Linear relaxation processes governed by fractional symmetric kinetic equations

The fractional symmetric Fokker-Planck and Einstein-Smoluchowski kinetic equations that describe the evolution of systems influenced by stochastic forces distributed with stable probability laws are derived. These equations generalize the known kinetic equations of the Brownian motion theory and involve symmetric fractional derivatives with respect to velocity and space variables. With the help of these equations, the linear relaxation processes in the force-free case and for the linear oscillator is analytically studied. For a weakly damped oscillator, a kinetic equation for the distribution in slow variables is obtained. Linear relaxation processes are also studied numerically by solving the corresponding Langevin equations with the source given by a discrete-time approximation to white Levy noise. Numerical and analytical results agree quantitatively.     

On the definition of entropy for non-equilibrium states

In this paper it is demonstrated how the mathematical theory of stability of motion can be applied to kinetic equations, describing irreversible processes in an isolated, homogeneous system. It turns out that functions having all the properties of entropy exist throughout the domain of definition of the kinetic equations. Since the kinetic equations depend only on variables defined outside equilibrium thermodynamics, it is possible to define entropy far beyond the range of validity of the thermodynamics of irreversible processes. It is shown that the commonly assumed properties of entropy are not sufficient, however, to single out just one entropy function.     

On a kinetic justification of the generalized nonequilibrium thermodynamics of multicomponent systems

The problem of the kinetic justification of the generalized thermodynamics of nonequilibrium processes using the method of moments for solving the kinetic equation for a multicomponent gas mixture is examined. Generalized expressions are obtained for the entropy density, entropy flux density, and entropy production as functions of an arbitrary number of state variables (moments of the distribution function). Different variants of writing the relations between fluxes and thermodynamic forces are considered, which correspond to the Onsager version for spatially homogeneous systems and, in a more general case, lead to the generalized thermodynamic forces of a complicated form, including derivatives of the fluxes with respect to time and spatial coordinates. Some consequences and new physical effects, following from the obtained equations, are analyzed. It is shown that a transition from results of the method of moments to expressions for the entropy production and the corresponding phenomenological relations of the generalized nonequilibrium thermodynamics is possible on the level of a linearized Barnett approximation of the Chapman–Enskog method.     

Barnett kinetic coefficients in dense charged and neutral media

An approach based on the nonlinear response theory is proposed for determining nonlinear transport properties of nonideal multielement charged and neutral media. A version of this theory developed for describing these properties involves the comparison of phenomenological conservation equations for dense media in the Barnett approximation and microscopic equations for operators of dynamic variables. The Mori algorithm used for formulating the microscopic equations makes it possible to obtain these equations for non-ideal media in the form of generalized nonlinear Langevin equations. As a result, macroscopic expressions are derived for nonlinear kinetic coefficients in the second order in thermal perturbations (of temperature, mass velocity, etc.). The expressions for nonlinear and linearized Barnett kinetic coefficients are compared.     

PROLONGATION STRUCTURES OF NONLINEAR SYSTEMS IN HIGHER DIMENSIONS

It is shown that the prolongation structure theory for nonlinear (evolution) equations with two independent variables can be generalized to the systems with many independent variables. By means of the nonlinear realization theory of gauge symmetries, the fundamental equations for prolongation structures and the requirements for the generalized Lax representations of the nonlinear systems in higher dimensions have been given. Based upon the invariances of the prvlongation structures or the generalized Lax representation under certain transformations, the general condition satisfied by the auto-Backlund transformations has been proposed and searching for a kind of auto-Backlund transformations has been transferred to solving the regular Riemann-Hilbert problem.     

Magnetic relaxation in a spin-1 Ising model near the second-order phase transition point

The magnetic relaxation of a spin-1 Ising model with bilinear and biquadratic interactions is formulated within the framework of statistical equilibrium theory and the thermodynamics of irreversible processes. Using a molecular-field expression for the magnetic Gibbs energy, the magnetic Gibbs energy produced in the irreversible process is calculated and time derivatives of the dipolar and quadrupolar order parameters are treated as fluxes conjugate to their appropriate generalized forces in the sense of Onsager theory. The kinetic equations are obtained by introducing kinetic coefficients that satisfy the Onsager relation. By solving these equations an expression is derived for the dynamic or complex magnetic susceptibility. From the real and imaginary parts of this expression, magnetic dispersion and absorption factor are calculated and analyzed near the second-order phase transition.     

