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Based on the non-Markov diffusion equation taking into account the spatial fractality and modeling for the generalized coefficient of particle diffusion $D^{\alpha\alpha'}(\mathbf{r},\mathbf{r}';t,t')=W(t,t')\bar{D}^{\alpha\alpha'}(\mathbf{r},\mathbf{r}')$ using fractional calculus the generalized Cattaneo--Maxwell--type diffusion equation in fractional time and space derivatives has been obtained. In the case of a constant diffusion coefficient, analytical and numerical studies of the frequency spectrum for the Cattaneo--Maxwell diffusion equation in fractional time and space derivatives have been performed. Numerical calculations of the phase and group velocities with change of values of characteristic relaxation time, diffusion coefficient and indexes of temporal $\xi$ and spatial $\alpha$ fractality have been carried out., Comment: 23 pages, 4 figures

We presented a general approach for obtaining the generalized transport equations with fractional derivatives by using the Liouville equation with fractional derivatives for a system of classical particles and Zubarev’s nonequilibrium statistical operator (NSO) method within Renyi statistics. New non-Markovian diffusion equations for particles in spatially heterogeneous environment with fractal structure and a generalized Cattaneo-type diffusion equation with taking into account nonlocality of space–time are obtained. Different models of frequency-dependent memory functions, which lead to known diffusion equations with nonlocality of space–time and their generalizations are studied.

A formulation of nonequilibrium thermo-field dynamics has been performed using the nonequilibrium statistical operator method by D.N. Zubarev. Generalized transfer equations for a consistent description of the kinetics and hydrodynamics of the dense quantum field system with strongly-bound states are derived.

We present a generalization of Zubarev’s nonequilibrium statistical operator method based on the principle of maximum Renyi entropy. In the framework of this approach, we obtain transport equations for the basic set of parameters of the reduced description of nonequilibrium processes in a classical system of interacting particles using Liouville equations with fractional derivatives. For a classical systems of particles in a medium with a fractal structure, we obtain a non-Markovian diffusion equation with fractional spatial derivatives. For a concrete model of the frequency dependence of a memory function, we obtain generalized Kettano-type diffusion equation with the spatial and temporal fractality taken into account. We present a generalization of nonequilibrium thermofield dynamics in Zubarev’s nonequilibrium statistical operator method in the framework of Renyi statistics.

To describe kinetic processes in gases and liquids far from equilibrium, we obtain the nonequilibrium statistical operator (NSO) and the generalized kinetic equations for the nonequilibrium one- and two-particle distribution functions based on the Zubarev NSO method and the maximum entropy principle for the Renyi entropy. We consider a q-generalization of the Liouville equation and obtain its general solution using the NSO method.

By using the Zubarev nonequilibrium statistical operator method, and the Liouville equation with fractional derivatives, a generalized diffusion equation with fractional derivatives is obtained within the Renyi statistics. Averaging in generalized diffusion coefficient is performed with a power distribution with the Renyi parameter $q$., 12 pages

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.

Chain of kinetic equations for non-equilibrium single, double and s-particle distribution functions of particles is obtained taking into account nonlin- ear hydrodynamic fluctuations. Non-equilibrium distribution function of non-linear hydrodynamic fluctuations satisfies a generalized Fokker-Planck equation. The method of non-equilibrium statistical operator by Zubarev is applied. A way of calculating of the structural distribution function of hydrodynamic collective variables and their hydrodynamic velocities (above Gaussian approximation) contained in the generalized Fokker-Planck equa- tion for the non-equilibrium distribution function of hydrodynamic collective variables is proposed., 31 pages

Generalized transport equations for description of diffusion-reaction processes in chemically active mixtures are obtained. The nonequilibrium statistical operator method by Zubarev is used and both strong and weak nonequilibrium processes are analyzed. In the approximation of the second order in fluctuations we get generalized equations of chemical kinetics for bimolecular reactions with generalized rate constants. In the case of spatial uniformity the integro-differential equation for the matrix of partial scattering functions is received, which are related to partial dynamic structure factors of chemically reactive system by the time Fourier transformation.

