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.     

Boundary conditions and non-equilibrium thermodynamics

The theory of non-equilibrium thermodynamics is applied to a system of two immiscible fluids and their interface. A singular energy density at the interface, which is related to the phenomenon of surface tension, is taken into account. Furthermore the momentum and the heat currents are allowed to be singular at the interface. Using the conservation laws and the Gibbs' relation for the surface, an expression for the singular entropy production density at the interface is obtained. The linear phenomenological laws between fluxes and thermodynamic forces occurring in this singular entropy production density are given. Some of these linear laws are boundary conditions for the solution of the differential equations governing the evolution of the state variables in the bulk.     

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.     

Phenomenological equations for multicomponent fluids

Ahydrodynamic equation of motion for each component of a multicomponent fluid is derived on the basis of nonequilibrium thermodynamics. Special care has been directed to the choice of state variables. In some limiting cases, this equation leads to customary phenomenological equations, such as the equation for diffusion and the Navier-Stokes equation. The viscosity is a consequence of nonlocal coupling of forces and fluxes. The reciprocity between the linear coefficients is examined closely.     

Kinetic phenomena in the gas mixture flow through nanodimensional capillaries: The effect of surface forces

The flow of a gaseous mixture in ultrafine capillaries whose size is comparable to the range of surface forces is studied with allowance for the effect of surface forces on gas component transport in the capillaries. The effect of surface forces is shown to be twofold. On the one hand, it is necessary to take into account the Boltzmann distribution of the molecular density in the capillary: it turns out that the contribution of this effect to the transport coefficient values may amount to 20%. On the other hand, an additional kinetic contribution to these coefficients may arise if the temperature in the gas is nonuniform. In this case, the temperature gradient may change the gas component transport coefficients several times, giving rise to a new mechanism of gas component separation that is associated with different potentials of interaction between the gas components and capillary walls. Also, surface forces modify the structure of phenomenological equations for gas component transport that implies entropy production in an inhomogeneous gas. However, the Onsager symmetry between cross coefficients is retained.     

Grad theory and extended thermodynamics

Grad-type approaches introduce an ansatz involving tensor Hermite functions with coefficients expresed in terms of moments of the ansatz. This formalism in usual form yields terms linear in first-order spatial derivatives in kinetic equations for the moments. Such terms disagree with alternative statistical derivations and phenomenological arguments. This disagreement is removed if different ansatzes are used to calculate entropy and moment equations. These are non-unique, and so Grad theory, while providing theoretical expressions for transport coefficients, does not serve uniquely to determine the structure of phenomenological equations.     

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.     

Non-linear constitutive and diffusion equations in the Burnett regime

A phenomenological method is presented to obtain the hydrodynamic equations for a multicomponent, isotropic, non-reactive fluid to any order in the spatial inhomogeneities. Two assumptions are made, the existence of a local equilibrium state and a non-linear dependence of the fluxes on the thermodynamic forces. In particular, the generalized form for the diffusion equation, to fourth order in the gradients, is obtained. Also, we derive the hydrodynamic equations for a binary mixture in a non-linear Burnett regime. The comparison of our results with others given in the literature and, in particular with those recently derived using the time-dependent correlation function formalism, is given. Finally some remarks are made in connection with the question about the existence of the transport coefficients beyond the Navier-Stokes regime.     

Diffusion in phases with several components and sublattices

Diffusion in multicomponent alloys is discussed in order to extend earlier treatments to phases with sublattices. Both a phenomenological approach and some physical models are applied. Care is taken so that the driving forces and the fluxes are chosen in a way obeying the requirements of irreversible thermodynamics. The Onsager reciprocal relation will thus be obeyed. Two kinds of models have been considered, exchange between two atoms and exchange between an atom and a vacancy.For phases with several sublattices a number of difficulties arise and only a few cases of great practical importance are considered. The connection between practical diffusion coefficients and phenomenological coefficients is given. The use of different concentration variables in Fick's second law is discussed.     

Onsager-Casimir symmetry properties of the Burnett equations

We demonstrate the approximate nature of the Onsager-Casimir relations for the example of the linearized Burnett equations for a dilute gas. For any discussion of Onsager relations the choice of a correct set of thermodynamic forces and fluxes is of course crucial. By retracing the Chapman-Enskog procedure, we show that the usual expressions for the thermodynamic forces require modifications at the Burnett level. However, inclusion of these terms does not remedy the violation of Onsager symmetry first noticed by McLennan. A modified version of the Onsager symmetry that involves thermodynamic forces derived from an entropy Lagrangian rather than from the entropy itself does remain valid on the Burnett level. Throughout, we allow for the presence of an external potential; the Burnett equations including potential terms are derived in an appendix for a set of variables particularly suited for our discussion. We stress that in discussing Onsager relations one should use the full thermodynamic fluxes rather than their dissipative parts only, in spite of the fact that only the latter contribute to the entropy production.     

