On quadratic stochastic operators generated by Gibbs distributions

We give a constructive description of quadratic stochastic operators which act to the set of all probability measures on some measurable space. Our construction depends on a probability measure $\mu$ and cardinality of a set of cells (configurations) which here can be finite or continual. We study behavior of trajectories of such operators for a given probability measure $\mu$ which coincides with a Gibbs measure. For the continual case we compare the quadratic operators which correspond to well-known Gibbs measures of the Potts model on $Z^d$. These investigations allows a natural introduction of thermodynamics in studying some models of heredity. In particular, we show that any trajectory of the quadratic stochastic operator generated by a Gibbs measure $\mu$ of the Potts model converges to this measure