An Enskog-Landau kinetic equation for a many-component system of charged hard spheres is proposed. It has been obtained from the Liouville equation with modified boundary conditions by the method of nonequilibrium statistical operator. On the basis of this equation the normal solutions and transport coefficients such as bulk kappa and shear eta viscosities, thermal conductivity lambda, mutual diffusion D^{\alpha\beta} and thermal diffusion D_T^\alpha have been obtained for a binary mixture in the first approximation using the Chapman-Enskog method. Numerical calculations of all transport coefficients for mixtures Ar-Kr, Ar-Xe, Kr-Xe with different concentrations of compounds have been evaluated for the cases of absence and presence of long-range Coulomb interactions. The results are compared with those obtained from other theories and experiment., Comment: 24 LaTeX209 pages, 3 EPS figures (4 files). To be published in Physica A

A general theoretical approach is developed for describing the dynamic properties of semiquantum fluids on the basis of the nonequilibrium statistical operator technique. A set of equations of generalized hydrodynamics is derived and the particular case of thermo-viscoelastic model of a fluid is analyzed in details in the hydrodynamic limit. The case of intermediate and large values of the wave vector is also discussed. The Markov approximation for transport kernels is used to obtain a closed set of equations for dynamic correlation functions. The problem is considered in the context of relation with the experimental data on neutron scattering and the theoretical results known previously in the literature.

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.

A new kinetic equation for dense gases and liquids with the interparticle potential of interaction in a form of the multistep function is proposed. The equation is consistently derived on the basis of Zubarev's nonequilibrium statistical operator method using a modification for weakening correlations principle which takes into account local conservation laws. In particular cases this equation coincides with the well known kinetic equations for hard-sphere and square-well potentials. Application of the obtained kinetic equation to real systems is discussed. A proof of a global H-theorem is given.

Using Zubarev's method of nonequilibrium statistical operator, the generalized hydrodynamic equations are obtained for a model of magnetic liquid in an inhomogeneous external field. In this model the “liquid” subsystem is treated as a classical one and the “magnetic” subsystem is described by quantum mechanical methods. The properties of the transport equations are analysed in the case of a weak nonequilibrium. The equations for time correlation functions and collective mode spectrum are also found in the same manner. It is shown that the generalized hydrodynamic equations reduce to the well-known results in the limiting cases when the dynamic variables of one subsystem are formally neglected. As an illustration, a simple model of spin relaxation is considered, and the frequency matrix and the matrix of memory functions are calculated. A comparison with previous works is made.

Zubarev's nonequilibrium statistical operator method is used to give a statistical description of a nonequilibrium plasma in its electromagnetic self-field. Generalized transport equations are obtained for the charged particles and the oscillators of the electromagnetic field with allowance made for the local conservation laws. The case of a nonequilibrium plasma is considered.

The problem of relaxation of a nonequilibrium state to the state of molecular hydrodynamics is considered for a classical system of interacting particles using the Zubarev nonequilibrium statistical operator method. The wave-vector and frequency dependencies of the dynamical structure factor and momentum-momentum transverse correlation function are investigated on the basis of the appropriate generalized transport equations. Comparison with the results of molecular hydrodynamics and molecular-dynamics simulations is given and the characteristic time intervals of the studied relaxation processes are determined.

The generalization of the Zubarev nonequilibrium statistical operator method for the case of Renyi statistics is proposed when the relevant statistical operator (or distribution function) is obtained based on the principle of maximum for the Renyi entropy. The nonequilibrium statistical operator and corresponding generalized transport equations for the reduced-description parameters are obtained. A consistent description of kinetic and hydrodynamic processes in the system of interacting particles is considered as an example., 13 pages, RevTeX4-format

The development of methods of construction of the theory of nonequilibrium processes for finite-temperature quantum-field systems is one of the important directions of modern theoretical physics. The modern methods of quantum field theory [1-8], on the one hand, and nonequilibrium statistical mechanics [9-16] on the other, are powerful tools in investigations of interacting quantum fields and particles. The unification of these methods can be helpful and have significant influence on the solution of many problems in the study of nonequilibrium states of finite-temperature quantum-field systems. In earlier years, the methods of quantum field theory (Green's function method, renormalization-group method) were helpful in statistical mechanics [17-20] (for example, in the theory of magnetism [21], in the theory of phase transitions [22,23]).

A consistent statistical description of kinetics and hydrodynamics of dusty plasma is proposed based on the Zubarev nonequilibrium statistical operator method. For the case of partial dynamics the nonequilibrium statistical operator and the generalized transport equations for a consistent description of kinetics of dust particles and hydrodynamics of electrons, ions and neutral atoms are obtained. In the approximation of weakly nonequilibrium process a spectrum of collective excitations of dusty plasma is investigated in the hydrodynamic limit., Comment: 31 pages, no figures