Bogolyubov Reduced Description Method Research Articles

Superradiance and solids: Reduced description method

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.

Dispersion equation for electromagnetic waves in a Dicke system

The self-organization process in a Dicke model is studied with taking into account electromagnetic field states. The interaction between two-level emitters provided by the field results in generating field waves of increasing amplitudes. The picture of superradiance development in a prolonged system can be described in terms of electromagnetic field in continuous medium formed by emitters. The generalized induction in the Dicke system is introduced on the basis of the material equation for charge current obtained using the Bogolyubov reduced description method. The dispersion equation for waves and its solutions are derived and analyzed. Non-singular expressions for material coefficients are proposed and used for calculating the wave characteristics.

Correlations in superradiance of a two-dimensional system

The process of cooperative spontaneous emission in a system of two-level emitters interacting via electromagnetic field is analyzed using the Bogolyubov reduced description method, which provides the possibility of studying the state of the generated field. Field macrostates are described with electric and magnetic field amplitudes and binary correlation functions of their components. Since the previously obtained set of differential equations describing the Dicke system evolution with considering field correlations is very cumbersome, we must find some way of simplifying the model to make it accessible for numerical modeling. The general picture of correlation development can be elucidated with using one- and two-dimensional models. The paper presents the set of evolution equations for Dicke system with fixed orientation of dipole moments of emitters forming a two-dimensional structure. The material equations and equilibrium correlations are discussed. The transition to the Cauchy problem for ordinary differential equations is proposed.

Kinetics of system of emitters and nonequilibrium electromagnetic field

A new step in research into the Dicke model kinetics is proposed on the basis of the Bogolyubov reduced description method. A more precise description of the emitter subsystem and electromagnetic field is considered. New reduced description parameters — emitter energy density and binary correlation functions of electric and magnetic field amplitudes — open the way to investigating non-uniform systems of two-level emitters and nonequilibrium electromagnetic field with its correlation properties. Kinetic equations for such systems have been obtained in terms of electrodynamics of continuous media consisting of two-level emitters. Material equations for corresponding media are analyzed.

Gravitation field dynamics in Jeans theory

Closed system of time equations for nonrelativistic gravitation field and hydrodynamic medium was obtained by taking into account binary correlations of the field, which is the generalization of Jeans theory. Distribution function of the systemwas built on the basis of the Bogolyubov reduced description method. Calculations were carried out up to the first order of a perturbation theory in interaction. Adiabatic and enthropic types of perturbations were corrected and two new types of perturbations were found.

Reduced description of nonequilibrium processes and correlation functions. Divergences and non-analyticity

A complete theory for investigation of time correlation functions is developed on the basis of the Bogolyubov reduced description method proceeding from his functional hypothesis. The problem of convergence in the theory of nonequilibrium processes and its relation to the non-analytic dependence of basic values of the theory on a small parameter of the perturbation theory are discussed. A natural regularization of integral equations of the theory is proposed. In the framework of a model of slow variables (hydrodynamics of a fluid, kinetics of a gas) a generalized perturbation theory without divergencies is constructed corresponding to a partial summation in a usual perturbation theory. Properties of Green functions are discussed on the basis of resolvent formalism for the Liouville operator. A generalized Ernst and Dorfman theory is elaborated allowing to study the peculiarities of correlation and Green functions and to solve the convergence problem in the reduced description method.