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Journal of Physics and Electronics

Bogolyubov reduced description method: some remarks and proves

Alexander I. Sokolovsky, Sergiy F. Lyagushyn, S. A. Sokolovsky

Published Dec 19, 2025• 0 citations
Publisher

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 {ηa(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 {ηa(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 {ηa} 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,ηa } form a Lie algebra. The kinetics of the slow variable model, in which {H0,ηa }~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 {ηa(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 La(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 {ηa(t)}). In our study of the model of slow variables, the operators of the reduced description parameters {ηa} 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.

DOI
10.15421/332501

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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 {ηa(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 {ηa(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 {ηa} 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,ηa } form a Lie algebra. The kinetics of the slow variable model, in which {H0,ηa }~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 {ηa(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 La(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 {ηa(t)}). In our study of the model of slow variables, the operators of the reduced description parameters {ηa} 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.
A complex of issues related to using the Peletminsky–Yatsenko model for the study of non-equilibrium processes is investigated. The problems are associated with the inability of performing the calculation of necessary averages when system states are described by a new, wider set of reduced description parameters, in particular, fluctuations. A matter system interacting with a non-equilibrium system of bosons is considered. The substance is described by two reduced description parameters η1 , η2 and calculations with the quasi-equilibrium statistical operator corresponding to these parameters are technically impossible; at the same time, the calculations can be performed with such operator in the case of the only parameter η1 . Overcoming this problem by considering only small values δη2 of the other reduced description parameter is proposed. An exact definition of δη2 is given, and the theory of perturbations by powers δη2 is developed for the calculation of the initial quasi-equilibrium statistical operator with accuracy up to second-order contributions. Since all further averages are calculated only with the named operator including only η1 , the simplified calculation procedure is carried out for it.On this basis, the theory of non-equilibrium processes in the system of two-level emitters interacting with the boson field is developed. The emitter system is described by excitation degree η1 and the value of emitter correlations η2. The system is considered as a system of spins. The mathematically identical Dicke model describing superradiance and Wagner model introduced for the analogous acoustic phenomenon description are studied. The quasi-equilibrium statistical operator calculation is constructed.
The reduced description method proceeds from Bogolyubov’s idea that at large times the non-equilibrium system evolution can be described with the limited number of parameters and principle of correlation weakening. The way to the right choice of such parameters and constructing the time evolution equations for them was opened by the works of Kharkiv school in statistical physics. A quarter of a century ago at Dnipropetrovsk National University the developed method was first applied to the Dicke system of two-level emitters interacting via electromagnetic field. The paper presents a short review of the results obtained. They include the analysis of Dicke Hamiltonian structure, the choice of reduced description parameters satisfying the Peletminskii – Yatsenko scheme conditions, the realization of this scheme for concentrated and prolonged Dicke systems resulting in the temporal equation for emitter subsystemstatistical operator. The reduced description method possibilities are illustrated with using the effective Hamiltonian concept, in particular taking into account emitter motion and higher correlations. Reformulating the theory in terms of electromagnetic field allowed to construct the electrodynamics of medium formed by two-level emitters with considering binary correlations of field variables and to put forward the problem of correlation development picture. A new method of investigating correlations in quasispin subsystem with using ferromagnetism theory approach was also proposed.
Time-resolved measurement techniques are opening a window on nonequilibrium quantum phenomena that is radically different from the traditional picture in the frequency domain. The simulation and interpretation of nonequilibrium dynamics are conspicuous challenges for theory. This paper presents an approach to quantum many-body dynamics that is based on a Hamiltonian formulation of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations of motion for reduced density matrices. These equations have an underlying symplectic structure, and we write them in the form of the classical Hamilton equations for canonically conjugate variables. Applying canonical perturbation theory or the Krylov-Bogoliubov averaging method to the resulting equations yields a systematic approximation scheme. The possibility of using memory-dependent functional approximations to close the Hamilton equations at a given level of the hierarchy is discussed. The geometric structure of the equations gives rise to reduced geometric phases that are observable even for noncyclic evolutions of the many-body state. The approach is applied to a finite Hubbard chain which undergoes a quench in on-site interaction energy UUU. Canonical perturbation theory, carried out to second order, fully captures the nontrivial real-time dynamics of the model, including resonance phenomena and the coupling of fast and slow variables.