Collisional Depolarization of the Luminescence of Asymmetric Top Molecules in the Gas Phase

To describe the collisional depolarization of the luminescence of asymmetric top polyatomic molecules, integral and integrodifferential forms of master equations, in which the effect of collisions is determined by the conditional probabilities of an instantaneous error of rotational phase variables, have been obtained. A symmetry analysis of the master equations has been performed, and it has been shown that in the general case the evolution of optically induced anisotropy is given by five independent relaxation components. The kinetic equations derived are initial equations for specific calculations of the anisotropy relaxation in various collisional models. They make it possible to study the influence of the angular momentum transfer efficiency on the orientational relaxation of anisotropy in a wide range of buffer medium densities: from free rotation to binary collisions in the gas and then to rotational diffusion.     

Thermodynamics of nonstationary and transient effects in a relativistic gas

We show how a system of generalized Fourier and Navier-Stokes equations, containing relaxation terms and couplings between heat flow and viscosity, can be consistently derived from phenomenological thermodynamics and from kinetic theory. The coefficients are given explicitly for a relativistic Boltzmann gas.     

On the theory of irreversible processes in molecular liquids and liquid crystals, nonequilibrium phenomena associated with the second and fourth rank alignment tensors

A theory for nonequilibrium phenomena primarily associated with the molecular orientation in liquids and liquid crystals is developed within the framework of irreversible thermodynamics. The alignment tensors of rank 2 and 4 are taken into consideration as additional macroscopic variables. With appropriate assumptions concerning the dependence of the internal energy, specific volume and the entropy on the alignment, a dynamic Landau theory is obtained. Nonlinear relaxation equations are derived for the alignment tensors. They possess zero and nonzero stable stationary solutions corresponding to the isotropic and nematic phases of a liquid crystal when the temperature is above or below the transition temperature. Some further applications of the relaxation equations are discussed, in particular fluctuations of the intensity of the depolarized component of scattered light.     

Fluctuations and stability of stationary non-equilibrium systems in detailed balance

A generalized thermodynamic potential for Markoffian systems with detailed balance and far from thermal equilibrium has been derived in a previous paper. It was shown that the principle of detailed balance is equivalent to a set of conditions fulfilled by this potential (“potential conditions”). The properties of this potential allow us to extend the validity of a number of thermodynamic concepts well known for systems in or near thermal equilibrium to stationary states far from thermal equilibrium. The concept of symmetry breaking phase transitions for these systems is introduced in analogy to thermal equilibrium systems by considering the dependence of the stationary probability density of the system on a set of externally controlled parameters {λ}. A functional of the time dependent probability density of the system is defined in close analogy to the Gibb's definition of entropy. This functional has the properties of a Ljapunov functional of the governing Fokker-Planck equation showing the stability of the stationary probability density. The Langevin equations connected with the Fokker-Planck equation are considered. It is shown that, by means of the potential conditions, generalized “thermodynamic” fluxes and forces may be defined in such a way that the smoothly varying part of the Langevin equations (kinetic equations) constitutes a linear relation between fluxes and forces. The matrix of coefficients is given by the diffusion matrix of the Fokker-Planck equation. The symmetry relations which hold for this matrix due to the potential conditions then lead to the Onsager-Casimir symmetry relations extended to systems with detailed balance near stationary states far from thermal equilibrium. Finally it is shown that under certain additional assumptions the generalized thermodynamic potential may be used as a Ljapunov function of the kinetic equations.     

The effect of rotation on plane waves in generalized thermo-microstretch elastic solid with one relaxation time for a mode-I crack problem

The present paper is aimed at studying the effect of rotation on the general model of the equations of the generalized thermo-microstretch for a homogeneous isotropic elastic half-space solid,whose surface is subjected to a Mode-I crack problem.The problem is studied in the context of the generalized thermoelasticity Lord-S hulman’s (L-S) theory with one relaxation time,as well as with the classical dynamical coupled theory (CD).The normal mode analysis is used to obtain the exact expressions for the displacement components,the force stresses,the temperature,the couple stresses and the microstress distribution.The variations of the considered variables through the horizontal distance are illustrated graphically.Comparisons of the results are made between the two theories with and without the rotation and the microstretch constants.     

