Memory kernel master equation

It is interesting to notice that Eq. (325) can also be derived from the Lindblad master equation using the same subordination approach as that adopted to derive Eq. (318). Here, however, the memory kernel of this master equation does not have the meaning of a correlation function. 

It is convenient to express the memory kernel of the generalized master equation in terms of the correlation functions of the variable of interest. To do that, rather than using Eq. (2.1) we prefer to have recourse to the corresponding interaction picture ... 

The relationship between the memory kernel P(E,E ) of the master equation and the energy diffusion coefficient D(E) of the diffusion equation is given by the second moment, Eq. (5.2).In many exjjeriments and simulations the average energy transfer jjer unit time can be measured. For kT < E < Q this is evaluated from, Eq. (5.3) to be... 

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). 

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