Theory Of Nonequilibrium Processes Research Articles

Taking into account the influence of correlations on the system dynamics in the reduced description method

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.

Thermodynamic substantiation of the direction of nonequilibrium processes in triadconjugations of machine parts based on the principles of maximum and minimum entropy

The article gives a thermodynamic substantiation of the direction of nonequilibrium processes in tribocouples of machine parts, in tribosystems, based on the principles of maximum and minimum entropy. It is clarified how nonequilibrium processes can be substantiated on the basis of the minimum and maximum function of entropy production: linear and nonlinear nonequilibrium processes and their different thermodynamics. The entropy production function is considered as a function of thermodynamic force flows and thermodynamic flows.
 The theory of nonequilibrium processes is based on the Liouville equation for classical tribosystems, taking into account external influences or perturbations. It is shown that in thermodynamic processes in tribosystems the principle of entropy maximization is realized as the second principle of synergetics.

On relaxation processes in a completely ionized plasma

Relaxation of the electron energy and momentum densities is investigated in spatially uniform states of completely ionized plasma in the presence of small constant and spatially homogeneous external electric field. The plasma is considered in a generalized Lorentz model which contrary to standard one assumes that ions form an equilibrium system. Following to Lorentz it is neglected by electron-electron and ion-ion interactions. The investigation is based on linear kinetic equation obtained by us early from the Landau kinetic equation. Therefore long-range electron-ion Coulomb interaction is consequentially described. The research of the model is based on spectral theory of the collision integral operator. This operator is symmetric and positively defined one. Its eigenvectors are chosen in the form of symmetric irreducible tensors which describe kinetic modes of the system. The corresponding eigenvalues are relaxation coefficients and define the relaxation times of the system. It is established that scalar and vector eigenfunctions describe evolution of electron energy and momentum densities (vector and scalar system modes). By this way in the present paper exact close set of equations for the densities valid for all times is obtained. Further, it is assumed that their relaxation times are much more than relaxation times of all other modes. In this case there exists a characteristic time such that at corresponding larger times the evolution of the system is reduced described by asymptotic values of the densities. At the reduced description electron distribution function depends on time only through asymptotic densities and they satisfy a closed set of equations. In our previous paper this result was proved in the absence of an external electric field and exact nonequilibrium distribution function was found. Here it is proved that this reduced description takes also place for small homogeneous external electric field. This can be considered as a justification of the Bogolyubov idea of the functional hypothesis for the relaxation processes in the plasma. The proof is done in the first approximation of the perturbation theory in the field. However, its idea is true in all orders in the field. Electron mobility in the plasma, its conductivity and phenomenon of equilibrium temperature difference of electrons and ions are discussed in exact theory and approximately analyzed. With this end in view, following our previous paper, approximate solution of the spectral problem is discussed by the method of truncated expansion of the eigenfunctions in series of the Sonine polynomials. In one-polynomial approximation it is shown that nonequilibrium electron distribution function at the end of relaxation processes can be approximated by the Maxwell distribution function. This result is a justification of Lorentz–Landau assumption in their theory of nonequilibrium processes in plasma. The temperature and velocity relaxation coefficients were calculated by us early in one- and two-polynomial approximation.

Similarity Theory of Energy Conversion Processes

On the basis of energodynamics as a unified theory of nonequilibrium processes of energy transfer and transformation, a theory of similarity of such processes is proposed. It includes dimensionless criteria of efficiency, constructive perfection and load of power plants and the criterial equations connecting these indicators. On its basis, universal load characteristics of thermal and non-thermal, cyclic and non-cyclic, direct and reverse machines are built, reflecting their efficiency with the load and losses from irreversibility. The examples confirming their justice are given.

Gas-kinetic model of heat conduction of heterogeneous substances

A theoretical approach is proposed for calculating thermal conductivity κ of an arbitrary type of porous structures as a function of porosity ξ, temperature T, density ρ, and a number of other parameters. The general computational algorithm is based on the theory of nonequilibrium processes. Its modification in the language of gas-kinetic approximation makes it possible to derive compact relations for κ and to easily estimate the corresponding dependences. Theoretical formulas are compared to experimental results and their good agreement is demonstrated for a specific example of refractory concrete, which is a very important substance for practical applications.

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.

