Uniform sampling of phase-space

Other algorithms have been developed for speeding up the uniform sampling of phase space points. For example, the efficient microcanonical sampling series of schemes exploits the possibility of sampling independently the spatial coordinates and momenta, simply by weighting the sampled geometries by their associated momentum space density. 

Completely uniform sampling of the conformational phase space is not exactly what is desired, because a lot of computational time will be wasted in simulating irrelevant regions of configurational phase space. Instead, a biasing potential, U, is required that ... 

Metropolis and Ulam determined how to generate these configurations that contribute to the sums in Eq. 15.5. The idea is simple instead of all microscopic states of phase space being sampled with uniform probability, more probable states are sampled more frequently than less probable ones. This is the importance sampling method, best described... 

Let us now consider the phase space if(P,Q), defined as a space of dimension 6N (3N dimensions for the coordinates q and 3N dimensions for the momenta p, where N is the total number of degrees of freedom) of all possible distributions so that the sum of their potential and kinetic energies is a constant. Microcanonical sampling is a sampling procedure whose key idea is to generate a (classical) uniform distribution in this phase space. Practically, it is based on an analytical frequency calculation at the reference geometry and on the use of a random number R (see the right-hand side of Scheme... 

To form a microcanonical ensemble random values for the P( and Qt are chosen so that there is a uniform distribution in the classical phase space of ( , Q) [17]. Two ways are described here to accomplish this. For one method, called microcanonical normal-mode sampling [18], random values are chosen for the mode energies Et, which are then transformed to random values for Pt and Qt. In the second method, called orthant sampling [11], random values for P and Q are sampled directly from the phase space. 

Let us now introduce the concept of degree of continuity of a phase. In the beginning of the IPN synthesis, polymer network I obviously exhibits continuity of both the network structure and its phase. When monomer II is uniformly swollen in, before polymerization of II, one phase also exists. Polymer network I is continuous (the sample is usually a swollen elastomer), and the monomer II is also distributed everywhere. Upon polymerization of II, phase separation takes place. Polymer network I is still continuous, but is partially or wholly excluded from some regions of space. Assuming the previous even distribution of monomer II, we have reason to believe polymer network II will exhibit some degree of chain continuity. Sometimes, polymer network II also appears to exhibit a degree of phase continuity. Usually, polymer network II has less continuity than polymer network I. A simple example of greater and lesser phase continuity in everyday life is chicken-wire in air. 

Speaking pictorially, the quasiergodic hypothesis implies that the trajectory of a single system, while it does not cover phase space, uniformly samples phase space. Questions regarding ergodlc properties of Hamiltonian systems are extremely complicated, and the subject is under active study today. But in practice an MD system is not quite Hamiltonian, because solving the equations of motion as finite difference equations Introduces a certain randomness Into the phase space trajectory. This property of the trajectory, which will be discussed In detail In Section III, presumably justifies the quasiergodic hypothesis for an MD system. 

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