Reduced density matrices for dissipative dynamics

The extension of the previous treatment to a system in a medium starts from the L-vN equation for t t) to derive the equation of motion of the RDOp p for the primary (or p-) region, that includes a dissipative term, given by the LiouviUe superoperator (t). This describes the interaction with a secondary (or s-) region. The PWT is applied only to the p-region so that the quantum and classical Hamiltonians of the previous section refer to the p-region the s-region is described in terms of its collective motions and a distribution of initial motion amplitudes [9,14]. 

The dissipative term in the L-vN equation has been derived in several ways. Expressions for the dissipative superoperator p (t) can be derived from the dynamics of the s-region, and could therefore be constructed from information about the electronic and atomic structure of this region. Three models of current interest are based on dissipative potentials [14,25], dissipative rate operators [26-29], and a second-order perturbation theory [4,6,30,31,40] with a coupling of p- and s-regions of the bilinear form = We develop the PWT treatment in the following, in terms applicable to some of these approaches. 

A popular choice for the dissipative superoperator follows from the so-called Lindblad-type expression [26,27], which amounts in our notation to 

The PWT of the resulting equation of motion can be obtained expressing a product [ABCjw in terms of each operator PWT, and keeping the first order 

The resulting equation of motion for pw also leads to a new equation for the phase space density y, after taking the trace over quantum variables. 

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