Master equations fields

Before trying to solve the master equation for growth processes by direct stochastic simulation it is usually advisable to first try some analytical approximation. The mean-field approximation often gives very good results for questions of first-order phase transitions, and at least it provides a qualitative understanding for the interplay of the various model parameters. 

Mean Field Approximation as a first order approximation, we will ignore all correlations between values at different sites and parameterize configurations purely in terms of the average density at time t p. The time evolution of p under an arbitrary rule [Pg.73]

The methodology of stochastic treatment of e-ion recombination kinetics is basically the same as for neutrals, except that the appropriate electrostatic field term must be included (see Sect. 7.3.1). This means the coulombic field in the dielectric for an isolated pair and, in the multiple ion-pair case, the field due to all unrecombined charges on each electron and ion. All the three methods of stochastic analysis—random flight Monte Carlo (MC), independent reaction time (IRT), and the master equation (ME)—have been used (Pimblott and Green, 1995). 

Previously, stochastic Schrodinger equations for a quantum Brownian motion have been derived only for the particle component through approximated equations, such as the master equation obtained by the Markovian approximation [18]. In contrast, our stochastic Schrodinger equation is exact. Moreover, our stochastic equation includes both the particle and the field components, so it does not rely on integrating out the field bath modes. 

Fig. 25 Comparison of the predictions of various models for current injection from a metal electrode into a hopping system featuring a Gaussian DOS of variance a = 15 meV as a function of the electric field at different temperatures. The ID continuum and the 3D master equation model have been developed by van der Holst et al. [127]. The calculations based upon the Burin-Ramer and the Arkhipov et al. models are taken from [175] and [170] respectively. Parameters are the sample length L, the intersite separation a and the injection barrierA. From [127] with permission. Copyright (2009) by the American Institute of Physics...

According to the second law, (W) = Q) > 0, which implies that the average switching field is positive, (// ) > 0 (as expected due to the time lag between the reversal of the field and the reversal of the dipole). The work distribution is just given by the switching field distribution p H ). This is a quantity easy to compute. The probability that the dipole is in the down state at field H satisfies a master equation that only includes the death process,... 

The basic remark is that linearity of the macroscopic law is not at all the same as linearity of the microscopic equations of motion. In most substances Ohm s law is valid up to a fairly strong field but if one visualizes the motion of an individual electron and the effect of an external field E on it, it becomes clear that microscopic linearity is restricted to only extremely small field strengths.23 Macroscopic linearity, therefore, is not due to microscopic linearity, but to a cancellation of nonlinear terms when averaging over all particles. It follows that the nonlinear terms proportional to E2, E3,... in the macroscopic equation do not correspond respectively to the terms proportional to E2, E3,... in the microscopic equations, but rather constitute a net effect after averaging all terms in the microscopic motion. This is exactly what the Master Equation approach purports to do. For this reason, I have more faith in the results obtained by means of the Master Equation than in the paradoxical result of the microscopic approach. 

In terms of the master equation for the Markov process the formal kinetics is nothing but the mean-field theory where the fluctuation terms like that on the r.h.s. of equation (2.2.43) are neglected. Strictly speaking, the macroscopic description, equation (2.1.2), were correct if the fluctuation terms vanished as V —> oo. In a general case the function P(N, t) does not satisfy the Poisson distribution [16, 27] in particular, °N (N> ... 

In such a representation of an infinite set of master equations for the distribution functions of the state of the surface and of pairs of surface sites (and so on) will arise. This set of equations cannot be solved analytically. To handle this problem practically, this hierarchy must be truncated at a certain level. In such an approach the numerical part needs only a small amount of computer time compared to direct computer simulations. In spite of very simple theoretical descriptions (for example, mean-field approach for certain aspects) structural aspects of the systems are explicitly taken here into account. This leads to results which are in good agreement with computer simulations. But the stochastic model successfully avoids the main difficulty of computer simulations the tremendous amount of computer time which is needed to obtain good statistics for the results. Therefore more complex systems can be studied in detail which may eventually lead to a better understanding of such systems. 

Real catalytic reactions upon solid surfaces are of great complexity and this is why they are inherently very difficult to deal with. The detailed understanding of such reactions is very important in applied research, but rarely has such a detailed understanding been achieved neither from experiment nor from theory. Theoretically there are three basic approaches kinetic equations of the mean-field type, computer simulations (Monte Carlo, MC) and cellular automata CA, or stochastic models (master equations). 

In this Section we introduce a stochastic alternative model for surface reactions. As an application we will focus on the formation of NH3 which is described below, equations (9.1.72) to (9.1.76). It is expected that these stochastic systems are well-suited for the description via master equations using the Markovian behaviour of the systems under study. In such a representation an infinite set of master equations for the distribution functions describing the state of the surface and of pairs of surface sites (and so on) arises. As it was told earlier, this set cannot be solved analytically and must be truncated at a certain level. The resulting equations can be solved exactly in a small region and can be connected to a mean-field solution for large distances from a reference point. This procedure is well-suited for the description of surface reaction systems which includes such elementary steps as adsorption, diffusion, reaction and desorption.The numerical part needs only a very small amount of computer time compared to MC or CA simulations. 

In Fig. 3, the simulation results for the same model problem are presented using the QCLE, the master equation, Tully s surface-hopping approach, the mean field approach, and adiabatic dynamics. The algorithmic details of each approach can be found elsewhere [2,40,79]. 

Fig. 3 Forward rate coefficient kAB(t) as a function of time for f3 = 1.0. The upper (blue) curve is the adiabatic rate, the purple curve is the result obtained by Tully s surface-hopping algorithm, the middle (black) curve is the quantum master equation result, the green curve is the QCL result, and the lowest dashed line (grey) is the result using mean-field dynamics.

Chemical kinetic rate methods including conventional transition state theory (TST), canonical variational transition state theory (CVTST) and Rice-Ramsper-ger-Kassel-Marcus in conjunction with master equation (RRKM/ME) and separate statistical ensemble (SSE) have been successfully applied to the hydrocarbon oxidation. Transition state theory has been developed and employed in many disciplines of chemistry [41 4]. In the atmospheric chemistry field, conventional transition state theory is employed to calculate the high-pressure-limit unimole-cular or bimolecular rate constants if a well-defined transition state (i.e., a tight... 

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. 

Of the three QM-FEP methods which will be outlined here the QM implementation of the dual topology method is the most similar to the techniques which are used with MM systems. However, as opposed to a dual-topology classical force field calculation in which the interactions are pairwise additive such that only a limited subset of terms needs to be recalculated, a QM method requires two full SCF calculations. This method is readily understood by considering basic FEP theory. In the equation below we give the basic master equations for the FEP method. 

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