Markovian truncation

In this Chapter we introduce a stochastic ansatz which can be used to model systems with surface reactions. These systems may include mono-and bimolecular steps, like particle adsorption, desorption, reaction and diffusion. We take advantage of the Markovian behaviour of these systems using master equations for their description. The resulting infinite set of equations is truncated at a certain level in a small lattice region we solve the exact lattice equations and connect their solution to continuous functions which represent the behaviour of the system for large distances from a reference point. The stochastic ansatz is used to model different surface reaction systems, such as the oxidation of CO molecules on a metal (Pt) surface, or the formation of NH3. 

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. 

The simplest problem of this kind is the escape from a truncated harmonic well in the low-friction Markovian limit. This problem was treated by Ben Jacob et al. The model is defined by... 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

Saiz-Urra, L., Gonzalez-Diaz, H.> 8c Uriarte, E. (2005). Proteins Markovian 3D-QSAR with spherically-truncated average... 

Truncated chain-type configuration of the residual bath (see Fig. 15.1, right panel). This variant again employs the band-diagonal construction of the chain model, but is terminated at a given order M by a star construction as described above, now taken to represent a Markovian closure acting on the end of the chain [32]. This model mimicks the Markovian closure which naturally terminates the chain as shown in the next section. 

Based upon the effective-mode construction, a systematic approximation procedure for the environment can be formulated in terms of a series of coarse-grained spectral densities [32,33]. These spectral densities are generated from successive orders of a truncated chain model with Markovian closure. Analytical expressions can be given in terms of Mori type continued fractions. Assuming that an - a priori arbitrarily complicated - reference spectral density can be obtained independently, e.g., from experiments or classical simulations, one can thus (1) extract those features of the spectral density that determine the interaction with the subsystem... 

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