Kinetic Theory of Quantum Electrodynamic Plasma in a Strong Electromagnetic Field: I. The Covariant Formalism

We present a covariant approach to the kinetic theory of quantum electrodynamic plasma in a strong electromagnetic field. The method is based on the relativistic von Neumann equation for the nonequilibrium statistical operator defined on spacelike hyperplanes in Minkowski space. We use the canonical quantization of the system on hyperplanes and a covariant generalization of the Coulomb gauge. The condensate mode associated with the mean electromagnetic field is separated from the photon degrees of freedom by a time-dependent unitary transformation of the dynamic variables and the nonequilibrium statistical operator. This allows using expansions of correlation functions and of the statistical operator in powers of the fine structure constant even in the presence of a strong electromagnetic field. We present a general scheme for deriving kinetic equations in the hyperplane formalism.     

Statistical mechanics of relaxation processes in alternating fields

Using the nonequilibrium statistical operator method and the projection technique, we derive the system of exact relaxation equations for a quantum system interacting with an alternating external field. These equations hold in the case where some of the basic dynamical variables describing a nonequilibrium state depend explicitly on time. We obtain the exact expression for the entropy production in an alternating external field. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 102–112, January, 2008.     

Nonequilibrium Statistical Operator Method and Generalized Kinetic Equations

We consider some principal problems of nonequilibrium statistical thermodynamics in the framework of the Zubarev nonequilibrium statistical operator approach. We present a brief comparative analysis of some approaches to describing irreversible processes based on the concept of nonequilibrium Gibbs ensembles and their applicability to describing nonequilibrium processes. We discuss the derivation of generalized kinetic equations for a system in a heat bath. We obtain and analyze a damped Schrödinger-type equation for a dynamical system in a heat bath. We study the dynamical behavior of a particle in a medium taking the dissipation effects into account. We consider the scattering problem for neutrons in a nonequilibrium medium and derive a generalized Van Hove formula. We show that the nonequilibrium statistical operator method is an effective, convenient tool for describing irreversible processes in condensed matter.     

Nonequilibrium statistical operator in the generalized molecular hydrodynamics of fluids

We discuss the important role of the Zubarev nonequilibrium statistical operator method in the generalized molecular hydrodynamics of fluids. Using this method allows developing a consistent approach of generalized collective excitations for simple, ion, polar, magnetic, and some other fluids. We construct a nonequilibrium statistical operator and derive the corresponding transport equations for a system that relaxes and passes into the state of molecular hydrodynamics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 91–101, January, 2008.     

Heisenberg evolution of quantum observables represented by unbounded operators

This paper deals with open quantum systems. In particular, we focus on the adjoint quantum master equations with initial conditions given by unbounded operators. Examples of this type of initial data are the position and momentum operators of quantum oscillators and the occupation number operator in many-body particle systems. The article establishes the existence and uniqueness of solutions of the operator equations governing the motion of unbounded observables under the Born-Markov approximations. To this end, we develop the relation between operator evolution equations arising in quantum mechanics and stochastic evolutions equations of Schrödinger type. Furthermore, we examine quantum dynamical semigroup properties of the Heisenberg evolutions of general classes of observables.     

Master equation of the reduced statistical operator of an atom in a plasma

We use the nonequilibrium Liouville equation to derive the master equation for the reduced statistical operator in a heat bath represented by a many-particle environment. Focusing on the case of a weak system-bath coupling, we consider the Born-Markov approximation of the master equation and compare the result to different approaches. The master equation is elaborated for the special case of an atom as a reduced system in a plasma background. We find that the dynamical structure factor determines the effect of the plasma on the reduced system. We consider the operator equation in the atomic eigenstate and in the phase-space representation, which yields two limiting cases: quantum mechanical behavior similar to the isolated atom for the lower strongly bound levels and a semiclassical one for highly excited Rydberg levels. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 31–62, January, 2008.     

