Rare events, complex systems

Since the particle occupancy depends both on reaction and diffusion the full dynamics of the automaton is complex however, if reaction is a rare event the system will be spatially homogeneous (o (r) independent of r) and the local particle distribution will differ little from the binomial distribution, pb(o p). 

There are a variety of ways to express absolute QRA results. Absolute frequency results are estimates of the statistical likelihood of an accident occurring. Table 3 contains examples of typical statements of absolute frequency estimates. These estimates for complex system failures are usually synthesized using basic equipment failure and operator error data. Depending upon the availability, specificity, and quality of failure data, the estimates may have considerable statistical uncertainty (e.g., factors of 10 or more because of uncertainties in the input data alone). When reporting single-point estimates or best estimates of the expected frequency of rare events (i.e., events not expected to occur within the operating life of a plant), analysts sometimes provide a measure of the sensitivity of the results arising from data uncertainties. 

Transition path sampling, based on a statistical mechanics in trajectory space, is a set of computational methods for the simulation of rare events in complex systems. In this chapter we give an overview of these techniques and describe their statistical mechanical basis as well as their application. 

The first of these points will be commented upon subsequently in connection with an interpretation of the statistical variability of fracture events. To answer this and the second question the highly developed tools of statistical analysis (e.g., 1—2) are available their application forms a backbone of industrial quality control and material design. Understandably, the complex input (mechanical, thermal, and environmental attack) acting upon a complex system (e. g., a highly structured polymer) has a complex, non-deterministic output. The determination and evaluation of only a small number of different data will necessarily reveal only a partial aspect of the fracture process. For this reason the questions as to cause and kinetics of fracture development can rarely be answered unambiguously. Any mathematical interrelations established between variables (e.g., time and stress) are valid for extrapolation only if the basis does not change. 

Molecular modeling of PT at dense interfacial arrays of protogenic surface groups in PEMs needs ab initio quantum mechanical calculations. In spite of fhe dramafic increase in computational capabilihes, it is still "but a dream" to perform full ab initio calculations of proton and water transport within realistic pores or even porous networks of PEMs. This venture faces two major obstacles structural complexity and the rarity of proton transfer events. The former defines a need for simplified model systems. The latter enforces the use of advanced compufahonal techniques that permit an efficient sampling of rare evenfs. ... 

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