Probability theory, fundamental limitation

The mathematics may help us to understand a system that is fundamentally hard to simulate. Examples of this include models involving queues in heavy traffic (Subsection 5.4) or problems in which a very small probability must be estimated. Long simulation runs may be needed to get any accuracy in such situations, whereas various limit theorems in probability theory often give very accurate results. 

The last assumption is very fundamental. It results in time-independent transition probabilities and makes a clean theory possible. It requires that the product of the time scale of the decay time for the tcf (called the correlation time and denoted x ) and the strength of the perturbation (in angular frequency units) has to be much smaller than unity (17-20). This range is sometimes denoted as the Redfield limit or the perturbation regime. 

As will be discussed in chapter 6, of fundamental importance in the theory of unimolecular reactions is the concept of a microcanonical ensemble, for which every zero-order state within an energy interval AE is populated with an equal probability. Thus, it is relevant to know the time required for an initially prepared zero-order state j) to relax to a microcanonical ensemble. Because of low resolution and/or a large number of states coupled to i), an experimental absorption spectrum may have a Lorentzian-like band envelope. However, as discussed in the preceding sections, this does not necessarily mean that all zero-order states are coupled to r) within the time scale given by the line width. Thus, it is somewhat unfortunate that the observation of a Lorentzian band envelope is called the statistical limit. In general, one expects a hierarchy of couplings between the zero-order states and it may be exceedingly difficult to identify from an absorption spectrum the time required for IVR to form a micro-canonical ensemble. 

In microfluid mechanics, the direct simulation Monte Carlo (DSMC) method has been applied to study gas flows in microdevices [2]. DSMC is a simple form of the Monte Carlo method. Bird [3] first applied DSMC to simulate homogeneous gas relaxation problem. The fundamental idea is to track thousands or millions of randomly selected, statistically representative particles and to use their motions and interactions to modify their positions and states appropriately in time. Each simulated particle represents a number of real molecules. Collision pairs of molecule in a small computational cell in physical space are randomly selected based on a probability distribution after each computation time step. In essence, particle motions are modeled deterministically, while collisions are treated statistically. The backbone of DSMC follows directly the classical kinetic theory, and hence the applications of this method are subject to the same limitations as kinetic theory. 

A principal aim of chemical analysis is to develop a theoretical model of the interaction between atoms and molecules. Experimental work of the previous two centuries has resulted in a highly successful empirical account of chemical reactivity, and efforts to formulate a rigorous, fundamental theory as a nonclassical many-body problem have lead to highly accepted and much used methods but these still have significant limitations. By the current approaches, chemical interaction is modeled in terms of probability-density distributions of independent electrons. Although the theory appears to work for one-particle problems, unforeseen effects emerge in the treatment of more complex systems [1]. In particular, the distribution of extranuclear electrons seems to obey an exclusion principle, not anticipated in the basic theory, and there is no fundamental understanding of three-dimensional molecular shapes, as observed experimentally. The pivotal role of entropy, which controls the course of chemical reactions, is theoretically equally unexpected. 

We have, then, two independent and, in important features, contradictory theories of acids and bases. This situation seems to be due to the neglect of two factors first, a portion of the experimental data, and second, the electronic theory of the covalent bond. Probably such neglect was natural and even necessary in the early stages of the development of each theory. Possibly neither would have accomplished as much as it has without some such limitation. However, as a result of this neglect, neither theory gives an insight into the fundamental nature of acids and bases. 

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