We prove the eventological H-theorem that complements the Boltzmann H-theorem from statistical mechanics and serves as a mathematical excuse (mathematically no less convincing than the Boltzmann H-theorem for the second law of thermodynamics.) for what can be called ``the second law of eventology'', which justifies the application of Gibbs and ``anti-Gibbs'' distributions of sets of events minimizing relative entropy, as statistical models of the behavior of a rational subject, striving for an equilibrium eventological choice between perception and activity in various spheres of her/his co-being.

We consider a previously proposed non-extensive statistical mechanics in which the entropy depends only on the probability, this was obtained from a f(\beta) distribution and its corresponding Boltzmann factor. We show that the first term correcting the usual entropy also arises from several f(\beta) distributions, we also construct the corresponding H-function and demonstrate that a generalized H-theorem is fulfilled. Furthermore, expressing this H-function as function of the simplest Maxwellian state we find, up to a first approximation some modified thermodynamic quantities for an ideal gas. In order to gain some insight about the behavior of the proposed generalized statistics, we present some simulation results for the case of a square-well and Lennard-Jones potentials, showing that an effective repulsive interaction is obtained with the new formalism.

Many-events-based and many-agent eventological models of supply and demand are offered. In these models new concepts of equilibrium intervals of prices and subsets of goods are introduced for the first time. These models follow from the eventological H-theorem (eventological generalization of Boltzmann’s H-theorem) which serves as a theoretical base for an application of eventological distributions of supply and demand in the mathematical description of many-agent and many-events-based market systems.

We define degree of dependence of two events A and B in a probability space by using Boltzmann-Shannon entropy function of an appropriate probability distribution produced by these events and depending on one parameter (the probability of intersection of A and B) varying within a closed interval I. The entropy function attains its global maximum when the events A and B are independent. The important particular case of discrete uniform probability space motivates this definition in the following way. The entropy function has a minimum at the left endpoint of I exactly when one of the events and the complement of the other are connected with the relation of inclusion (maximal negative dependence). It has a minimum at the right endpoint of I exactly when one of these events is included in the other (maximal positive dependence). Moreover, the deviation of the entropy from its maximum is equal to average information that carries one of the binary trials defined by A and B with respect t...