Memory kernel density matrix

Hazards of parameter mismatch. In the present model all results are solvable. We know the exact memory kernels in (152) and we can solve for the time evolution from (156). However, in realistic applications, one must ordinarily resort to approximations. In this situation, we get equations of the type (150) but with incorrect values of the parameters in the memory function. In order to display the possible disasters that may ensue, we modify the functions /i and /2 arbitrarily and look at the time evolution of the density matrix. 

Another problem is that memory kernels seem to be delicate entities. Erroneous kernels can destroy the physical sense of the time evolution of an initially acceptable density matrix. We do not have a general criterion to help us judge from the Master Equation with memory if the evolution is acceptable. In the Markovian limit, we know that the Lindblad form is certain to preserve the physical interpretation. It is a challenge for the theory of irreversibility in quantum systems to find such a criterion when memory effects are important. 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

In this section the generalized Langevin equation (GLE) for density correlation functions for molecular liquids is derived based on the memory-function formalism and on the interaction-site representation. In contrast to the monatomic liquid case, all functions appearing in the GLE for polyatomic fluids take matrix forms. Approximation schemes are developed for the memory kernel by extending the successful frameworks for simple liquids described in Sec. 5.1. 

Note that the term open system refers here to exchange of energy and phase with the environment, as the number of particles is conserved throughout. The reduced density matrix p t) evolves coherently under the influence of the nuclear Hamiltonian, Hnuc, and the non-adiahatic effects enter the equation via the dissipative Liouvillian superoperator jSfn- The latter is also termed memory kernel , as it contains information about the entire history of the environmental evolution and its interaction with the system. The definition of the memory kernel is by no means unique nor straightforward. One possible solution is to start from the microscopic Hamiltonian of the total system, eqn (1). Using the projector formalism, it is possible to separate the evolution of the system, i.e., the... 

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