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.
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.
On the basis of our previous study of the quantum system of atoms and the electromagnetic field, a system of non-relativistic stationary and non-interacting small one-electron atoms located in a region much smaller in size than the characteristic wavelength of the field is considered. Atoms are considered in the two-level approximation and interact with the field by dipole-electric interaction. We carefully analyze the state space of the atom and the operators in it in the spin s = 1 / 2 formalism, which leads to the Dicke Hamiltonian in terms of the spin formalism. Non-equilibrium states of the system are investigated using the reduced description method and described by the occupation numbers of photon states ηαk , the degree of excitation of the atoms η1 , and the value of correlations between them η2. The basis of our consideration is the Peletminsky–Yatsenko model, the foundations of which and, in particular, the approach to solving the Cauchy problem, based on the notion of effective initial conditions, are thoroughly discussed. Since the Dicke model deals with the dynamics of a system of particles with spin, the technique of calculating the average values of products of spin operator proposed by Vax, Larkin, and Pikin is developed to a certain extent. Averages in a state with a statistical operator of a system of atoms with given excitations are considered. The obtained results are maximally simplified in the case of spin s = 1 / 2 . The impossibility of calculations with a quasi-local statistical operator when describing the state of the system with parameters ηαk , η1, η2 is overcome with using a somewhat simplified approach of our previous work, in which, instead of a parameter η2, the system is described by a small deviation δη2 = η2 - η20 of the parameter η2 from its value η20 when describing the system only by the parameters ηαk, η1. Developed approaches to calculations with spin operators are used to calculate the quasi-equilibrium statistical operator when choosing of the quantities ηαk, η1 , and δη2 as reduced description parameters and the right-hand sides of the time equations for these parameters.
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.
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.
Superradiance, both optical and acoustic, is one of the most important methods of generating the coherent pulses and obtaining the information about emitters via resonant phenomena. The process of ordering in a system of excited emitters, which results in a powerful short radiation pulse, is described with correlation functions of emitter and field parameters. Their behavior is problem of interest, especially in a spatial system. The known theoretical approaches to such questions are based on using some ways of correlator decoupling, at the same time the validity and exactness of applied operations remain without evaluation. The authors propose a way to solve this problem in the framework of the Bogolyubov reduced description method in the Peletminskii–Yatsenko formulation. Additionally to the usual small parameter of the method (the interaction constant), a new small parameter (the difference between correlation functions in the theory using it as an independent reduced description parameter and the process picture with one less parameter) is introduced. The method is developed for a concentrated Dicke model. The conventional application of quasispin operators for two-level objects allows to describe the system of emitters as a spin system and use the perfect methods of calculating the averages of spin operator combinations, which were developed in the ferromagnetism theory.
The process of self-ordering in the famous Dicke model was studied in the framework of eliminating the boson variables. But the reduced description method enables us to obtain also the picture of electromagnetic field evolution provided field amplitudes and correlation functions are included into the number of reduced description parameters. In the Dicke Hamiltonian structure the interaction term includes the operators of emitter dipole moments or dipole moment density (polarization) since a spatial system is under consideration. Thus operator evolution equations are based on using such operators and their derivatives. The chain of evolution equations for averaged field amplitudes and binary correlation functions are obtained with using the statistical operator calculated in a perturbation theory in quasispin-photon interaction assumed to be small. The problem of chain decoupling does not arise since at any step we have a closed set of equations. The sets should be solved on the basis of material equations for current density and their generalizations for more complicated correlation functions. The way to constructing such equations and estimating the material parameters which are necessary for the numerical modeling of the development of correlations is discussed in the paper.
Spontaneous ordering in a system of excited emitters with a powerful short radiation pulse predicted by Dicke attracted the great attention of scientists for decades as a bright example of the nontrivial behavior of a nonequilibrium system. The Bogolyubov reduced description method has some advantages and provides new possibilities in the theoretical investigation of the Dicke process. The paper presents the authors’ results of modeling such processes concerning both optical ones in the presence of crystal lattice influence and acoustic superradiance, which is possible in solids. The right choice of reduced description parameters allows the account of atom motions (phonons), cavity influence, and excitations in the emitter system (excitons) in the electromagnetic radiation process. The mathematical analogy between the Dicke system and a spin system in a magnetic field opens the way to applying the reduced description technique to acoustic phenomena previously investigated with the boson variable elimination method. Such a new approach to superradiance in crystals with paramagnetic impurities and with competing mechanisms of relaxation (Wagner model) is outlined. Acoustic superradiance is discussed in connection with the problem of structural phase transitions.