Nonequilibrium Green’s Functions in the Atomic Representation and the Problem of Quantum Transport of Electrons Through Systems With Internal Degrees of Freedom

We develop the theory of quantum transport of electrons through systems with strong correlations between fermionic and internal spin degrees of freedom. The atomic representation for the Hamiltonian of a device and nonequilibrium Green’s functions constructed using the Hubbard operators allow overcoming difficulties in the perturbation theory encountered in the traditional approach because of a larger number of bare scattering amplitudes. Representing the matrix elements of effective interactions as a superposition of terms each of which is split in matrix indices, we obtain a simple method for solving systems of very many equations for nonequilibrium Green’s functions in the atomic representation. As a result, we obtain an expression describing the electron currents in a device one of whose sites is in tunnel coupling with the left contact and the other, with the right contact. We derive closed kinetic equations for the occupation numbers under conditions where the electron flow leads to significant renormalization of them.     

Electrical Conductivity of Charged Particle Systems and Zubarev’s Nonequilibrium Statistical Operator Method

One of the fundamental problems in physics that are not yet rigorously solved is the statistical mechanics of nonequilibrium processes. An important contribution to describing irreversible behavior starting from reversible Hamiltonian dynamics was given by D. N. Zubarev, who invented the method of the nonequilibrium statistical operator. We discuss this approach, in particular, the extended von Neumann equation, and as an example consider the electrical conductivity of a system of charged particles. We consider the selection of the set of relevant observables. We show the relation between kinetic theory and linear response theory. Using thermodynamic Green’s functions, we present a systematic treatment of correlation functions, but the convergence needs investigation. We compare different expressions for the conductivity and list open questions.     

Statistical Irreversible Thermodynamics in the Framework of Zubarev’s Nonequilibrium Statistical Operator Method

We describe the formalism of statistical irreversible thermodynamics constructed based on Zubarev’s nonequilibrium statistical operator (NSO) method, which is a powerful and universal tool for investigating the most varied physical phenomena. We present brief overviews of the statistical ensemble formalism and statistical irreversible thermodynamics. The first can be constructed either based on a heuristic approach or in the framework of information theory in the Jeffreys-Jaynes scheme of scientific inference; Zubarev and his school used both approaches in formulating the NSO method. We describe the main characteristics of statistical irreversible thermodynamics and discuss some particular considerations of several authors. We briefly describe how Rosenfeld, Bohr, and Prigogine proposed to derive a thermodynamic uncertainty principle.     

Algebraic Aspects of the Dynamics of Quantum Multilevel Systems in the Projection Operator Technique

Using the projection operator method, we obtain approximate time-local and time-nonlocal master equations for the reduced statistical operator of a multilevel quantum system with a finite number N of quantum eigenstates coupled simultaneously to arbitrary classical fields and a dissipative environment. We show that the structure of the obtained equations is significantly simplified if the free Hamiltonian dynamics of the multilevel system under the action of external fields and also its Markovian and non-Markovian evolutions due to coupling to the environment are described via the representation of the multilevel system in terms of the SU(N) algebra, which allows realizing effective numerical algorithms for solving the obtained equations when studying real problems in various fields of theoretical and applied physics.     

Broken translation invariance in quasifree fermionic correlations out of equilibrium

Using the C^? algebraic scattering approach to study quasifree fermionic systems out of equilibrium in quantum statistical mechanics, we construct the nonequilibrium steady state in the isotropic XY chain whose translation invariance has been broken by a local magnetization and analyze the asymptotic behavior of the expectation value for a class of spatial correlation observables in this state. The effect of the breaking of translation invariance is twofold. Mathematically, the finite rank perturbation not only regularizes the scalar symbol of the invertible Toeplitz operator generating the leading order exponential decay but also gives rise to an additional trace class Hankel operator in the correlation determinant. Physically, in its decay rate, the nonequilibrium steady state exhibits a left mover-right mover structure affected by the scattering at the impurity.     

Applying the Linblad equation to quantum dissipative systems

For quantum systems with linear dissipation, we obtain the representation of the Linblad equation in the canonical form via Hermitian operators. Based on this representation, we derive equations for the entropy density and for the statistical projection operator. We consider the quantum harmonic oscillator with linear dissipation as an example. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 2, pp. 288–294, August, 2006. An erratum to this article is available at .     

