Liouville, stochastic equation

The dynamical behaviors of p(At) v and p(At)av av, have to be determined by solving the stochastic Liouville equation for the reduced density matrix the initial conditions are determined by the pumping process. For the purpose of qualitative discussion, we assume that the 80-fs pulse can only pump two vibrational states, say v = 0 and v = 1 states. In this case we obtain... 

Brown, D. W., Lindenberg, K. and West, B. J. Energy transfer in condensed media. II. Comparison of stochastic Liouville equations, J.Chem.Phys., 83 (1985), 4136-4143... 

The stochastic Liouville equation, in the form relevant for the ESR line shape calculation, can be written in a form reminiscent of the Redfield equation in the superoperator formulation, Eq. (19) (70-73) ... 

Equation (57) is the stochastic equation of motion for r(/), in which the matrix element 2(/) is a random process. This is similar to Eq. (2). This may be written as a stochastic Liouville equation in the form... 

The stochastic equation of motion of v(t), Eq. (77), can be transformed into a stochastic Liouville equation of the type Eq. (7) if a Markovian process can be properly defined to generate the process of H(t). Then we again obtain Eq. (63) for the conditional expectation V(t) defined by Eq. (60). The line shape function is then given by... 

Introduced by R. Kubo, J. Mathem. Phys. 4, 174 (1963) under the title Stochastic Liouville Equation . The adjective stochastic is here used in the sense of pertaining to stochastic phenomena in contrast to our use as a synonym of random - as in the title of this chapter. 

Consider the non-adiabatic transition a — b shown schematically in Figure 5.2, where a and b may denote the electronic states of D A and D A, respectively. To describe the dynamic processes of the system, one starts with the stochastic Liouville equation... 

It has been shown that for the case in which the pumping and probing lasers do not overlap, one can use the GLRT. In this section, it shall be shown how the GLRT can be applied to calculate the ultrafast time-resolved spectra. For this purpose, start from the stochastic Liouville equation to describe the EOM for the density matrix system embedded in a heat bath... 

It is possible to perform more precise calculations that simultaneously account for the coherent quantum mechanical spin-state mixing and the diffusional motion of the RP. These employ the stochastic Liouville equation. Here, the spin density matrix of the RP is transformed into Liouville space and acted on by a Liouville operator (the commutator of the spin Hamiltonian and density matrix), which is then modified by a stochastic superoperator, to account for the random diffusive motion. Application to a RP and inclusion of terms for chemical reaction, W, and relaxation, R, generates the equation in the form that typically employed... 

The stochastic Liouville equation is highly useful when applied at high field, as techniques exist to reduce in size the typically large matrices it produces, and it has thus been used to simulate electron and nuclear spin polarizations in magnetic resonance experiments.A relatively recent book describes the approach in detail. However, for determining field dependences, such reductions are not possible, meaning that the sizes of the matrices are too large for even modern computers, and so this approach is seldom used for the simulation of field effects. 

In order to better model the effects of J and D without the complexity of the stochastic Liouville equation, Monte Carlo approaches have recently been employed. 

Quantitative calculations of CIDNP intensities can be performed with the stochastic Liouville equation (Eq. 9.3), ... 

Since the Bia value are less than 10 mT for most organic radical pairs, the magnetically induced changes due to the HFCM are usually saturated below 0.1 T. It is noteworthy that no analytical prediction of the magnetic field dependence of Yc (B) and Ye (B) is possible in the case of the HFCM. On the other hand, the quantitative Yc (B) and Ye (B) values can only be obtained by numerical calculations with the stochastic Liouville equation [27]. 

Theoretical analysis of MFEs due to the d-type TM has been carried out with the aid of stochastic Liouvile equations (SLEs). Although exact treatments with SLEs are beyond the scope of this book, an introduction to SLEs will be given in Chapter 11. Applying a SLE to R B) of Eq. (10-9) for a uniform population of triplet exciplexes (r/x =Py = i z), Serebrennikov and Minaev obtained the following result [5] ... 

Theoretical Analysis with the Stochastic Liouville Equation... 

The time dependence of p is given by the stochastic Liouville equation (SLE) ... 

Dekker has studied multiplicative stochastic processes. In his work the stochastic Liouville equation was solved explicitly through first order in an expansion in terms of correlation times of the multiplicative Gaussian colored noise for a general multidimensional weakly non-Markovian process. He followed the suggestions of refs. 17 and 18 and applied, Novikov s theorem. In the general multidimensional case, however, he improved the earlier work by San Miguel and Sancho. ... 

Both vibrational and rotovibrational relaxation can be described analyti-caDy as multiplicative stochastic processes. For these processes, RMT is equivalent to the stochastic Liouville equation of Kubo, with the added feature that RMT takes into account the back-reaction from the molecule imder consideration on the thermal bath. The stochastic Liouville equation has been used successfully to describe decoupling in the transient field-on condition and the effect of preparation on decay. When dealing with liquid-state molecular dynamics, RMT provides a rigorous justification for itinerant oscillator theory, widely applied to experimental data by Evans and coworkers. This implies analytically that decoupling effects should be exhibited in molecular liquids treated with strong fields. In the absence of experimental data, the computer runs described earlier amount to an independent means of verifying Grigolini s predictions. In this context note that the simulation of Oxtoby and coworkers are semistochastic and serve a similar purpose. 

The Fokker-Planck equation to be assodated with Eqs. (4.1) belongs to the family of stochastic Liouville equations. Its expUcit expression is... 

Similar to fluorescence depolarization and NMR, two limiting cases exist in which the molecular motion becomes too slow or too fast to further effect the ESR lineshape (Fig. 8) (35). At the fast motion limit, one can observe a narrow triplet centered around the average g value igxx + gyy + giz with a distance between lines of aiso = Axx- -Ayy- -A2,z)l3, where gu and Ajj are principal values of the g-tensor and the hyperflne splitting tensor A, respectively. At the slow motion limit, which is also referred to as the rigid limit, the spectrum (shown in Fig. 8) is a simple superposition of spectra for all possible spatial orientations of the nitroxide with no evidence of any motional effects. Between these limits, the analysis of the ESR lineshape and spectral simulations, which are based on the Stochastic Liouville Equation, provide ample information on lipid/protein dynamics and ordering in the membrane (36). 

Photoinduced charge separation processes in the supramolecular triad systems D -A-A, D -A -A and D -A-A have been investigated using three potential energy surfaces and two reaction coordinates by the stochastic Liouville equation to describe their time evolution. A comparison has l n made between the predictions of this model and results involving charge separation obtained experimentally from bacterial photosynthetic reaction centres. Nitrite anion has been photoreduced to ammonia in aqueous media using [Ni(teta)] " and [Ru(bpy)3] adsorbed on a Nafion membrane. 

The model proposed by Stillman and Freed (SF) in their 1980 paper [33] is very versatile. By choosing carefully (i) the coupling forces between molecule variables (x,) and augmented ones (x,), and (ii) the potential function in the final equilibrium distribution, one can easily recover a variety of mathematical forms, reflecting different physical cases. The SF procedure starts from considering a system coupled to a second one in a deterministic way (interaction potential) the latter, in the absence of any coupling is described by a FP operator. The first step to obtain a description of the full system is to write the stochastic Liouville equation (SEE), according to Kubo [44] and Freed [45]... 

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