Ohmic bath

Ky represent the frequency and the linear coupling of the yth vibrational mode, respectively. Here, we consider specifically an Ohmic bath described by the spectral density... 

The coupling parameters df. are sampled according to a specified spectral density, which is here taken to be Ohmic [89-91], More generally, the external bath itself can be taken to be non-Markovian. An example of this scheme is given in Fig. 8 of Sec. 5.1, for an Ohmic bath at zero temperature, i.e., exhibiting no thermal fluctuations. Here, the damping effect is generated by quantum fluctuations at T = 0 [90,91],... 

In the case of an ohmic bath, s = 1, the integrand in k(T) scales as 1 /ujp, p = 1, 2 and has thus a logarithmic divergence at the lower integration limit. Thus, the MF contribution would vanish. In other words, no gap would exist on this approximation level. 

After ruling out slow modulation as a possible approach to complexity, we are left with the search for a more satisfactory approach to complexity that accounts for the renewal BQD properties. Is it possible to propose a more exhaustive approach to complexity, which explains both non-Poisson statistics and renewal at the same time We attempt at realizing this ambitious task in Section XVII. In Section XVII.A we show that a non-Ohmic bath can regarded as a source of memory and cooperation. It can be used for a dynamic approach to Fractional Brownian Motion, which, is, however, a theory without critical events. In Section XVIII.B we show, however, that the recursion process is renewal and fits the requests emerging from the statistical analysis of real data afforded by the researchers in the BQD held. In Section XVII.C we explain why this model might afford an exhaustive approach to complexity. 

We would like to attract the attention of the reader to the case when the environment is a source of anomalous diffusion. Paz et al. [116] studied the decoherence process generated by a supra-ohmic bath, but they did not find any problem with the adoption of the decoherence theory. It is convenient to devote some attention to the case when the fluctuation E, is a source of Levy diffusion [59]. If the fluctuation E, is an uncorrelated Levy process, the characteristic function again decays exponentially, and the only significant change is that the... 

In conclusion, we think that in this section an attractive model for BQD complexity emerges. The non-Ohmic bath creates a trajectory x t) with infinite memory. When this trajectory crosses the point x = 0 and for a given time remains in the positive semiplane (x > 0), it activates the photon emission. Then, as a result of this diffusion process the trajectory can re-cross the point x = 0 again, so as to enter the negative semiplane, where it can sojourn for another amount of time. In this region the photon emission process is turned off. 

The relaxation induced by the bath is seen to be entirely detennined by the properties of this spectral function. In particular, a Ohmic bath is defined to have the property... 

In theoretical studies, one usually deals with two simple models for the solvent relaxation, namely, the Debye model with the Lorentzian form of the frequency dependence, and the Ohmic model with an exponential cut-off [71, 85, 188, 203]. The Debye model can work well at low frequencies (long times) but it predicts nonanalytic behavior of the time correlation function at time zero. Exponential cut-off function takes care of this problem. Generalized sub- and super-Ohmic models are sometimes considered, characterized by a power dependence on CO (the dependence is linear for the usual Ohmic model) and the same exponential cut-off [203]. All these models admit analytical solutions for the ET rate in the Golden Rule limit [46,48]. One sometimes includes discrete modes or shifted Debye modes to mimic certain properties of the real spectrum [188]. In going beyond the Golden Rule limit, simplified models are considered, such as a frequency-independent (strict Ohmic) bath [71, 85, 203], or a sluggish (adiabatic)... 

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