A general stochastic model

In Section 9.1.1 we have introduced a stochastic model for the description of surface reaction systems which takes correlations explicitly into account but neglects the energetic interactions between the adsorbed particles as well as between a particle and a metal surface. We have formulated this by master equations upon the assumption that the systems are of the Markovian type. In the model 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) arise. This chain of equations cannot be solved analytically. To handle this problem practically this hierarchy was truncated at a certain level. The resulting equations can be solved numerically exactly in a small region and can be connected to a mean-field solution for large distances from a reference point. 

Therefore the model avoids two main difficulties the large amount of computer time which is normally needed for simulations and the loss of structural information which occurs in simple theoretical models (mean-field models) which do not take into account the structural aspects of the adsorbate layer. Mean-field-kind models fail in the prediction of phase transitions of the second order because at these points the long-range correlations appear. They also fail in describing the system s behaviour in the neighbourhood of the point of first-order kinetic phase transition. 

We have applied our model to the A - - B2 0 reaction and compared it in Section 9.1.2 with results of computer simulations. We found that the results are in very good agreement with each other. Disordered surfaces were treated within the stochastic approach in Sections 9.1.3 and 9.1.4. Lastly, in this Section we introduce energetic interactions into the model defined earlier. We define a standard model in order to compare different surface reactions which are modeled using this theoretical ansatz. We show that in the case when energetic interactions are neglected, the model reduces to that presented earlier in Section 9.1.1. 

Definitions. We consider a lattice with coordination number z. To each lattice site is given a lattice vector 1. The state of the site I is represented by the lattice variable ct/ which may depend on the state of the catalyst site (e.g., promoted or not) and on its coverage with a particle. Here we deal only with the simple case in which all sites are identical and therefore ai depends only on its coverage (the other case is explained in Section 9.1.4. Therefore cri G 0,A,B. where 0 represents a vacant site, A is a site which is occupied by an A particle and so on. Next we want to define a variable which is unity for the case that I and n are the nearest neighbour sites on the lattice and zero otherwise. Sometimes we will use the abbreviation ai = A, cr = an = and a n = v. The states of the neighbourhood z sites) of site I are denoted by cr f. 

Monomolecular steps. We study the processes which depends only on one lattice site. Examples for such steps are the creation of a particle (0 A), 

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J. Mai, V. N. Kuzovkov, and W. von Niessen, A General Stochastic Model for the Description of Surface Reaction Systems, Physica A, 203... 

Thus OAS is a general stochastic model, with discount rates derived from the standard benchmark term structure of interest rates. This is an advantage over more traditional methods in which a single discount rate is used. The calculated spread is a spread over risk-free forward rates, accounting for both interest-rate uncertainty and the price of default risk. As with any methodol-ogy, OAS has both strengths and weaknesses however, it provides more realistic analysis than the traditional yield-to-maturity approach. Hence, it has been widely adopted by investots since its introduction in the late 1980s. 

In the past, the equivalence between the size distribution generated by the Smoluchowski equation and simple statistical methods [9, 12, 40-42] was a source of some confusion. The Spouge proof and the numerical results obtained for the kinetics models with more complex aggregation physics, e.g., with a presence of substitution effects [43,44], revealed the non-equivalence of kinetics and statistical models of polymerization processes. More elaborated statistical models, however, with the complete analysis made repeatedly at small time intervals have been shown to produce polymer size distributions equivalent to those generated kinetically [45]. Recently, Faliagas [46] has demonstrated that the kinetics and statistical models which are both the mean-field models can be considered as special cases of a general stochastic Markov process. 

A similar representation of a pictorial vector model for CIDEP processes is more difficult to formulate, but recently Monchick and Adrian (100) have succeded in casting the stochastic Liou-ville model of CIDEP into the form of a "Block-type" equation with diffusion which led to a generalized vector model of the radical-pair mechanism to give a clear qualitative picture of both CIDNP and CIDEP. 