Beyond the navier-stokes regime in hydrodynamics

In this letter we show that if one suitably modifies the conventional linear relationship between forces and fluxes in irreversible thermodynamics, then all the higher-order hydrodynamic equations, namely, the Burnett, super-Burnett, etc. equations, are consistent with both the entropy balance equation and the local equilibrium assumption. Some implications of this result are also pointed out.     

Transport phenomena in a capillary-porous medium II. Entropy production and phenomenological equations

By means of the balance equation of internal energy and the assumption of local equilibrium there is derived the balance equation of entropy. In the entropy production such terms that have connection with the work of reactive forces, with the energy of change of mass, and with the work of binding forces also appear in addition to the usual expressions. It is shown that entropy production can be written in two equivalent ways by means of tensors of different orders. Every form of entropy production in linear area leads to a different system of phenomenological equations. Transfer relations between the two systems are found. Also an expression is found for mass flux which depends explicitly neither on the temperature gradient nor on the gradient of chemical potentials. A relationship is found between viscosity tensor and the tensor which describes the swelling of a body, and friction between a fluid and the solid phase. Dependence of dispersion coefficient on concentration is derived.     

Generalized entropy and transport coefficients of hadronic matter

We use the generalized entropy four-current of the Müller-Israel-Stewart (MIS) theories of relativistic dissipative fluids to obtain information about fluctuations around equilibrium states. This allows one to compute the non-classical coefficients of the entropy 4-flux in terms of the equilibrium distribution functions. The Green-Kubo formulae are used to compute the standard transport coefficients from the fluctuations of entropy due to dissipative fluxes.     

TheH theorem and irreversible thermodynamics

Following a method first introduced by Prigogine, theH theorem is written as the law of increase of entropy for a slightly inhomogeneous gas. It is shown that the local rate of entropy production for such a gas is simply a homogeneous quadratic form of the generalized forces associated with the various irreversible processes with coefficients possessing all the properties of the phenomenological coefficients of irreversible thermodynamics. The local rate of entropy production is explicitly evaluated for a simple monatomic gas and is compared with the corresponding expression of irreversible thermodynamics.     

Fluctuations in nonlinear Markovian systems

We study nonlinear irreversible processes by statistical mechanical methods from a general point of view. Assuming that the macroscopic variables behave approximatively Markovian we derive evolution equations for the mean values as well as for the fluctuations about the mean. The mean values obey nonlinear transport equations and the fluctuations obey linear nonstationary Langevin equations. The equations of motion are completely specified by the entropy and the transport coefficients as functions of the macroscopic state. The theory provides a statistical mechanical basis for some phenomenological approaches put forward recently.     

The reciprocal relations between cross phenomena in boundless gaseous systems

The Onsager-Casimir reciprocal relations obtained for gaseous systems on the basis of the linearized Boltzmann equation in a general form [F. Sharipov, Onsager-Casimir reciprocal relations based on the Boltzmann equation and gas-surface interaction law. Single gas., Phys. Rev. E 73 (2006) 026110] are applied to boundless domains. Under additional assumptions, expressions of the kinetic coefficients obtained previously are significantly simplified and can be easily applied in practice. In contrast to all previous works on this topic, it is shown that for a given set of thermodynamic forces there are several sets of thermodynamic fluxes providing the same entropy production. As a consequence, several forms of the kinetic coefficients satisfying the reciprocal relations do exist for a given set of thermodynamic forces. An illustrative example confirms that in many situations the kinetic coefficients are neither odd nor even with respect to the time-reversal and hence the reciprocal relations between them should be written in more general form than that obtained by Casimir.     

Relativistic analysis of four-body final states

The constraints of unitarity and analyticity on four-body final states are studied. It is shown that unitarity alone forces the amplitudes to be coherent and have singular behaviour. The implementation of unitarity with total energy analyticity yields a set of relativistic linear integral equations for the four-body amplitude. This is the minimal set consistent with quantum mechanics and also is the full dynamical set of equations with two-body separable interactions. These equations will provide important ingredients for the phenomenological analysis of four-body final states using the isobar model.     

Non-equilibrium thermodynamics of interfacial systems: II. Boundary conditions for fluids with spin

The theory of non-equilibrium thermodynamics is applied to a system of two immiscible fluids whose molecules possess internal angular momentum (spin) which are separated by an interface. The conservation laws and the Gibbs relation are used to derive the entropy production at the interface. The resulting linear laws relating the fluxes and forces represent boundary conditions on the hydrodynamic equations for the bulk phases. A limiting case is considered and boundary conditions derived by previous authors are obtained.