Phase space configurations

The small number of variables needed for thermodynamic state description is certainly surprising from a microscopic molecular dynamic viewpoint. For the complete molecular-level description of an arbitrary state (phase-space configuration) of the order of 1023 particles, we should expect to require an enormously complex nonequilibrium function independent variables (i.e., positions rt and velocities r,-), time evolution until equilibrium is achieved, we find that a vastly simpler description is possible for the resulting equilibrium state state properties R, R2.i.e., for a pure substance,... 

Often it is not the rate kernel itself that is of interest, but rather the decay of the system from some initial nonequilibrium state. A case in point is the study of atom recombination following a photodissociation event. The initial state is presumed to correspond to a specific phase-space configuration of the pair of atomic radicals in an equilibrium solvent. The decay of this initial state is then monitored in the experiment. This is discussed in more detail in Section XII here we simply show how the kinetic theory can be formulated to accommodate this situation. 

The point (phase space point or configuration) is accepted or rejected according to the criterion... 

An individual point in phase space, denoted by F, corresponds to a particular geometry of all the molecules in the system. There are many points in this phase space that will never occur in any real system, such as configurations with two atoms in the same place. In order to describe a real system, it is necessary to determine what configurations could occur and the probability of their occurrence. 

A sequence of successive configurations from a Monte Carlo simulation constitutes a trajectory in phase space with HyperChem, this trajectory may be saved and played back in the same way as a dynamics trajectory. With appropriate choices of setup parameters, the Monte Carlo method may achieve equilibration more rapidly than molecular dynamics. For some systems, then, Monte Carlo provides a more direct route to equilibrium structural and thermodynamic properties. However, these calculations can be quite long, depending upon the system studied. 

The simulation of a molecular system at a finite temperature requires the generation of a statistically significant set of points in phase space (so-called configurations), and the properties of a system can be obtained as averages over these points. For a many-particle system, the averaging usually involves integration over many degrees of freedom. 

Labeling each cell of the one-dimensional lattice by i G Z, where Z is the set of integers, the collection of all configurations S defines the CA phase space, and is denoted by T =. ... 

A partial analogy between the dynamics of CA and the behaviors of continuous dynamical systems may be obtained by exploiting a fundamental property of CA systems namely, continuity in the Cantor-set Topology. We recall from section 2.2.1 that the collection of all one-dimensional configurations, or the CA phase space, r = where E = 0,1,..., fc 9 cr and Z is the set of integers by which each site of the lattice is indexed, is a compact metric space homeomorphic to the Cantor set under the metric... 

Ordinarily, the term phase space refers to the conjunction of configuration and momentum space. We use it here for configuration-velocity space. 

Let us leave, for a moment, the domain of discrete torsion angles and consider torsion-angle moves in a torsion continuum. Much effort has been directed to the development of efficient phase-space sampling methods for MC [17,23,37,128-138]. The simplest off-lattice move - being a pivotal move [128,131] of a single torsion angle at a time - hardly samples configuration... 

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