62 ISSN 0021-3640, JETP Letters, 2016, Vol. 103, No. 1, pp. 62–75. © Pleiades Publishing, Inc., 2016. Original Russian Text © V.V. Kocharovsky, Vl.V. Kocharovsky, S.V. Tarasov, 2016, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2016, Vol. 103, No. 1, pp. 67–80. Bose–Einstein Condensation in Mesoscopic Systems: The Self-Similar Structure of the Critical Region and the Nonequivalence of the Canonical and Grand Canonical Ensembles V. V. Kocharovsky a, b , Vl. V. Kocharovsky a, c, *, and S. V. Tarasov a a Institute of Applied Physics, Russian Academy of Sciences, ul. Ul’yanova 46, Nizhny Novgorod, 603950 Russia b Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843-4242, USA c Lobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950 Russia * e-mail: kochar@appl.sci-nnov.ru Received November 10, 2015 The analytical theory of Bose–Einstein condensation of an ideal gas in mesoscopic systems has been briefly reviewed in application to traps with arbitrary shapes and dimension. This theory describes the phases of the classical gas and the formed Bose–Einstein condensate, as well as the entire vicinity of the phase transition point. The statistics and thermodynamics of Bose–Einstein condensation have been studied in detail, includ- ing their self-similar structure in the critical region, transition to the thermodynamic limit, effect of boundary conditions on the properties of a system, and nonequivalence of the description of Bose–Einstein condensa- tion in different statistical ensembles. The complete classification of universality classes of Bose–Einstein condensation has been given. DOI: 10.1134/S0021364016010070 1. PROBLEM OF THE CRITICAL REGION OF BOSE–EINSTEIN CONDENSATION The problem of the construction of the micro- scopic theory of second-order phase transitions in the critical region has remained unsolved for more than a century. The desired theory should allow tracing the evolution of the system at a transition from a disor- dered phase to an ordered phase. The problem is exceptionally difficult because the process of phase transitions involves the simultaneous action of a num- ber of factors, each providing a hardly solvable theo- retical problem on its own. The mentioned factors are (a) the existence of many particles in a mesoscopic system, (b) the microscopic interaction between parti- cles and the presence of long-range correlations, (c) a critical dependence on the dimension of space and the absence of a solution of the three-dimensional prob- lem, (d) the presence of unstable modes, (e) the non- linear saturation of instability at a macroscopic level, (f) spontaneous symmetry breaking, (g) the presence of constraints in the Hilbert space of the system (including those dictated by the broken symmetry because of Noether’s theorem), (h) anomalously strong fluctuations of the parameters of the system, etc. This problem has not yet been solved by known methods of mean field theory, perturbation theory, theory of oscillations and waves, quantum field the- ory, standard diagrammatic technique, and renormal- ization group. The construction of the microscopic theory of phase transitions in the critical region becomes possi- ble only with a new method involving the exact reduc- tion of the Hilbert space with constraints, the non- polynomial diagrammatic technique, partial operator contractions, exact recurrence equations for them, the method of the characteristic function for the joint probability distribution of noncommuting observ- ables, and exact equations for physical (calculated on the reduced Hilbert space) Green’s functions and the order parameter, which differ from the standard Dyson equations for approximate (calculated on the extended Hilbert space without constraints) Green’s functions and order parameter. This method, which fully includes the entire set of the above factors, was described in detail in review [1]. In this review, we focus only on a small but signifi- cant part of this problem central for theoretical phys- ics. Exactly, we describe the role of constraints imposed by broken symmetry on the Hilbert space of the system in the origin and picture of the phase tran- sition by example of Bose–Einstein condensation in mesoscopic systems. In this case, the global gauge symmetry is broken and the constraint corresponding to it according to Noether’s theorem is the conserva- SCIENTIFIC SUMMARIES