"The inhomogeneous product-potential social systems"
This article contains theory of potential networks as very important type of locally interrupted system with relations and reactions. The potential networks were deduced from locally interrupting system by using so call "principle of maximum non-ergodicity". The system are balanced in social sense if the set of psychological reactions on the graph of relation satisfy the principle of maximum non-ergodicity and this system of reactions are product potential on the so call "two-steps" graph of relations. In real life survive only relatively stable groups and we can observe only stable groups that in social science call balanced. So reason why social system (group) is stable based on hidden potentiality of reactions: only social systems with potential system of reactions (potential fields) are stable (balanced in social sense). This conclusion is not a big surprise for natural (physical) systems. For instance the system consisting of a star and a single planet is stable because the gravitational interaction is potential (friction is absent). The potentiality of gravitational interaction means that work done along any closed path is zero. But for social science and, particularly for human groups, a similar property comes as a surprise. The main problem that was presented in this paper is problem of existence of potential fields. For solving this problem was used method of smooth fields on the solid domain and product integrals. For of smooth fields will be write the system of infinitesimal equations (system of partial differential equations) that must be hold for all potential fields and then was found solutions of infinitesimal equations. Then smooth potential field on domain will be transformed into discrete potential marking on embedded graph of relation by using product integrals. The finally will be found system differential equations that transfer any initial fields of reactions into potential.