Correlation function dichotomous fluctuations

Section IX is devoted to the important task of revealing the physical reason of the breakdown of the equivalence between density and trajectory perspective in the non-Poisson case. First of all, in Section IX.A, we show that several theoretical approaches to the generalized fluctuation-dissipation process rest implicitly on the assumption that the higher-order correlation functions of a dichotomous noise are factorized. In Section IX.B we show that the non-Poisson condition violates this factorization property, thereby explaining the departure of the density from the trajectory approach in the non-Poisson case. 

This is a crucial aspect subtly related to the main goal of this review. In Section VIII we have seen that the non-Poisson condition violates the DF property. This is a prescription for constructing the correlation functions of the dichotomous fluctuation E,(f), serving the purpose of defining the corresponding diffusion process. In Section XII we have illustrated the problems associated to the derivation of these higher-order correlation functions from a density perspective. It is worth while to address this issue again. 

We apply to this equation the same remarks as those adopted for the comparison among Eqs. (316)—(318). We note, first of all, that the structure of Eq. (325) is very attractive, because it implies a time convolution with a Lindblad form, thereby yielding the condition of positivity that many quantum GME violate. However, if we identify the memory kernel with the correlation function of the 1/2-spin operator ux, assumed to be identical to the dichotomous fluctuation E, studied in Section XIV, we get a reliable result only if this correlation function is exponential. In the non-Poisson case, this equation has the same weakness as the generalized diffusion equation (133). This structure is... 

When the correlation function of inverse power law, the relaxation of the pointer can be evaluated with the trajectory method, and it is proven to be an exponential decay followed by oscillations whose intensity decays with an inverse power law [59]. 

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