SHTYK V. Investigation of evolution equations of exactly solvable models of statistical mechanics

The thesis is devoted to the investigation of the initial value problem of the quantum BBGKY hierarchy for many-particle systems by the non-equilibrium cluster expansion method. We construct a new representation of the solution of the initial-value problem to the quantum BBGKY hierarchy of equations as an expansion over particle clusters whose evolution are described by the corresponding-order cumulant (semi-invariant) of the evolution operators of finitely many-particle quantum systems. The criterion of cumulant representation of the solution of the BBGKY hierarchy is proved. For the initial data from the space of sequences of the trace operators the existence and uniqueness theorem is proved. We also investigate the initial-value problem of the nonlinear Liouville hierarchy. For the initial data from the space of integrable functions the existence of a strong solution of the Cauchy problem is proved. It was also shown that the nonlinear Liouville hierarchy is basic in the substantiation of the validating of the correlation-weakening principle.