Correlation effects in the diffusion and electrical conductivity of an interacting lattice gas

A simple and effective self-consistent scheme is proposed for the determination of the average potentials which allows thermodynamic functions and other characteristics of lattice systems in equilibrium to be calculated with high accuracy and short computing time. This scheme has been used for analysis of the expressions for the coefficients of diffusion and electrical conductivity obtained on the basis of the modern statistical theory of nonequilibrium processes. Results of the simulations are correlated with the data of Monte Carlo simulations obtained using parallel vector algorithms on a Cray TZE computer of the Max Plank Society (Germany).

Thermodynamic approach to nonequilibrium chemical fluctuations

A generalized thermodynamic description of one-variable complex chemical systems is suggested on the basis of the Ross, Hunt, and Hunt (RHH) theory of nonequilibrium processes. Starting from the stationary solution of a chemical Master Equation, two complimentary, related sets of generalized state functions are introduced. The first set of functions is derived from a generalized free energy F̌X, and is used to compute the moments of stationary and non-Gaussian concentration fluctuations. Exact expressions for the cumulants of concentration are derived; a connection is made between the cumulants and the fluctuation–dissipation relations of the RHH theory. The second set of functions is derived from an excess free energy φ(x); it is used to express the conditions of existence and stability of nonequilibrium steady states. Although mathematically distinct, the formalisms based on the F̌X and φ(x) functions are physically equivalent: both lead to the same type of differential expressions and to similar global equations. A comparison is made between the RHH and Keizer’s theory of nonequilibrium processes. An appropriate choice of the integration constants occurring in Keizer’s theory is made for one-variable systems. The main differences between the two theories are: the constraints for the two theories are different; the stochastic and thermodynamic descriptions are global in RHH, whereas Keizer’s theory is local. However, both theories share some common features. Keizer’s fluctuation–dissipation relation can be recovered by using the RHH approach; it is valid even if the fluctuations are nonlinear. If the thermodynamic constraints are the same, then Keizer’s theory is a first-order approximation of RHH; this approximation corresponds to a Gaussian description of the probability of concentration fluctuations. Keizer’s theory is a good approximation of RHH in the vicinity of a stable steady state: near a steady state the thermodynamic functions of the two theories are almost identical; the chemical potential in the stationary state is of the equilibrium form in both theories. Keizer’s theory gives a very good estimate of the absolute values of the peaks of the stationary probability density of RHH theory. Away from steady states the predictions of the two theories are different; the differences do not vanish in the thermodynamic limit. The shapes of the tails of the stationary probability distributions are different; and hence the predictions concerning the relative stability are different for the two theories.

A nonequilibrium polydisperse flow in an axisymmetric nozzle with condensate particle coagulation and breakup

A kinetic description is given for the flow of a polydisperse medium, which is based on the general theory for nonequilibrium processes; the conversion to generalized transport equations is discussed and the closure of the infinite equation chain. An example is given of numerical calculations on such a flow in a nozzle on the basis of particle coalescence and breakup.

A thermodynamic approach to viscoelasticity and plasticity

In the present paper a constitutive equation of rate type is derived from Onsager's theory of non-equilibrium processes. Some simple solutions of the equation are discussed, and it is found that the material at hand exhibits different kinds of relaxation phenomena, normal stress effects, non-Newtonian behaviour, plastic properties as well as the creep and elasticity at small loads.

Asymptotic behavior of Green's functions and correlation functions in the statistical theory of nonequilibrium processes

