Markov approximation

Markov approximation for Green s functions of molecular subsystems in the condensed phase... 

On the other hand, there exist well-developed methods for calculating states of subsystems using the Markov approximation for the reduced density matrix... 

Importantly, the value of the results gained in the present section is not limited to the application to actual systems. Eq. (4.2.11) for the GF in the Markov approximation and the development of the perturbation theory for the Pauli equation which describes many physical systems satisfactorily have a rather general character. An effective use of the approaches proposed could be exemplified by tackling the problem on the rates of transitions of a particle between locally bound subsystems. The description of the spectrum of the latter considered in Ref. 135 by means of quantum-mechanical GF can easily be reformulated in terms of the GF of the Pauli equation. 

Figure 23. Radial segment density profile through a cross-section of a highly curved spherical vesicle. The origin is at r = 0, and the vesicle radius is very small, i.e. approximately r = 25 (in units of segment sizes). The head-group units, the hydrocarbons of the tails and the ends of the hydrocarbon tails are indicated. Calculations were done on a slightly more simplified system of DPPC molecules in the RIS scheme method (third-order Markov approximation), i.e. without the anisotropic field contributions...

Thirdly one needs a drastic step to turn this integral equation into a differential equation. This is the Markov approximation , which comes in two varieties. The first variety consists in replacing ps(t — t) with ps(t). The error is of relative order rc/rm (where l/rm is the unperturbed rate of change due to S s) and of absolute order a2T2/rm. In this approximation one may as well omit the S s in the exponent of (4.13) and the result is the same as (3.19). The second variety takes the zeroth order variation of ps into account by setting ps(t — r) = e T sps(t). The result is the same as was obtained in (3.14) by means of the interaction representation, and the only requirement is arc <[Pg.444]

Fig. 3 Absorption spectra of one sample of B850 calculated with different methods for an Ohmic spectral density with / = 3.37 and usc = 0.027 eV. Left Methods with Markov approximation, i.e. Redfield theory with and without secular approximation and TL method. Right TL with and without Markov approximation, TNL and modified Redfield method. (Reproduced from Ref. [37]. Copyright 2006, American Institute of Physics.)...

The RC is an ideal system to test theoretical ideas (memory effect, coherence effect, etc.), fundamental approximations (isolated line approximation, Markov approximation, etc.), and techniques (generalized linear response theory, Forster-Dexter theory, Marcus theory, etc.) for treating ultrafast phenomena. As mentioned above, this ideality is mainly due to the fact that the electronic energy level spacing in RC is small (typically from 200 to 1500 cm-1), and the interactions between these electronic states are weak. 

Although a theoretical approach has been desecrated as to how one can apply the generalized coupled master equations to deal with ultrafast radiationless transitions taking place in molecular systems, there are several problems and limitations to the approach. For example, the number of the vibrational modes is limited to less than six for numerical calculations. This is simply just because of the limitation of the computational resources. If the efficient parallelization can be realized to the generalized coupled master equations, the limitation of the number of the modes can be relaxed. In the present approach, the Markov approximation to the interaction between the molecule and the heat bath mode has been employed. If the time scale of the ultrashort measurements becomes close to the characteristic time of the correlation time of the heat bath mode, the Markov approximation cannot be applicable. In this case, the so-called non-Markov treatment should be used. This, in turn, leads to a more computationally demanding task. Thus, it is desirable to develop a new theoretical approach that allows a more efficient algorithm for the computation of the non-Markov kernels. Another problem is related to the modeling of the interaction between the molecule and the heat bath mode. In our model, the heat bath mode is treated as... 

Let us plug this result in Eq. (39). The exponential decay of the kernel is compatible with the Markov approximation, which yields... 

On the other hand, the adoption of the Markov approximation, although yielding no mathematical inconsistencies, might correspond to annihilating important physical effects. Let us illustrate this important fact with a process related to the Anderson localization issue. Let us depict the Anderson localization as an ordinary fluctuation-dissipation process. Let us consider the Langevin equation... 

In conclusion, in this section we have proved that the Markov approximation requires some caution. The Markov approximation may be incompatible with the quantum mechanical nature of the system under study. It leads to the Pauli master equation, and thus it is compatible with the classical picture of a particle randomly jumping from one site to another, a property conflicting, however, with the rigorous quantum mechanical treatment, which yields Anderson localization. 

In conclusion, the condition of ordinary statistical physics makes the decoherence theory a valuable perspective, as well as an attractive way of deriving classical from quantum physics. The argument that the Markov approximation itself is subtly related to introducing ingredients that are foreign to quantum mechanics [23] cannot convince the advocates of decoherence theory to abandon the certainties of quantum theory for the uncertainties for a search for a new physics. The only possible way of converting a philosophical debate into a scientific issue, as suggested by the results that we have concisely reviewed in this section, is to study the conditions of anomalous statistical mechanics. In the next sections we shall explore with more attention these conditions. 

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