RYABUKHA T. Non-equilibrium cluster expansions in theory of infinite dynamical systems

The thesis is devoted to the investigation of the initial value problem of the Bogolyubov chain of equations for infinite systems by the non-equilibrium cluster expansion method. We construct new representations of the solutions in the form of the expansions over the particle clusters, whose evolution is governed by the cumulant (semi-invariant) of the evolution operator of the Liouville equation. The existence theorem of the solution in the cumulant representation of the initial value problem for the Bogolyubov chain of equations in the space of sequences of integrable functions has been proved. For the initial data from the space of sequences of functions, bounded with respect to the configuration variables and exponentially decreasing with respect to the momentum variables, we establish the nature of divergent terms of the expansion for the solution in the cumulant representation and we suggest its regularization method. We develop a dual non-equilibrium cluster expansion method and on its basis weconstruct a new representation of the solution of the initial value problem for the dual Bogolyubov chain of equations in the space of sequences of bounded functions. We investigate the existence of functionals for the average values of observables determined by the constructed expansions over the cumulants.