

Using the assumption that the system is thermally equilibrated and therefore has Boltzmann-distributed energies, the probability density for finding the system at the point (je, v) in combined configuration and momentum space will be... 

The usual Metropolis Monte Carloprocedure when applied to the study of water at low temperatures is found to lead to bottlenecks in configuration space. In the Monte Carlo (M.C.) method different configurations of the system are sampled according to the Boltzmann distribution ... 

The formula requires an explanation. Here, there is the full mechanical energy U + K = E, being a function of the system s state, i.e., dependent on particle coordinates and components of molecular speeds (or momentums). The factor dx = dxdydzdvjtv dv is an element of a configuration space (eq. 1.3.38). Constant C is not yet determined, but can be found from normalization. This is the Maxwell-Boltzmann distribution. 

Reiteration of such a procedure gives a Markov chain of molecular configurations distributed in the phase space of the system, with the probability density proportional to the Boltzmann weight factor corresponding to the canonical NVT statistical ensemble. 

As the density of the fluid is increased above values for which a linear term in the expansion of equation (5.1) is adequate (crudely above values for which a third virial coefficient is adequate to describe the compression factor of a gas), the basis of even the formal kinetic theory is in doubt. In essence, the difficulty arises because it becomes necessary, at higher densities, to consider the distribution function, in configuration and momentum space, of pairs, triplets etc. of molecules in order to formulate an equation for the evolution of the single-particle distribution function. Such an equation would be the generalization to higher densities of the Boltzmann equation, discussed in Chapter 4 (Ferziger Kaper 1972 Dorfman van Beijeren 1977). 

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