Rovenchak A. Bose-statistics and fractional types of statistics in the many-body theory and related problems

Approaches to analyze many-boson systems and fractional statistics systems are developed. Analogies with physical models are used for interdisciplinary studies. The methods for calculation of the Bose-system spectrum are proposed and in the case of helium-4 the correct long-wavelength behaviour without introducing an effective mass For bosons in traps the influence of the finite number of particles on thermodynamic functions is studied. A way to effectively take into account items with three and four creation-annihilation operators in the Hamiltonian is proposed. Thermodynamic functions of the harmonic oscillators obeying the Gentile statistics are calculated. New types of fractional statistics are proposed, namely the Polychronakos statistics with a complex parameter and two-parametric fractional statistics. The latter are used to model anyons and weakly-interacting finite Bose-systems. In the case of the complex parameter, phase transitions are predicted in the system of harmonic oscillators and estimates for their possible experimental observation are made. Analogies with the Bose-distribution are applied in several related problems of the interdisciplinary nature. A series of results are obtained in the problem of integer partitions, in particular, the number of restricted plane partitions is estimated for the first time. New parameters are proposed for the classification of texts, which is linked to studies of complex systems.