Stochastic mean-field approach

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. 

A. Stochastic Difference in Time Definition A Stochastic Model for a Trajectory Weights of Trajectories and Sampling Procedures Mean Field Approach, Fast Equilibration, and Molecular Labeling Stochastic Difference Equation in Length Fractal Refinement of Trajectories Parameterized by Length... 

The following three sections describe the Bohmian quantum-classical approach [22,23] that uniquely solves the quantum back-reaction branching problem, the stochastic mean-field approximation [20] (SMF) that both resolves the back-reaction problem and incorporates the quantum decoherence and Franck-Condon overlap effects into NA-MD, and the quantized mean-field method [21] (QMF) that takes into account ZPE. The Bohmian and QMF approaches are illustrated by a model designed to capture some features of the O2 dissociation on a Pt surface. The concluding section summarizes the features of the methods and discusses further avenues for development and consideration. 

The stochastic mean-field [20] (SMF) method simultaneously resolves the following two major issues with NA MD. First, decoherence effects within the quantum subsystem that take place due to its interaction with an environment are included. Second, decoherence naturally leads to the asymptotic branching of NA trajectories. That is, the implementation of the decoherence effect in the SMF approach automatically resolves the branching problem. By extending the ordinary quantum-classical MF approximation, the SMF approach accounts for the quantum features of the environment in the Lindblad formulation. The Lindblad formulation is exact for a bath of harmonic oscillators and is an approximation for other types of solvents. While the quantum nature of the environment is treated by SMF within an approximation, its classical properties are included exactly by classical MD with a true Hamiltonian. 

To recapitulate, the Bohmian quantum-classical, stochastic mean-field and quantized mean-field approaches described above are capable of reproducing quantum solvent effects that are crucial in simulation of NA chemical processes. The approaches are computationally simple and are particularly suitable for studies of large chemical systems. 

In this paper the problem of transient behaviour of stochastic models of inhomogeneous chemical systems will be considered. First, the main results concerning compartmental models will be presented. Next, a qualitative approach will be proposed. Finally, some remarks concerning "mean-field approach and explosion in inhomogeneous system will be made. 

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). 

Ellis et al. [30] discussed the quantum fluctuations about the mean-field solution that would correspond in field theory to quantum fluctuations in the lightcone and could be induced by higher-genus effects in the string approach. Such effects would result in stochastic fluctuations in the velocity of light as of the order of... 

These results show that simplified molecular dynamics simulations can qualitatively account for micellization quite well. However, the computation time necessary for even such simple models is too great to allow the model to be useful for the calculation of other micellar properties or phase behavior or for an in-depth study of solubilization. Stochastic dynamics simulations, in which the solvent effects are accounted for through a mean-field stochastic term in the equations of motion, can also be used to study surfactant self-assembly [22], but the most efficient approach to date is still the one based on lattice Monte Carlo simulations, which are discussed next. 

Let us now consider stochastic motion in an OB system. In general, noise in an OB system may result from fluctuations of the incident field, or from thermal and quantum fluctuations in the system itself. We shall consider the former. The fluctuations of the intensities of the input or reference signals give rise respectively to either multiplicative or additive noise driving the phase. Both types of fluctuations can be considered within the same approach [108], Here we discuss only the effects of zero-mean white Gaussian noise in the reference signal ... 

The stochastic Liouville equation is highly useful when applied at high field, as techniques exist to reduce in size the typically large matrices it produces, and it has thus been used to simulate electron and nuclear spin polarizations in magnetic resonance experiments.A relatively recent book describes the approach in detail. However, for determining field dependences, such reductions are not possible, meaning that the sizes of the matrices are too large for even modern computers, and so this approach is seldom used for the simulation of field effects. 

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