Master equation method

Master equation methods are not tire only option for calculating tire kinetics of energy transfer and analytic approaches in general have certain drawbacks in not reflecting, for example, certain statistical aspects of coupled systems. Alternative approaches to tire calculation of energy migration dynamics in molecular ensembles are Monte Carlo calculations [18,19 and 20] and probability matrix iteration [21, 22], amongst otliers. 

We have already commented that the master equation method is not suitable, at present, to handle reactive products because, inter alia, the dimensionality of the problem increases with reactions of products. There is no difficulty, in principle, to including reactive products in Monte Carlo simulation, since the time of reaction and the positions of the products can be recorded. In practice, however, this requires a greatly expanded computational effort, which is discouraging. 

For both cluster ions, the ZTRID rates for Equations (24) and (25) at room temperature (Table 6) were analyzed by the master equation method, the ab initio calculated IR intensities being used, to give the fitted values of Eq displayed in the Table. In addition, the temperature dependence of Equation (25) was fitted to the master equation simulation (the E approach), to give an independent value of Ef (0.31 eV) in excellent agreement with the value from the approach (0.32 eV). These values were corrected to 298 K by use of Equations (3) and (4) to give the BDE values displayed as AH (ZTRID). 

When the coupling to the leads is weak, electron-electron interaction results in Coulomb blockade, the sequential tunneling is described by the master equation method [169-176] and small cotunneling current in the blockaded regime can be calculated by the next-order perturbation theory [177-179], This theory was used successfully to describe electron tunneling via discrete... 

In the single site model with two (spin-up and spin-down) levels it is possible to make the direct comparison between our Ansatz and the master equation methods. For the latter, we used the well known master equations for quantum dots [180,181],... 

It is quite clear from the presented figures that our Ansatz and the master equation method give essentially the same results in the limit of weak coupling... 

It is important that our Ansatz can be applied straightforwardly to multilevel systems in the case when the exact eigenstates of an isolated system are unknown and the usage of the master equation method is not easy. In this paper we consider the simplest example of such a system, namely a double site case. 

The general strategy of attack to the molecular dynamics outlined by this and earlier chapters appears especially promising to shed further light into this stimulating field of research In the special case of H-bonded liquids, a natural development of the ideias outlined here implies the replacement of the discrete variable t) with a continuous variable, which in turn involves the replacement of the master equation method with a suitable Fokker-PIanck equation. Moreover, this improvement of the theory is fundamental to exploring the short-time dynamics when the details of the correlation functions on the time scale of structure V of water must be accounted for. 

There are several theoretical approaches that can be used to calculate the dynamics and correlation properties of two atoms interacting with the quantized electromagnetic held. One of the methods is the wavefunction approach in which the dynamics are given in terms of the probability amplitudes [9]. Another approach is the Heisenberg equation method, in which equations of motion for the atomic and held operators are found from the Hamiltonian of a given system [10], The most popular approach is the master equation method, in which the equation of motion is found for the density operator of an atomic system weakly coupled to a system regarded as a reservoir [7,8,41], There are many possible realizations of reservoirs. The typical reservoir to which atomic systems are coupled is the quantized three-dimensional multimode vacuum held. The major advantage of the master equation is that it allows us to consider the evolution of the atoms plus held system entirely in terms of atomic operators. 

The time scales of the structural transitions in (NaCl)35Cl mean that it is impossible to use conventional molecular dynamics to investigate the interfunnel dynamics. Instead we use the master equation method outlined in Section III.D. To reduce the computational expense and numerical difficulties we recursively removed from our sample those minima that are only connected to one other minimum—these dead-end minima do not contribute directly to the probability flow between different regions of the PES. The resulting pruned sample had 1624 minima and 2639 transition states. RRKM theory in the harmonic approximation was used to model the individual rate constants,. ... 

Abstract The intermonomer interaction effect on electromagnetically induced transparency (BIT) in dipole-coupled dimer models with different orientations and intermonomer distances is investigated. The absorption properties are evaluated using the imaginary part of the dynamic polarizability a calculated by the quantum master equation method. It is found that BIT can be observed even in the dimer systems with near-degenerate excited states originating in an intermonomer interaction by adjusting the incident field frequency. 

Here a simple extension of the master equation method developed for macromolecules in solution [8.4, 8.22] is used to model correlated internal motions in liquid crystals. By explicitly generating all of the possible conformations in a mesogen and weighing these conformers according to their equilibrium probabilities imposed by the nematic mean field [8.12, 8.14], those improbable conformations that were obtained based on the assumption of independent rotations about different C-C bonds may be effectively eliminated. Thus, internal rotations about different axes are considered to be highly correlated. A similar approach has been used to model correlated internal motions in lamellar mesophases of lyotropic liquid crystals [8.20]. All of the studies still retain the simplifying assumption of decoupling internal rotations from the reorientation of the whole molecule. First, the decoupled model of correlated internal motions is considered. 

First, it is necessary to find the configuration conditional probability p(i,t /,0) = Piioit) in Eq. (8.30) in order to obtain the internal correlation functions. This is achieved by following the master equation method of Wittebort and Szabo [8.4]. Conformational transitions between N distinct configurations occur via one-bond, two-bond, or three-bond motion [8.24] in the chain. These bond motions are characterized by phenomenological rate constants and fcs, respectively. In general, there is more than... 

Chachaty and co-workers [8.20, 8.37, 8.38] were first to describe correlated internal motions in alkyl chains of surfactant molecules that form lyotropic liquid crystals. The last section described an extension of the master equation method of Wittebort and Szabo [8.4] to treat spin relaxation of deuterons on a chain undergoing trans-gauche jump rotations in liquid crystals. This method was also followed by Chachaty et al. to deal with spin relaxation of nuclei in surfactants. However, they assumed that the conformational changes occur by trans-gauche isomerization about one bond at a time. In their spectral density calculations (see Section 8.3.1), they used a transition rate matrix that was constructed from the jump rate Wi, W2, and Ws about each bond. Since W3 is much smaller than Wi and W2, the time scale of internal motions was practically governed by Wi and W2 of each C-C bond. Since... 

Miller JA, Klippenstein SJ (2006) Master equation methods in gas phase chemical kinetics. J Phys Chem A 110 10528-10544... 

Rohertsmi SH, Pilling MJ, Jitaiiu LC, Hillier IH (2007) Master equation methods for multiple well systems applicatimi to the 1-,2-pentyl system. Phys Ghent Ghem Phys 9 4085—4097... 

Different stochastic methods have been considered to overcome the accretion limit problem. The main focus has been on the master equation method [47 9] and macroscopic Monte Carlo simulations [49-51]. Of the two, the master equation method can be more easily coupled to rate equations, which handle the gas phase chemistry in the most straightforward way, since it treats the gas and surface kinetics with equations of similar form. 

Choose the master equation method (FEP, TI, or slow growth), the number of windows to be calculated, the amount of equilibration to be performed at each window, and the amount of data collection to be performed at each window. Typically, if high precision results are desired for a simulation, the total data collection time over the simulation will be several hundred picoseconds or more. Parameters associated with standard MD or MC will also need to be set (step size, effective temperature, etc.). 

---