Bogolyubov reduced description method: some remarks and proves

Abstract

Bogolyubov's reduced description method is based on his functional hypothesis. The reduced description of system's nonequilibrium state occurs over long-time scales t>>τ₀. The synchronization time τ₀ determines the set of reduced description parameters {η_a(t)}, which fully describe the state of the system. The problem of finding the statistical operator of the system at the time of reduced description ρ(η)  and the effective initial values of the reduced description parameters {η_a(0)}  is posed and solved. This problem is solved only if there is a small parameter in the theory, with which the perturbation theory is built. The mathematical structure of the operators of the reduced description parameters {η_a} and the Hamilton operator H of the system  plays the main role in this. The paper investigates the Peletminskii–Yatsenko model, in which , H=H₀+H₁, H₀~ λ⁰ , H₀~ λ¹ (λ<<1), and the operators {H₀,η_a } form a Lie algebra. The kinetics of the slow variable model, in which {H₀,η_a }~λ¹ (λ<<1), is also investigated. In these models, integral equations for the statistical operator ρ(η) and the effective initial values of the reduced description parameters {η_a(0)} are derived. Their solutions are investigated in the main and first orders of the perturbation theory for a small parameter λ. The right-hand sides of the time equations for the reduced description parameters L_a(η) are investigated with accuracy up to the second-order contributions for λ. The paper thoroughly discusses the basic concepts of the reduced description method with remarks that supplement existing literature. In the Peletminskii–Yatsenko model, complex relations are proved that are given in the literature without justification or proved too complicatedly (in particular, this concerns the derivation of the integral equation for the initial values of the parameters {η_a(t)}). In our study of the model of slow variables, the operators of the reduced description parameters {η_a} are not specified, which leads to the implementation of the reduced description with rather complex algebra and operator analysis calculations. At the same time, a reasonable degree of details in the construction of the reduced description of the system is chosen.