Introduction. 1: Evolution of States of Quantum Systems of Finitely Many Particles. 1. Principal Concepts of Quantum Mechanics. 2. Evolution of States of Quantum Systems with Arbitrarily Many Particles. 3. Evolution of States in the Heisenberg Representation and in the Interaction Representation. Mathematical Supplement I. References. 2: Evolution of States of Infinite Quantum Systems. 4. Bogolyubov Equations for Statistical Operators. 5. Solution of the Bogolyubov Equations. 6. Gibbs Distributions. Mathematical Supplement II. Mathematical Supplement III. References. 3: Thermodynamic Limit. 7. Thermodynamic Limit for Statistical Operators. 8. Statistical Operators in the Case of Quantum Statistics. 9. Bogolyubov's Principle of Weakening of Correlations. Mathematical Supplement IV. References. 4: Mathematical Problems in the Theory of Superconductivity. 10. Frohlich Model. 11. Bogolyubov's Compensation Principle for 'Dangerous' Diagrams. Compensation Equations. 12. Bardeen--Cooper--Schrieffer (BCS) Hamiltonian. 13. Microscopic Theory of Superfluidity. Mathematical Supplement V. References. 6: Green's Functions. 14. Green's Functions. Equations for Green's Functions. 15. Investigation of the Equations for Green's Functions in the Theory of Superconductivity and Superfluidity. 16. Green's Functions in the Thermodynamic Limit. References. 6: Exactly Solvable Models. 17. Description of the Hamiltonians of Model Systems. 18. Functional Spaces of Translation--Invariant Function. 19. ModelHamiltonians in the Spaces of Translation Invariant Functions. 20. Model BCS Hamiltonian in the Space hT. Equivalence of General and Model Hamiltonians in the Space of Pairs. 21. Equations for Green's Functions and their Solutions. Mathematical Supplement VI. References. 7: Quasiaverages. Theorem on Singularities of Green's Functions of 1/q2-Type. 22. Quasiaverages. 23. Green's Functions and their Spectral Representations. 24. Theorem on Singularities of Green's Functions of 1/q2-Type. References. Subject Index.
The purpose of the present paper is to consider the problem of quantization close to a nontrivial classical field. It is well-known that careless introduction of the classical fields leads to violation of the conservation laws and to the zero-mode problem, which reflects incompatibility between the adopted perturbation theory and the symmetry properties of the system. Bogolyubov group variables are some operators, which can be treated as parameters of the system symmetry group. For example, let’s consider a system with translational symmetry. After Bogolyubov transformation a static classical solution σ(x) transforms in σ(x− â), where â is an operator, canonical conjugate to an operator of the total linear momentum. Group variables are cyclic due to Hamiltonian symmetry, so the conservation laws are accomplished. For a system with adiabatic coupling the problem of combining of the conservation laws with perturbation theory has been solved by N.N. Bogolyubov in 1950, when he studied the polaron problem [1]. The further investigations have shown that Bogolyubov group variables have a wide range of application. These variables have been used in the strong coupling theory [2–8], the relativistic theory of gravitation [9] and other fields of the theoretical physics [10–22]. The Bogolyubov method has been independently rediscovered in the middle of 1970s and named collective coordinate method [23–26]. Connection between various variants of this method has been discussed in [10]. It was revealed however that consideration of systems with a nonstationary classical components is a rather nontrivial problem, because explicit Hamiltonian structure as the time-translation generator becomes clear only after solving the motion equations. So we have to construct such Bogolyubov’s scheme that group variables are defined in
We now consider systems for which the Hamiltonian contains time-dependent interaction terms. In such cases we cannot reduce the time-dependent Schro¨dinger equation to an eigenvalue equation. However, if the time-dependent terms can be regarded as small perturbations, we can develop a form of perturbation theory that can be used to calculate the effects of time-dependent terms. Consider a system for which the Hamiltonian may be written as the sum of a dominant unperturbed part Hˆ0 and a small perturbation Hˆ ′ as Hˆ = Hˆ0 + λHˆ ′(t) , (10.1.1) where the unperturbed Hamiltonian Hˆ0 is independent of time, whereas the perturbation Hˆ ′ may depend on time. The parameter 0 < λ < 1 has been introduced to keep track of various orders of approximations as in the case of time-independent perturbation theory discussed in Chapter 8. The basic idea is that a small perturbation to the Hamiltonian will produce a small change in the wave function. For a time-dependent perturbation, this change means that a system, initially in a particular eigenstate of Hˆ0, will find itself in a time-dependent admixture of the eigenstates. Thus the time-dependent perturbation causes the system to undergo transition to different eigenstates of Hˆ0. So for time-dependent perturbations our interest lies in calculating the probabilities for transition to different states.