Master equation analytical solutions

In the examples of the previous section, and in most other applications, the coefficients rn and gn are not just a collection of numbers, but are given as simple analytic functions r(n), g(n) of the variable n. If that were not so, there could be no hope to find explicit solutions (unless the number of states is very small). However, it also implies that the special equations (1.3) and (1.5) at the boundaries are to be taken seriously and cannot be incorporated in the general equation by the simple trick described in (1.4) and (1.6) without spoiling the analytic character. Hence it is necessary, in the case of two boundaries, to write the master equation in three separate lines,... 

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. 

For linear systems, the differential equation for the jth cumulant function is linear and it involves terms up to the jth cumulant. The same procedure will be followed subsequently with other models to obtain analogous differential equations, which will be solved numerically if analytical solutions are not tractable. Historically, numerical methods were used to construct solutions to the master equations, but these solutions have pitfalls that include the need to approximate higher-order moments as a product of lower moments, and convergence issues [383]. What was needed was a general method that would solve this sort of problem, and that came with the stochastic simulation algorithm. 

Figure 28 shows the N distribution at t = tmax and for times t > tmax. In Fig. 28a we have also reported the instantaneous Treanor s distributions at the relevant 0i(t) s values. These distributions, which are analytical solutions of the system of vibrational master equations including v-v rates only, should represent upper limits to the actual Nv distributions, which include all v-t deactivating processes. 

The analytic solution of the master equation decomposes the flow of probability into a series of exponentially decaying modes, each of which has a characteristic decay constant. It is instructive to look at how the contributions from these modes vary across the spectrum of time scales. From Eq. (1.41) mode j makes an important contribution to the probability evolution of minimum / if is large in magnitude. The mode... 

No other analytic solution to the master equation for a weak collision system over the whole range of pressures has yet been found. A solution is known, at the low pressure limit only for a rather limited exponential probability model of a unimolecular reaction [77.T2 80.F1], and Troe has developed empirical schemes for determining the pressure range over which the fall-off exhibits curvature and for joining smoothly the high and low pressure limiting solutions [77.Q 79.T2]. 

An analytical solution to the master equation is only possible for very simple systems. The master equation, however, can readily be simulated by using stochastic kinetics or more specifically kinetic Monte Carlo simulation. Several Monte Carlo algorithms exist. More details on kinetic Monte Carlo simulation can be found in the Appendix. 

The master equation, however, can only be solved analytically for very simple systems such as the gas-phase reaction A—>B. The analysis of these systems typically requires numerical simulation of a lattice-based kinetic Monte Carlo model. The lattice gas model can then be used to formulate the respective transition probabilities in order to solve the master equationThe groups of both Zhdanov[ ° ° ] and Kreuzerl ° l have been instrumental in demonstrating the application of lattice gas models to solve adsorption and desorption processed from surfaces. Once a lattice model has been formulated there are three types of solution ... 

Obviously, the preceding hierarchy of master density equations can also be closed at v = N. However, the product density equations may allow closure at a considerably lower value of r, which makes them much more attractive to solve than the master density equations. As pointed out earlier, even an analytical solution to the master density equation is not particularly valuable because of its combinatorial complexity. 

We now return to the master differential equation, Eqn. 107. We require an analytical solution to this nonlinear problem to derive an expression for the flux and hence an expression for the current. Much of the previous work described in the literature employed a numerical finite-difference approach to this problem, although Albery and coworkers and Lyons et presented an approximate analytical... 

The problem of fluctuation lissipation relations in multivariable systems is analyzed in [15] the mathematics needed for that task goes beyond the level chosen for this book, and hence only a brief verbal smnmary is presented. A statistical ensemble is chosen, which consists of a large number of replicas of the system, such as for example the Selkov model, each characterized by different composition vectors. There exists a master equation for this probability distribution of this ensemble, which serves as a basis for this approach an analytical solution of this master equation is given in [15]. 

It is possible to simplify the master equation in some limiting conditions and to obtain analytical solutions. 

Under various restrictive conditions, it is possible to obtain analytical solutions to the master Equation [3.7], These analytical solutions are provided here without... 

If (4.17, 34, 35) are incorporated into the master equation (4.18) and by using (4.22, 23, 28) in the mean value and variance equations (4.29, 31) as well as in the Fokker-Planck equation (4.32), the explicit form for all kinds of equations of motion is obtained. Relevant cases for which solutions can be obtained either analytically or by numerical methods will now be considered. 

In the next section, we present a simple example that accepts an analytic solution of the master equation, in order better to illustrate the derivation and solution of Eq. 13.15. 

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