Dobrushin R.L., Sinai Y.G. (eds.) Multicomponent Random Systems

New York: Marcel Dekker Inc, 1980. - 624 p.Over the last decade there has taken shape and, we can say, ripened a new interdisciplinary scientific field--a theory of multicomponent random systems. The main object of study here is multidimensional systems comprising a large number of homogeneous locally interacting components. These components may be of different real nature depending on the field of their application. In physical applications these are atoms in points of crystal lattices; in cybernetic applications, interacting finite-state
automata, logic-informational elements, and queue systems; in biological applications, cells, neurons, etc.. It is believed that the range of the real phenomena, the mathematical description of which naturally leads to the models of the discussed type, is extremely wide.
As a branch of applied mathmatics, the theory of multicomponent random systems came into being on the intersection of the theory of probability, statistical physics, information theory, mathematical biology. Some of its basic notions, e.g., the notion of the Markov process with interaction was created in parallel by representatives of these fields of science. But the most important proved to be the influence of statistical physics, apparently because the scientific experience accumulated by this field is incomparably deeper than that of younger sciences. The decisive factor was the development of mathematically rigourous statistical physics. At first its evolution was motivated by the desire to substantiate logically the fundamental physical notions, however, the gradual transition to a formal level, irrelevant to a direct use of physical intuition, showed that the basic ideas of classical statistical physics are connected with the fact that statistical physics studies one of the classes of multicomponent random systems with local interactions. Then the approach based on the Gibbs distribution became natural for the description of random fields of a general type independent of their nature. Singularities of the phase transition type may be described as jump discontinuities of system (macroscopic — in terms of physics) characteristics under the continuous variation of element (microscopic) characteristics and are equally relevant both to the systems of physical and informational-cybernetic nature. The methods of the nonequilibrium statistical physics are applicable to the description of the dynamics of informational-cybernetic and biological systems etc.
In its mathematical methods the theory of multicomponent random systems has borrowed much from the theory of probability and particularly from the theory of random processes. In its tiim, it has lent to the theory of probability the new idea that the transition from one-dimensional random processes to multidimensional random fields provides qualitatively new opportunities. The influence of cybernetic and biological applications essentially widened the range of situations subject to study.
The present book contains a set of original scientific researches devoted to various problems of the theory of dynamic systems and multicomponent random systems, both theoretical and motivated by concrete applications. We believe that all the basic
ideas and methods of this theory are reflected here and therefore the reader will get a due notion of its content.