Generalized thermoelsticity of the thermal shock problem in an isotropic hollow cylinder and temperature dependent elastic moduli

In this paper, we construct the equations of generalized thermoelasicity for a non-homogeneous isotropic hollow cylider with a variable modulus of elasticity and thermal conductivity based on the Lord and Shulman theory. The problem has been solved numerically using the finite element method. Numerical results for the displacement, the temperature, the radial stress, and the hoop stress distributions are illustrated graphically. Comparisons are made between the results predicted by the coupled theory and by the theory of generalized thermoelasticity with one relaxation time in the cases of temperature dependent and independent modulus of elasticity.     

Nonequilibrium Sources of Hydrodynamic Fluctuations Mutual Influence of Hydrodynamic Motion and Nonequilibrium Fluctuations on the Kinetic Coefficients

From generalized expressions for the entropy production the kinetic coefficients in the hydrodynamic equations are determined taking into account the mutual influence of the hydrodynamic motion and the fluctuations. Nonlinear effective and turbulent kinetic coefficients are obtained. The usefulness of the introduced coefficients is demonstrated for boundary layer and tube flows.     

Microscopic derivation of kinetic equations for interacting optical two-level systems

Kinetic equations are derived for optical two-level atoms interacting with a molecular subsystem treated as a thermostat. It is assumed that the kinetics is determined by the electric dipole interaction perturbed by the thermal motion. Dynamic parts of the kinetic equations coincide with the corresponding terms of optical Bloch equations, whereas nonlinear relaxation and shift terms have the specific form and are absent in the phenomenological generalized Bloch equations. It is shown that the relaxation kinetics can be substantially different from the exponential one and depends on the initial state of the system. In particular, the inversion relaxation is frozen at small deviations from the equilibrium. The possibility of observation of the optical bistability is discussed.     

Collisional Relaxation of Luminescence Anisotropy in Rarefied Gases in the Condensed Phase

The orientational relaxation of optically induced anisotropy in rarefied gases and at a damped rotation has been investigated. It has been found that the anisotropy relaxation in rarefied gases is described by a reduced kinetic equation depending only on free rotation integrals. The behavior of the integral anisotropy of luminescence for free symmetric and asymmetric top molecules has been elucidated. The law of luminescence depolarization has been obtained for asymmetric top molecules in the Gordon J-diffusion model. It represents the sum of two Stern–Volmer-type dependences, whose relative contribution is determined by the orientation of the dipole moments of transitions with absorption and emission of light in the molecular coordinate system and by the principal moments of inertia of the molecular top. It has been established that in the limit of a strongly damped rotation, kinetic equations of the general form reduce to equations of rotational diffusion. A number of modified diffusion equations correctly describing the contribution of inertial effects to the orientational relaxation of anisotropy have been obtained.     

Modeling of the non-equilibrium effects by high electric fields in small semiconductor devices

For the purpose of the future theory of energy and momentum relaxation in semiconductor devices, the introduction of two temperatures and two mean velocities for electron and phonons is required. A new model, based on an asymptotic procedure for solving the generalized kinetic equations of electrons and phonons is proposed, which gives naturally the displaced Maxwellian at the leading order. After that, balance equations for the electron number, energy densities, and momentum densities are constructed, which now constitute a system of five equations for the chemical potential of electrons, the temperatures and the drift velocities. In the drift-diffusion approximation the constitutive laws are derived and the Onsager relations recovered.     

Spin transfer from the point of view of the ferromagnetic degrees of freedom

Spintronics is the generic term that describes magnetic systems coupled to an electric generator, taking into account the spin attached to the charge carriers. For this topical review of Spin Caloritronics, we focus our attention on the study of irreversible processes occurring in spintronic devices that involve both the spins of the conduction electrons and the ferromagnetic degrees of freedom. The aim of this report is to clarify the nature of the different kinds of power dissipated in metallic ferromagnets contacted to an electric generator, and to exploit it in the framework of the theory of mesoscopic non-equilibrium thermodynamics. The expression of the internal power (i.e. the internal entropy production multiplied by the temperature) dissipated by a generic system connected to different reservoirs, allows the corresponding kinetic equations to be derived with the introduction of the relevant phenomenological kinetic coefficients. After derivation of the kinetic equations for the ferromagnetic degrees of freedom (i.e. the Landau–Lifshitz equation) and the derivation of the kinetic equations for the spin-accumulation effects (within a two-channel model), the kinetic equations describing spin transfer are obtained. Both spin-dependent relaxation (usual spin-accumulation) and spin-precession in quasi-ballistic regime (transverse spin-accumulation) are taken into account. The generalization of the Landau–Lifshitz equation to spin-accumulation is then performed with the introduction of two potential energy terms that are experimentally accessible.