The stochastic model accepts a Markov type connection between both elementary states. So, with ai2Ar, we define the transition probability from type I to type II, whereas the transition probability from type II to a type I is a2iAr. By Pi(x,t) and P2(x, t) we note the probability of locating the microparticle at position x and time T with a type I or respectively a type II evolution. With these introductions and notations, the general stochastic model (4.71) gives the particularization written here by the following differential equation system ... 

In the second paper, Thomas considers a related problem but incorporates a general stochastic demand function and backlogging of excess demand. Specifically, Thomas considers a periodic review, finite horizon model with a fixed ordering cost and stochastic, price-dependent demand. The paper postulates a simple policy, referred to by Thomas as (5,5,p), which can be described as follows. The inventory strategy is an (5, S) policy If the inventory level at the beginning of period t is below the reorder point, st, an order is placed to raise the inventory level to the order-up-to level, St. Otherwise, no order is placed. Price depends on the initial inventory level at the beginning of the period. Thomas provides a counterexample which shows that when price is restricted to a discrete set this policy may fail to be optimal. Thomas goes on to say ... 

A general theory of the equilibrium polycondensation of an arbitrary mixture of monomers, described by the FSSE model, has been developed [75]. Proceeding from rigorous thermodynamic considerations a branching process has been indicated which describes the chemical structure of condensation polymers and expressions have been derived which relate the probability parameters of this stochastic process to the thermodynamic parameters of the FSSE model. 

In summary, models can be classified in general into deterministic, which describe the system as cause/effect relationships and stochastic, which incorporate the concept of risk, probability or other measures of uncertainty. Deterministic and stochastic models may be developed from observation, semi-empirical approaches, and theoretical approaches. In developing a model, scientists attempt to reach an optimal compromise among the above approaches, given the level of detail justified by both the data availability and the study objectives. Deterministic model formulations can be further classified into simulation models which employ a well accepted empirical equation, that is forced via calibration coefficients, to describe a system and analytic models in which the derived equation describes the physics/chemistry of a system. 

A general modeling strategy that has been successfully employed to model the joint velocity PDF in a wide class of turbulent flows30 is to develop stochastic models which... 

The formulation outlined above allows for a simple stochastic implementation of the deterministic differential equation (35). Starting with an ensemble of trajectories on a given adiabatic PES W, at each time step At we (i) compute the transition probability pk k, (h) compare it to a random number ( e [0,1], and (iii) perform a hop if pt t > C- In Ih se of a pure A -level system (i.e., in the absence of nuclear dynamics), the assumption (37) holds in general, and the stochastic modeling of Eq. (35) is exact. Considering a vibronic problem with coordinate-dependent however, it can be shown that the electronic... 

To ensure that the original information structure associated with the decision process sequence is honored, for each of the products whose demand is uncertain, the number of new constraints to be added to the stochastic model counterpart, replacing the original deterministic constraint, corresponds to the number of scenarios. Herein lies a demonstration of the fact that the size of a recourse model increases exponentially since the total number of scenarios grows exponentially with the number of random parameters. In general, the new constraints take the form ... 

G. N. Bochkov and Y. E. Kuzovlev, Non-linear fluctuation relations and stochastic models in nonequilibrium thermodynamics. 1. Generalized fluctuation-dissipation theorem. Physica A 106, 443-J79 (1981). 

In this final section we shall outline a few examples of the application of stochastic models to systems of physical chemical interest. Two of these appear in the author s previous review67 but the others do not. These examples are chosen to be representative of the general approach. 

Returning to the general case in which kb is permitted to be nonzero, we comment on one more feature of this stochastic model, namely that the equilibrium distribution, considered as a function of kfjkb, displays a first order transition in the limit K = oo. With dNi(oo)/dt = 0 the equilibrium solution to Eqs. (1)—(3) is seen to be of the form Af(oo) = A(kfjkby. The conservation condition, Eq. (6), gives the value of A, and we have... 

Let us apply general stochastic equations (2.2.15) to the simple A+B —> C reaction with particle creation - the model problem discussed more than once ([84] to [93]). A relevant set of kinetic equations reads... 

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