# Petrenko S. M.

### Hahn-Jordan decomposition as an equilibrium state of the conflict system

Koshmanenko V. D., Petrenko S. M.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 64-77

The notion of conflict system is introduced in terms of couples of probability measures. We construct several models of conflict systems and show that every trajectory with initial state given by a couple of measures $\mu, \nu$ converges to an equilibrium state specified by the normalized components $\mu_+, \nu_+$ of the classical Hahn – Jordan decomposition of the signed measure $\omega = \mu - \nu$.

### Quasicontinuous approximation in classical statistical mechanics

Petrenko S. M., Rebenko A. L., Tertychnyi M. V.

Ukr. Mat. Zh. - 2011. - 63, № 3. - pp. 369-384

A continuous infinite systems of point particles with strong superstable interaction are considered in the framework of classical statistical mechanics. The family of approximated correlation functions is determined in such a way that they take into account only those configurations of particles in the space $\mathbb{R}^d$ which, for a given partition of $\mathbb{R}^d$ into nonintersecting hypercubes with a volume $a^d$, contain no more than one particle in every cube. We prove that so defined approximations of correlation functions pointwise converge to the proper correlation functions of the initial system if the parameter of approximation a tends to zero for any positive values of an inverse temperature $\beta$ and a fugacity $z$. This result is obtained for both two-body and many-body interaction potentials.