Pauli-Master equation

Generalized first-order kinetics have been extensively reviewed in relation to teclmical chemical applications [59] and have been discussed in the context of copolymerization [53]. From a theoretical point of view, the general class of coupled kinetic equation (A3.4.138) and equation (A3.4.139) is important, because it allows for a general closed-fomi solution (in matrix fomi) [49]. Important applications include the Pauli master equation for statistical mechanical systems (in particular gas-phase statistical mechanical kinetics) [48] and the investigation of certain simple reaction systems [49, ]. It is the basis of the many-level treatment of... 

Figure A3.13.14. Illustration of the quantum evolution (pomts) and Pauli master equation evolution (lines) in quantum level structures with two levels (and 59 states each, left-hand side) and tln-ee levels (and 39 states each, right-hand side) corresponding to a model of the energy shell IVR (liorizontal transition in figure...

We shall use the projection operator method to derive the Pauli master equation. With the Liouville equation, we separate the Liouville operator into two parts ... 

One drawback of the name is that it is not specific enough to forbid the use of it for equations of a different type, which creates confusion.810 Sometimes the more specific name Pauli master equation is used. 5,0 The other drawback is that it cannot be translated. In the Polish literature the name M-equation has been introduced (following a suggestion of M. Kac), which does not suffer from these drawbacks. I shall also use it occasionally but it may be too late to make this more international usage prevail. 

The outline of this review is as follows. In Section II we discuss the Pauli master equation and we show how to derive it from a coherent quantum mechanical process through the addition of a dephasing process. 

Section III is devoted to illustrating the first theoretical tool under discussion in this review, the GME derived from the Liouville equation, classical or quantum, through the contraction over the irrelevant degrees of freedom. In Section III.A we illustrate Zwanzig s projection method. Then, in Section III.B, we show how to use this method to derive a GME from Anderson s tight binding Hamiltonian The second-order approximation yields the Pauli master equation. This proves that the adoption of GME derived from a Hamiltonian picture requires, in principle, an infinite-order treatment. The case of a vanishing diffusion coefficient must be considered as a case of anomalous diffusion, and the second-order treatment is compatible only with the condition of ordinary... 

The master equation approach has a long story tracing back to the pioneering work of Pauli [6] and van Hove [7], motivated by the need of reconciling quantum mechanical coherence with the evident randomness of statistical physics. In this section we illustrate this transition from coherent to incoherent behavior with a simple, but paradigmatic, model that will be repeatedly adopted in this review. Let us imagine a quantum system with two states, 11) and 2). The incoherent process described by the Pauli master equation is illustrated by the following set of two equations... 

On behalf of the discussion that we shall present in this review, it is convenient to rewrite the Pauli master equation of Eq. (1) in the following... 

Of course the diagonal density elements pj 1 (f), p22(t) are identified with the probabilities p (t) and pi(t), respectively. One of the major tasks of this review is to point out the difficulties that are currently met with the extension of this interpretation to the case where the Markov condition of the ordinary Pauli master equation does not apply. 

In the special case where the site energies are random fluctuations, this is the Anderson model [20,21]. It is well known that Anderson used this model to prove that randomness makes a crystal become an insulating material. Anderson localization is subtly related to subdiffusion, and consequently this important phenomenon can be interpreted as a form of anomalous diffusion, in conflict with the Markov master equation that is frequently adopted as the generator of ordinary diffusion. It is therefore surprising that this is essentially the same Hamiltonian as that adopted by Zwanzig for his celebrated derivation of the van Hove and, hence, of the Pauli master 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 the special case where there are only two sites, the CTRW procedure, supplemented by the Poisson assumption of Eq. (69), yields the Pauli master equation of Eq. (2). This means that the Pauli master equation is compatible with a random picture, where a particle with erratic motion jumps back and forth from the one to the other state, with a condition of persistence expressed by the exponential waiting time distribution of Eq. (69). Recent fast technological advances are allowing us to observe in mesoscopic systems analogous intermittent properties, with distinct nonexponential distribution of waiting times. This is the reason why in this section we focus our attention on how to derive a v(/(t) with a non-Poisson character. 

Energy migration among a number of chromophores with inhomogeneously broadened spectra can be modeled using a Pauli master equation approach [10, 27, 70, 71, 103-107] as long as the excitation is localized as it hops from... 

The Pauli Master equation approach to calculating RET rates is particularly useful for simulating time-resolved anisotropy decay that results from RET within aggregates of molecules. In that case the orientation of the aggregate in the laboratory frame is also randomly selected at each Monte Carlo iteration in order to account for the rotational averaging properly. 

Consider a closed system characterized by a constant temperature T. The system is prepared in such a way that molecules in energy levels are distributed in departure from their equilibrium distribution. Transitions of molecules among energy levels take place by collisional excitation or deexcitation. The redistribution of molecular population is described by the rate equation or the Pauli master equation. The values for the microscopic transition probability kfj for transition from ith level toyth level are, in principle, calculable from quantum theory of collisions. Let the set of numbers vr be vibrational quantum numbers of the reactant molecule and vp be those of the product molecule. The collisional transitions or intermolecular relaxation processes will be described by ... 

For a system satisfying these assumptions, the excitation dynamics may be described by a Pauli master equation [14]. 

The theory presented here is limited to the transport of incoherent excitations, describable with a Pauli master equation. In addition, chromophore diffusion resulting from Brownian forces on the chains is assumed to be negligible on the time scale of excitation transport. However, no assumptions are made about the density of the material, so the theory should be applicable to homogeneous melts and to polymers dispersed in amorphous solids, as well as to chains in solution. 

A semiquantitative model of the effects of L on Ip>F rid Ip was given by Webber and Swenberg (11). These authors analyzed the dependence of exciton annihilation on parameters such as lattice size, lattice dimensionality and the exciton fusion rate using a Pauli-Master equation. Their treatment is quite similar to the formalism of Paillotin et (12) used in modeling the fluorescence quantum yield decreases observed in photosynthetic membranes (13) with increasing excitation... 

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