This thesis is dedicated to the study of the thermodynamics of systems of particles, which can be described by fractional statistics that generalize the Bose-Einstein and Fermi-Dirac statistics. Abelian and non-Abelian anyons are considered in the work. For the latter, a permutation of particles changes not only the phase, but also the wave function itself. Two types were analyzed for the non-Abelian anyons, depending on different values of hard-core parameter - with a soft and hard core. For the first type, a detailed description of the thermodynamics of the system of anyons, which have, in addition to the electric charge, also a magnetic charge, has been worked out. The study is based on the usage of the second virial coefficient and its correction. So, for the second type of anyons - non-Abelian - the question of the second virial coefficient was investigated. The study of non-Abelian anyons is significantly complicated by the fact that, depending on the type of the hard-core parameter, the expression for the second virial coefficient is different. And if in the case of a hard core it has a fairly convenient form of recording, then for a soft one it takes on a very cumbersome appearance, and therefore it is very difficult to work with it, both analytically and numerically. Therefore, to simplify the description, it was decided to propose an original way of expressing the relationship between the parameters of their virial coefficients and the parameters of two-parametric fractional statistics in order to describe the system of anyons through the latter one. Obviously, this was done taking into account the number of statistics parameters. For anyons with a hard core, non-additive (with the Tsallis q-exponential) and incomplete modifications of Polychronakos and Haldane-Wu statistics were used. The values of the parameters of fractional statistics were calculated, according to which anyons of this type can be described through them. In addition to what was written above, two variants of virial expansion, which can be found in the scientific literature, were also analyzed and compared. One of them is based on the expansion in the equation of state in a power series of the density. Another approach is based on the expansion of the partition function in a series by degrees of fugacity. The work presents the derivation of the relationship between both virial coefficients. As a separate issue, the problem for a Fermi system with a weak contact interaction described by non-additive Polychronakos statistics was solved. The work shows in detail how to obtain the relationship between the parameters of the statistics and interaction parameters of a real Fermi system using virial expansion. The work also shows in detail the effect of using deformation to generalize the Fermi distribution using the non-additive Tsallis q-exponential instead of the usual one in the expression for the occupation numbers. This modification is applied in two models, which differ from each other in terms of the choice of variable in the exponent power, that is, the change in the Gibbs factor. In the first case, it is the chemical potential, and in the second it is the activity, or in other words, the fugacity. The low- and high-temperature limiting cases are considered in detail. The dependences of the chemical potential and activity on temperature are derived for two models both for the limits. For better clarity, all results are supported by appropriate analytical and numerical calculations, as well as figures and tables.
