Master equation relaxation

A3.13.3.1 THE MASTER EQUATION FOR COLLISIONAL RELAXATION REACTION PROCESSES... 

The fimdamental kinetic master equations for collisional energy redistribution follow the rules of the kinetic equations for all elementary reactions. Indeed an energy transfer process by inelastic collision, equation (A3.13.5). can be considered as a somewhat special reaction . The kinetic differential equations for these processes have been discussed in the general context of chapter A3.4 on gas kmetics. We discuss here some special aspects related to collisional energy transfer in reactive systems. The general master equation for relaxation and reaction is of the type [H, 12 and 13, 15, 25, 40, 4T ] ... 

Tabor M, Levine R D, Ben-Shaul A and Steinfeld J I 1979 Microscopic and macroscopic analysis of non-linear master equations vibrational relaxation of diatomic molecules Mol. Phys. 37 141-58... 

Levitt M H and Di Bari L 1994 The homogeneous master equation and the manipulation of relaxation networks Bull. Magn. Reson. 16 94-114... 

The relaxation of a system to equilibrium can be modeled using a master equation... 

The relaxation of a thermodynamic system to an equilibrium configuration can be conveniently described by a master equation [47]. The probability of finding a system in a specific state increases by the incoming jump from adjacent states, and decreases by the outgoing jump from this state to the others. From now on we shall be specific for the lattice-gas model of crystal growth, described in the previous section. At the time t the system will be found in the state. S/ with a probability density t), and its evolution... 

In Eqs. (II. 1)—(II.4) we have assumed that there is only one system oscillator. In the case where there exists more than one oscillator mode, in addition to the processes of vibrational relaxation directly into the heat bath, there are the so-called cascade processes in which the highest-frequency system mode relaxes into the lower-frequency system modes with the excess energy relaxed into the heat bath. These cascade processes can often be very fast. The master equations of these complicated vibrational relaxation processes can be derived in a straightforward manner. 

We do not know which donor D has been excited. We therefore assume that immediately after irradiation at t = 0 all sites i have the same excitation probability Pt(0), while all traps T are in the ground state, hence P/(0) = 0. These probabilities change with time because of energy migration, relaxation processes, and trapping. The excitation probability Pt(t) is governed by the following master equation ... 

Immediately when the dynamic interpretation of Monte Carlo sampling in terms of the master equation, Eq. (31), was realized an application to study the critical divergence of the relaxation time in the two-dimensional Ising nearest-neighbor ferromagnet was attempted . For kinetic Ising and Potts models without any conservation laws, the consideration of dynamic universality classespredicts where z is the dynamic exponent , but the... 

Two special cases of the theory illustrate the important features. The first is the relaxation of an ensemble of noninteracting harmonic oscillators in contact with a heat bath, and subject to nearest neighbor transitions in the discrete translational energy space. The master equations which describe the evolution of the ensemble can be written13... 

The Montroll-Shuler equation can also predict how fast a molecule which is created in a highly excited vibrational state will decay to the equilibrium state. This is of interest in connection with chemiluminescence phenomena. In certain cases one finds experimentally that this relaxation is much faster than what one would expect from the master equation of Montroll and Shuler and improved versions of this equation. One possible mechanism for this fast relaxation is that although most of the collisions in which the diatomic molecule participates are between the diatomic molecule and an inert gas atom, there will also be some collisions between diatomic molecules. In the latter case we have the situation where two diatomic molecules in quantum state n collide producing, with fairly high probability, molecules in quantum states n I and n + 1, respectively. The number of such collisions is, of course quite small compared to the number of collisions of the first kind, but since they are so extremely efficient they may still be of importance. This mechanism, we believe, was first suggested in connection with chemiluminescence by Norrish in a Faraday Society discussion.5 The equations describing this relaxation had, however, been discussed several years earlier by Shuler6 and Osipov.7... 

First of all, we define the transition rates for our stochastic model using an ansatz of Kawasaki [39, 40]. In the following we use the abbreviation X for an initial state (07 for mono- and oion for bimolecular steps), Y for a final state (ct[ for mono- and a[a n for bimolecular steps) and Z for the states of the neighbourhood ( cr f 1 for mono- and a -1 a -1 for bimolecular steps). If we study the system in which the neighbourhood is fixed we observe a relaxation process in a very small area. We introduce the normalized probability W(X) and the corresponding rates 8.(X —tY Z). For this (reversible) process we write down the following Markovian master equation... 

As a final note, it has to be stressed out that Eqs. (4.49) and (4.50) and Eqs. (4.52) and (4.53) hold for an arbitrary stochastic process. These evolution equations cannot give any information about whether or not the process is Markovian.135 The master equation concept has been used to analyze some examples of multistate relaxation processes.139... 

The renormalized contributions come from the influence of bath. The terms Dj represents the shift of the /-th mode in the bath for the m-th electronic level. The master equation (10), as it stands, holds true after relaxation process in the bath appears, so the contribution from the bath corresponds to relaxed bath states. It is worth mentioning that although the constant term disappears in the interaction term (6) for the vertical transition, it is... 

The effects of the vibronic coupling competing with vibrational and/or vibronic relaxation on the femtosecond pump-probe stimulated emission spectra of molecules in condensed phases have been investigated. Taking into account vibronic and vibrational relaxation and vibronic coupling in molecular terms, the coupled master equations have been briefly derived for... 

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... 

In general the relaxation to equilibrium of E(t) is nonexponential, since the rate matrix in the master equation has an infinite number of (in principle) nondegenerate eigenvalues if there are an infinite number of states n). There are, however, two instances where the relaxation is approximately exponential. In the first instance one assumes that the initial nonequilibrium state has appreciable population only in the first two oscillator eigenstates, and further that k,. 0 k,, m and k0. t k0 m for m > 2. If one neglects terms involving these small rate constants, the master equation reduces to a pair of coupled rate equations for a two-level system ... 

It seems that the conventional approach to the quantum mechanical master equation relies on the equilibrium correlation function. Thus the CTRW method used by the authors of Ref. 105, yielding time-convoluted forms of GME [96], can be made compatible with the GME derived from the adoption of the projection approach of Section III only when p > 2. The derivation of this form of GME, within the context of measurement processes, was discussed in Ref. 155. The authors of Ref. 155 studied the relaxation process of the measurement pointer itself, described by the 1/2-spin operator Ez. The pointer interacts with another 1/2-spin operator, called av, through the interaction Hamiltonian... 

Pfadenhauer and McCain (179) and Tarasov and Buslaev (180) studied the hexafluoroscandate ion in terms of two dynamic processes affecting the line-shape of the resonance, namely chemical exchange involving fluoride ions on the one hand and " Sc relaxation on the other hand. The latter two authors tackled the problem by solving the Kubo-Sack master equation for chemical exchange using... 

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