General Discussion and Literature Survey

In this section, we shall discuss the results presented in the preceding sections as a whole and supply the relation of the development presented in this article to previous investigations. No attempt will be made, however, to give an exhaustive survey of the literature on complex reaction systems. Such a survey may be obtained from standard works on chemical kinetics [Benson (6), Frost and Pearson (5), Laidler (SI)] and from review literature such as the Annual Report on the Progress in Chemistry and the Annual Review of Physical Chemistry. 

The monomolecular reaction systems of chemical kinetics are examples of linear coupled systems. Since linear coupled systems are the simplest systems with many degrees of freedom, their importance extends far beyond chemical kinetics. The linear coupled systems in which we are interested may be characterized, in general terms, as arising from stochastic or Markov processes that are continuous in time and discrete in an appropriate space. In addition, the principle of detailed balancing is observed and the total amount of material in the system is conserved. The system is characterized by discrete compartments or states and material passes between these compartments by first order processes. Such linear systems are good models for a large number of processes. 

The application of this model in physics and chemistry has had a long history. We shall give some examples of the early works. The work of Einstein S2) on the theory of Brownian motion is based on a random walk process. Dirac S3) used the model to discuss the time behavior of a quantum mechanical ensemble under the influence of perturbations this development enables one to discuss the probability of transition of a system from one unperturbed stationary state to another. Pauli 34) [also see Tolman 35)], in his treatment of the quantum mechanical H-theorem, is concerned with the approach to equilibrium of an assembly of quantum states. His equations are identical with those of a general monomolecular 

A typical recent example of the application of the model is the irreversible thermodynamics of Cox (36). A recent paper of interest to the readers of this article is that of Thomsen (37) in which he attempts to establish the convention that microscopic reversibility is to mean that the matrix K is symmetric, and that detailed balancing is to mean that the matrix KD is symmetric. 

This model is also used for many problems outside of physics and chemistry, for example, card shuffling, power supply, and parking lot congestion problems [Feller (SS)]. A more exhaustive survey of the varied application of this model can be found in Feller (39), Bharucha-Reid (40), Bellman 41), Chandrasekhar 42), and many other books on stochastic processes. 

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