On N. N. Bogolyubov's works in classical and quantum statistical mechanics
Nicolai Nicolajewich Bogolyubov (born 21. 8. 1909 in Nishni-Novgorod (Gorki)) founded new directions in nonlinear mechanics, statistical physics, and quantum field theory. The authors list these results, the ideas of his proofs and the prehistories. This begins with the theorem on the existence of invariant measure for compact dynamical systems and its history (Boltzmann's ergodic hypothesis, and von Neumann and Birkhoff). For the Hamiltonian systems, the required measure is the volume of phase space. Together with his teacher N. M. Krylov he showed that the invariant measure exists for any dynamical system. They introduced the concept of ergodic sets and showed that the invariant measure can be decomposed into invariant measures that are concentrated on ergodic sets and cannot be decomposed. Fine properties of complete continuous positive operators were used. A model for the transition to statistical equilibrium in a system connected with a heat bath is being considered. In his monograph `Problems of Dynamic Theory in Statistical Physics' the classical statistical mechanics is formulated in terms of sequences of distribution functions and equations for these (=Bogolyubov equations). The states of infinite systems were determined in terms of the states of finite systems by the thermodynamic limit procedure. These papers gave modern form to classical statistical mechanics and opened a new stage. The method of justification of the thermodynamic limit procedure suggested by Bogolyubov in 1949 was reanimated in the sixties and applied to the case of grand canonical ensemble. In his `Lectures in Quantum Statistics' (1949) he introduced a new concept of states in infinite systems in terms of an infinite sequence of statistical operators. A new approach to the approximate quantization was elaborated and applied to the polar theory of metals and the theory of magnetism. It is shown that the methods developed by Bogolyubov for classical equilibrium systems can be applied to quantum systems, too. Bogolyubov's microscopy theory for superfluidity obtained an approximate Hamiltonian -- a quadratic form with respect to operators of creation and annihilation. The Hamiltonian was diagonalized by the use of canonical transformation and the spectrum of elementary excitation was found. A complete theory of superconductivity based on an investigation of systems of interacting electrons and phonions determined the ground state consisting of paired electrons with opposite moments and spins, its energy, and the energy of elementary excitations. Bogolyubov created a rigorous theory of superfluidity and superconductivity and discovered that these two fundamental phenomena are based on the process of condensation of Bose- particles and fermion pairs. A new method of finding exact solutions for a broad class of model systems is given. 64 references.
scientific article; zbMATH DE number 635710
scientific article