ИСТИНА

here are two main classes of classical (non-quantum) dynamics for $N$-particle systems. Namely, deterministic hamiltonian systems and interacting Markov processes. Formally speaking, the first one is a particular case of the second. However, deep results for both systems differ strongly in the extent and in the methods. This is mainly due to the fact that the classical theory of Markov processes developped independently of its deterministic counterpart, using the assumption that the conditional probability measures $P^{(n)}(x,.)$ are absolutely continuous with respect to some fixed measure on the state space $X$, for some fixed $n$ and all $x\in X$, This condition greatly simplified the proofs of convergence to equilibrium in the ergodic case. It is even more important that any weakening of this condition on $P^{(n)}(x,.)$ gave rise to serious results and theories, see for example \cite{MT,nummelin,Orey}. For purely hamiltonian many particle dynamics one should first formalize the famous Ludwig Boltzmann hypothesis concerning convergence to one of the equilibrium measures - Liouville or Gibbs, or possibly to other measures. There are two evident possibilities for such formalization: 1) for given purely hamiltonian dynamics one should find a class of random initial conditions such that for any initial condition from this class there will be convergence to one of these measures, 2) find interaction of the given hamiltonian system with external world (reservoir) and prove convergence, for any initial conditions, to one (depending on the choice of the reservoir and on the interaction) of the equilibrium measures. We consider here the second possibility but our goal here is more challenging. Namely, we would like to find the minimal possible interaction with the external world which provides the desired convergence. It is big surprise that interaction of only one of $N$ particles with extermal media is sufficient for this. We believe that it is what Boltzmann really meant. We give review and generalization of the results in this direction, and discuss some new and more general problems. Mathematical tools we need are quite numerous: linear and non\={l}inear analysis of hamiltonian many particle systems, techniques for Markov chains with continuum state space, random operator dynamics in Banach spaces.