Damping matrix

Thus, eq. (10-7) implies that the time development of the amplitudes a,(t) is governed by a linear operator which may be represented by a diagonal matrix. We shall refer to this matrix as the damping matrix. 

Fig. 3. The schematic representation of a virtual two photon process which is allowed when there is a nonzero matrix element of the radiative damping matrix rnm connecting two states n and m. This process is negligible unless n and m are nearly degenerate...

Hu (sb) and T(sb) are the molecular Hamiltonian and the damping matrix, respectively, within the states s, 4>b, while A is the matrix of nonradiative decay rates which has only one nonzero matrix element As = Ass within this BO basis set. The states jx, 1 which diagonalize (25) are then the resonant states, each with some energy Er and some radiative and nonradiative decay rates Tr and Ar, respectively. When these resonances are overlapping, i.e.,... 

To make connection with the macroscopic law, we must analyze the structure of the damping matrix. The local density fields can change by both motion of the center of mass of the molecules and by reactive events that change the number of molecules of a given species. Thus we have... 

The dynamic contribution to the damping matrix is also slightly modified because now appears. We have... 

In Appendix B we show that the action of the pseudo-Liouville operator on the singlet field generates a coupling to the doublet field, and so on. One may show that if the phase-space density fields are defined as in (7.5), if density fields up to the th order are included explicitly in the description, the random forces corresponding to all fields lower than the nth are zero. Hence only the damping matrix corresponding to this /ith-order field is nonzero. As an example, consider the case of singlet and doublet fields that are explicitly treated. In this case, (7.14) reduces to two coupled equations of the form... 

From (B.14), we see that when + acts on the doublet field n 2), a contribution proportional to the triplet field n gs(123) is obtained. Thus according to the general formulation set out in Section VII.B, we consider a description based on the two fields 6 ab> abs)- Since the only nonzero damping matrix occurs in the triplet field equation, we may immediately write the following two coupled equations for the phase-space correlation functions. For the doublet field, we have... 

From this we see that the diagonal elements of the damping matrix V are real if the set A is chosen such that X is diagonal. This is always possible. Note that X is diagonal if and only if... 

The same arguments can be applied to the matrix M(f) so that the damping matrix is... 

Let us now consider the damping matrix / . Because A and A2 have different time-reversal symmetry and the random forces F = QiLA and F2 = QiLA2 also have different time-reversal symmetries, it follows from Eq. (11.5.15) that the kinetic coefficients have the symmetry relation... 

Note that there is no energy dissipating mechanism (zero damping matrix) included in this model. Figure 1.2 shows the free vibration response of the mass M50 in the middle of the chain with an zero initial condition except that the mass at the right end (Afioo) has an initial displacement of 0.01 m. The response appears to be random even though the system is deterministic with a simple initial condition and zero excitation. Figures 1.3 and 1.4 zoom into... 

For dynamic problems, besides the stiffness matrix, it is necessary to calculate the inertia matrix and the damping matrix. The inertia matrix is calculated in a similar way to the stiffness matrix, while the damping matrix cannot be obtained in an accurate way at the current state of the art. 

Here the mass and damping matrices are added to the equation. Mass matrices are obtained in a straightforward manner by the Finite Element Method or by other techniques, but in the current state of the art, the way of obtaining approaches for the damping matrix is not sufficiently established. 

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