Configurational integral Conservation

If one is to examine the approach to calculating the steady, planar, onedimensional gaseous detonation velocity, one should consider a system configuration similar to that given in Chapter 4. For the configuration given there, it is important to understand the various velocity symbols used. Here, the appropriate velocities are defined in Fig. 1. With these velocities, the integrated conservation and static equations are written as... 

NEMD simulations were started from well-equilibrated, independent, equilibrium configurations. The potential parameters e and a were chosen to be 480 K and 3.405 A, respectively. The simulations were performed at a reduced temperature of k T/e = 1 and at a density of po = 0.64. The equations of motion (Eqs. [202]) were integrated with the additional variable I with a time step of 0.5 fs to ensure an energy conservation (see Eq. [203]) of one part in 10 ... 

Integration of the energy-conserving delta functions over all possible configurations at a temperature T yields the following transfer rate ... 

Fi is the force on particle i caused by the other particles, the dots indicate the second time derivative and m is the molecular mass. The forces on particle i in a conservative system can be written as the gradient of the potential energy, V, C/, with respect to the coordinates of particle /. In most simulation studies, U is written as a sum of pairwise additive interactions, occasionally also three-particle and four-particle interactions are employed. The integration of Eq. (1) has to be done numerically. The simulation proceeds by repeated numerical integration for tens or hundreds of thousands of small time steps. The sequence of these time steps is a set of configurations, all of which have equal probability. The completely deterministic MD simulation scheme is usually performed for a fixed number of particles, iV in a fixed volume V. As the total energy of a conservative system is a constant of motion, the set of configurations are representative points in the microcanonical ensemble. Many variants of these two basic schemes, particularly of the Monte Carlo approach exist (see, e.g.. Ref. 19-23). 

The required integration over deformation histories is accomplished by integrating numerically microscopic particle trajectories for large global ensembles simultaneously with the macroscopic equations of mass and momentum conservation. The term trajectories in the previous sentence refers to both real space trajectories, i.e., positions r, t) and to configurational phase space trajeetories, i.e., in the case of a dumbbell model, connector vector Q. 

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