The book develops the ideas of Gibbs and Poincaré about the thermal equilibrium of mechanical systems. Although Gibbs's ideas are widely known, many of the problems they posed have not yet been solved. On the contrary, Poincaré's profound results in kinetics were unclaimed and generally unknown to specialists in statistical mechanics. The range of issues discussed in this book is grouped around three related topics: weak convergence of probability measures (density of which is the solution of the Liouville equation), the hierarchy of randomness of Hamilton's dynamical systems, the perturbation theory of an ensemble of weakly interacting subsystems. The obtained results make it possible to better understand the nature of irreversible behavior of thermodynamic systems, to give a new interpretation of the second principle of thermodynamics about the growth of entropy, as well as to give a strict conclusion of the canonical Gibbs distribution, not based on the ergodic hypothesis. The text of the book is structured in the form of essays: four chapters are largely independent of each other. Each chapter has a commentary and bibliography. Additions are devoted to the properties of invariant measures with smooth density, the conditions for the existence of additional conservation laws - the first integrals of Hamilton's equations, as well as the phenomenon of diffusion in nonlinear dynamical systems. The book is intended for mathematicians, mechanics and physicists interested in classical statistical mechanics and questions of substantiation of thermodynamics.