Phase-space integration conservation equations

One property of the exact trajectory for a conservative system is that the total energy is a constant of the motion. [12] Finite difference integrators provide approximate solutions to the equations of motion and for trajectories generated numerically the total energy is not strictly conserved. The exact trajectory will move on a constant energy surface in the 61V dimensional phase space of the system defined by. 

For N particles in a system there are 2N of these first-order equations. For given initial conditions the state of the system is uniquely specified by the solutions of these equations. In a conservative system F is a function of q. If q and p are known at time t0, the changes in q and p can therefore be determined at all future times by the integration of (12) and (13). The states of a particle may then be traced in the coordinate system defined by p(t) and q(t), called a phase space. An example of such a phase space for one-dimensional motion is shown in figure 3. 

Thus, in order to derive the disperse-phase mass- and momentum-conservation equations, it suffices to multiply the GPBE by m(i) and vm( ), respectively, and to integrate over the phase-space variables. 

Let us now discuss in detail the question of moment conservation during time integration. Consistently with Chapter 8, the source terms due to phase-space processes are set to zero so that only transport terms in real space are considered in this discussion. When Eq. (D.23) is integrated using an explicit Euler scheme, the volume-average moment of order k in the cell centered at X at time (n + l)Af is directly calculated from the volume-average moment of order k at time n Af from the following equation ... 

The transitional mode Hamiltonian is given by the last four terms in equation (7.36). In the work of Wardlaw and Marcus, the phase space volume for the transitional modes was calculated versus the center of mass separation R, so that R is assumed to be the reaction coordinate. In recent work, Klippenstein (1990, 1991) has considered a more complex reaction coordinate. The multidimensional phase space volume for the transitional modes can not be determined analytically, but must be evaluated numerically, for example, by a Monte Carlo method of integration (Wardlaw and Marcus, 1984). The density of states is then obtained by dividing the phase space volume by h", where n is the dimensionality of the integral, and differentiating with respect to the energy. The total sum of states of the transition state is obtained by convoluting the density of the transitional modes with the sum of the conserved modes, N(E,J) so that... 

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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