Expressions are obtained for the asymptotic Green's and correlation functions of arbitrary operators which are not integrals of the motion. The derivation of the expressions presupposes only the existence of a set of operators (coarse-grain variables) such that the variation in time of their mean values determines the slow evolution of the system. It is shown that the correlation functions of the coarse-grain variables, the fluxes, and their "random" components can be expressed in terms of the same transport coefficients. The nonequilibrium statistical operator method is used to derive dynamical equations with sources for the asymptotic Green's functions of the coarse-grain variables. In the statistical theory of nonequilibrium processes [1, 2], one can, using asymptotic solutions of the Liouville equation, derive macroscopic equations describing the slow variation in time of the coarse-grain variables which specify the nonequilibrium state of the system. The transport coefficients in these equations can be expressed in terms of the correlation functions of the random fluxes. It is also of interest to consider the "inverse" problem of obtaining the low-frequencylas ymptotic behavior of the Green's functions and the correlation functions of various dynamical variables using the equations for the coarse-grain variables [3-5]. On the one hand, this asymptotic behavior is universal, since at large times the evolution of the system is basically determined by the variation of the coarse-grain variables. On the other hand, the asymptotic Green's functions relate the low-frequency component of the fluctuation spectrum of the dynamical variables, which is measured, for example, in inelastic neutrorl scattering experiments, to the macroscopic transport coefficients. There exist two main approaches to the calculation of the low-frequency asymptotic behavior of Green's functions. The method based on introducing into the Hamiltonian additional terms describing the interaction of the system with external fields was proposed by Bogolyubov [3] and developed later in various papers [6-10]. Sources appear in the hydrodynamic equations, and the variations with respect to them of the hydrodynamic quantities are related to the asymptotic Green's functions. The indirect linear response method of Kadanoff-Martin [4, 11] consists of comparing the expressions for the response of the system to a small perturbation with the relaxation of the system to thermal equilibrium after an external field has been switched off. It is assumed that the relaxation of the system can be described by the equations of phenomenological hydrodynamics. This method was applied, for example, to a Bose superfluid in [12] and to magnetic systems in [13]. Both approaches make it possible to obtain expressions for the asymptotic Green's functions of the coarse-grain variables themselves if the macroscopic equations are known. However, it is obvious that once the slowly varying coarse-grain variables have been chosen (the set of them is determined by the physics of the considered problem) the asymptotic behavior of the Green's functions of all operators must follow from the microscopic theory without recourse to the phenomenologieat equations. In the present paper, using the Liouville equation, we obtain general expressions for the asymptotic behavior of the Green's functions and the time correlation functions of arbitrary operators that are not integrals of the motion. In the special case when the coarse-grain variables are these operators the asymptotic behavior of the Green's functions corresponds to the results of the Kadanoff-Martin theory. At the end of the paper, Zubarev's nonequilibrium statistical operator method [1, 14] is used in a different approach to the calculation of the asymptotic Green's functions based on the introduction of sources into the dynamical equations for the coarse-grain variables. It is shown in particular that sources can be introduced in the standard manner for an arbitrary set of coarseInstitute of Civil Aviation Engineers, Moscow. Translated from Teoreticheskaya i Matematicheskaya

Critères de stabilité et d'évolution des systèmes électrochimiques hors d'équilibre

The Glansdorff-Prigogine linear stability theory of non equilibrium processes as well as the associated evolution criterion, are extended to polarized systems. The equilibrium stability conditions are derived from the negative definite form of the second differential of entropy. Additional inequalities are obtained giving rise to dipole stability conditions. The non equilibrium stability condition is formulated by using a Liapounoff function including both the second differential of entropy production and a negative term related to the electromagnetic energy. The explicit form is then calculated by means of the excess balance equations for mass and energy as well as the Maxwell equations applied to the disturbance of the electromagnetic field. When convection occurs, the excess balance equation for momentum is also required.

Boundary conditions for statistical operators in the theory of nonequilibrium processes and quasiaverages

Boundary conditions for statistical operators in the theory of nonequilibrium processes and quasiaverages

Brownian Motion of Polyatomic Molecules: The Coupling of Rotational and Translational Motions

The coupling of rotational and translation Brownian motions is examined from several points of view. The first is a phenomenological theory based upon generalized Langevin equations of motion and a Markoff integral equation. Next, a more detailed statistical-mechanical theory is fashioned after the pattern of Kirkwood's theory for nonequilibrium processes in monatomic liquids. Both schemes lead to a generalized Fokker—Planck—Chandrasekhar equation for the singlet-distribution function. This equation includes terms that account for separate rotational and translational motions as well as two mutually symmetric contributions which are descriptive of their coupling. The friction tensors associated with the uncoupled components of these motions are found to be proportional to the autocorrelations of the environmental force and torque which act upon a given molecule. The frictional coupling is related to the cross correlation of force and torque. From the principle of microreversibility it is possible to establish a reciprocal relationship between the two coupling tensors. A third approach is to derive the generalized Fokker—Planck equation from the Boltzmann equation for a dilute solution of rotating molecules. This has been done for the model of perfectly rough spheres and also for ``loaded spherocylinders.'' In both cases explicit formulas are obtained for the various friction tensors. The last section of the paper is devoted to the application of these theories to problems of diffusion.