Nonlinear generalized master equations and accounting for initial correlations

We develop a new method based on using a time-dependent operator (generally not a projection operator) converting a distribution function (statistical operator) of a total system into the relevant form that allows deriving new exact nonlinear generalized master equations (GMEs). The derived inhomogeneous nonlinear GME is a generalization of the linear Nakajima-Zwanzig GME and can be viewed as an alternative to the BBGKY chain. It is suitable for obtaining both nonlinear and linear evolution equations. As in the conventional linear GME, there is an inhomogeneous term comprising all multiparticle initial correlations. To include the initial correlations into consideration, we convert the obtained inhomogeneous nonlinear GME into the homogenous form by the previously suggested method. We use no conventional approximation like the random phase approximation (RPA) or the Bogoliubov principle of weakening of initial correlations. The obtained exact homogeneous nonlinear GME describes all evolution stages of the (sub)system of interest and treats initial correlations on an equal footing with collisions via the modified memory kernel. As an application, we obtain a new homogeneous nonlinear equation retaining initial correlations for a one-particle distribution function of the spatially inhomogeneous nonideal gas of classical particles. In contrast to existing approaches, this equation holds for all time scales and takes the influence of pair collisions and initial correlations on the dissipative and nondissipative characteristics of the system into account consistently with the adopted approximation (linear in the gas density). We show that on the kinetic time scale, the time-reversible terms resulting from the initial correlations vanish (if the particle dynamics are endowed with the mixing property) and this equation can be converted into the Vlasov-Landau and Boltzmann equations without any additional commonly used approximations. The entire process of transition can thus be followed from the initial reversible stage of the evolution to the irreversible kinetic stage.     

Running tails as codimension two quasi-solitons in excitation taxis waves with negative refractoriness

Using a Lindblad dissipation dynamics [Lindblad G. On the generators of quantum dynamical semigroups. Commun Math Phys 1976;48:119–130 and see also Gorini V, Frigerio A, Verri M, Kossakowski A, Sudarshan ECG. Properties of quantum Markovian master equations. Rep Math Phys 1978;13:149–173; Alicki R, Messer J. Nonlinear quantum dynamical semigroups for many-body open systems. J Stat Phys 1983;32:299–312.] for biological rate equations we derive a one-component discrete dynamics for the spread of Avian Influenza. Numerical solutions of the difference equations are calculated and compared with measurement data.     

Statistical hydrodynamics of magnetic fluids. I. the nonequilibrium statistical operator method

The Zubarev nonequilibrium statistical operator is used to describe the generalized hydrodynamic state of a magnetic fluid in an external magnetic field. The magnetic fluid is modeled with “liquid-state” and “magnetic” subsystems described using the classical and quantum statistics methods respectively. Equations of the generalized statistical hydrodynamics for a magnetic fluid in a nonhomogeneous external magnetic field with the Heisenberg spin interaction are derived for “liquid-state” and “magnetic” subsystems characterized by different nonequilibrium temperatures. These equations can be used to describe both the weakly and strongly nonequilibrium states. Some limiting cases are analyzed in which the variables of one of the subsystems can be formally neglected. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 1, pp. 132–153, April, 1998.     

Two-time Green’s functions in superconductivity theory

Based on the method of the equations of motion for two-time Green’s functions, we derive superconductivity equations for different types of interactions related to the scattering of electrons on phonons and spin fluctuations or caused by strong Coulomb correlations in the Hubbard model. We derive an exact Dyson equation for the matrix Green’s function with the self-energy operator in the form of the multiparticle Green’s function. Calculating the self-energy operator in the approximation of noncrossing diagrams leads to a closed system of equations corresponding to the Migdal-Eliashberg strong-coupling theory. We propose a theory of high-temperature superconductivity due to kinematic interaction in the Hubbard model. We show that two pairing channels occur in systems with a strong Coulomb correlation: one due to the antiferromagnetic exchange in interband hopping and the other due to the coupling to spin and charge fluctuations in hopping within one Hubbard band. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 129–146, January, 2008.